Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.47.0-wmf.4 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 2812021 2811805 2026-05-29T15:31:23Z Codename Noreste 2969951 /* Create an autopatrolled user group? */ new topic ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2812021 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] == I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&#126;2026-28640-56|&#126;2026-28640-56]] ([[User talk:&#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC) :What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:35, 12 May 2026 (UTC) : Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC) ::Sounds good -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:13, 15 May 2026 (UTC) :::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC) == Proposal to rehost Wikinews here == As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance. I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:30, 14 May 2026 (UTC) :I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) ===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) == [[Wikiversity:Deletion policy]] proposed as policy == [[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]]. This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]]. Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy. === Voting === *{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:33, 18 May 2026 (UTC) *{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC) === Comments === == May 2026 Wikimedia Café meetups regarding the Wikimedia Foundation Annual Plan == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 15px; padding-right: 15px;">[[File:Wikimedia Café logo in plain SVG format.svg|75px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of May. Both sessions will focus on the [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2026-2027 the 2026-2027 Wikimedia Foundation Annual Plan]. Participants may attend either or both sessions. #'''Saturday, 30 May 2026 at 15:00 UTC''' ([https://zonestamp.toolforge.org/1780153200 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''Sunday, 31 May 2026 at 05:00 UTC''' ([https://zonestamp.toolforge.org/1780203600 timestamp converter]), at a time friendly to Asia and the Pacific Café participants are highly encouraged to read in advance [https://en.wikipedia.org/wiki/User:Sohom_Datta/annual_plan_guide at least this summary of the plan]. Optionally, Café participants are encouraged to read portions of the plan that interest them and [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_Annual_Plan/2026-2027 ask questions or provide feedback on the Annual Plan talk page]. Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#May_2026_meetings_with_a_focus_on_Wikimedia_Foundation_Annual_Plan/2026-2027 tables of timestamp conversions for both sessions], [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#Agenda._This_will_be_an_approximately_1_hour_Caf%C3%A9_session,_and_is_extendible_for_an_additional_30_minutes_if_needed. the agenda], and [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 19:46, 21 May 2026 (UTC) == Vote now in the 2026 U4C election == <section begin="announcement-content" /> Eligible voters are asked to participate in the 2026 [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee]] election. More information–including an eligibility check, voting process information, candidate information, and a link to the vote–are available on Meta at the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|2026 Election information page]]. The vote closes on 2 June 2026 at [https://zonestamp.toolforge.org/1780358400 00:00 UTC]. Please vote if your account is eligible. Results will be available by 14 June 2026. -- In cooperation with the U4C,<section end="announcement-content" /> [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 17:15, 27 May 2026 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> == Create an autopatrolled user group? == 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) owt1lkjx7wgcbj4ur269q9jw5lpuke4 2812022 2812021 2026-05-29T15:45:06Z Koavf 147 /* Create an autopatrolled user group? */ Reply 2812022 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] == I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&#126;2026-28640-56|&#126;2026-28640-56]] ([[User talk:&#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC) :What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:35, 12 May 2026 (UTC) : Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC) ::Sounds good -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:13, 15 May 2026 (UTC) :::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC) == Proposal to rehost Wikinews here == As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance. I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:30, 14 May 2026 (UTC) :I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) ===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) == [[Wikiversity:Deletion policy]] proposed as policy == [[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]]. This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]]. Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy. === Voting === *{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:33, 18 May 2026 (UTC) *{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC) === Comments === == May 2026 Wikimedia Café meetups regarding the Wikimedia Foundation Annual Plan == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 15px; padding-right: 15px;">[[File:Wikimedia Café logo in plain SVG format.svg|75px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of May. Both sessions will focus on the [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2026-2027 the 2026-2027 Wikimedia Foundation Annual Plan]. Participants may attend either or both sessions. #'''Saturday, 30 May 2026 at 15:00 UTC''' ([https://zonestamp.toolforge.org/1780153200 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''Sunday, 31 May 2026 at 05:00 UTC''' ([https://zonestamp.toolforge.org/1780203600 timestamp converter]), at a time friendly to Asia and the Pacific Café participants are highly encouraged to read in advance [https://en.wikipedia.org/wiki/User:Sohom_Datta/annual_plan_guide at least this summary of the plan]. Optionally, Café participants are encouraged to read portions of the plan that interest them and [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_Annual_Plan/2026-2027 ask questions or provide feedback on the Annual Plan talk page]. Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#May_2026_meetings_with_a_focus_on_Wikimedia_Foundation_Annual_Plan/2026-2027 tables of timestamp conversions for both sessions], [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#Agenda._This_will_be_an_approximately_1_hour_Caf%C3%A9_session,_and_is_extendible_for_an_additional_30_minutes_if_needed. the agenda], and [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 19:46, 21 May 2026 (UTC) == Vote now in the 2026 U4C election == <section begin="announcement-content" /> Eligible voters are asked to participate in the 2026 [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee]] election. More information–including an eligibility check, voting process information, candidate information, and a link to the vote–are available on Meta at the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|2026 Election information page]]. The vote closes on 2 June 2026 at [https://zonestamp.toolforge.org/1780358400 00:00 UTC]. Please vote if your account is eligible. Results will be available by 14 June 2026. -- In cooperation with the U4C,<section end="announcement-content" /> [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 17:15, 27 May 2026 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> == Create an autopatrolled user group? == 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) 8glk2foge18a2a95emcg67kjj90nqli 2812033 2812022 2026-05-29T16:39:21Z Codename Noreste 2969951 /* Create an autopatrolled user group? */ reply to Koavf ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2812033 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] == I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&#126;2026-28640-56|&#126;2026-28640-56]] ([[User talk:&#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC) :What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:35, 12 May 2026 (UTC) : Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC) ::Sounds good -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:13, 15 May 2026 (UTC) :::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC) == Proposal to rehost Wikinews here == As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance. I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:30, 14 May 2026 (UTC) :I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) ===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) == [[Wikiversity:Deletion policy]] proposed as policy == [[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]]. This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]]. Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy. === Voting === *{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:33, 18 May 2026 (UTC) *{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC) === Comments === == May 2026 Wikimedia Café meetups regarding the Wikimedia Foundation Annual Plan == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 15px; padding-right: 15px;">[[File:Wikimedia Café logo in plain SVG format.svg|75px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of May. Both sessions will focus on the [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2026-2027 the 2026-2027 Wikimedia Foundation Annual Plan]. Participants may attend either or both sessions. #'''Saturday, 30 May 2026 at 15:00 UTC''' ([https://zonestamp.toolforge.org/1780153200 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''Sunday, 31 May 2026 at 05:00 UTC''' ([https://zonestamp.toolforge.org/1780203600 timestamp converter]), at a time friendly to Asia and the Pacific Café participants are highly encouraged to read in advance [https://en.wikipedia.org/wiki/User:Sohom_Datta/annual_plan_guide at least this summary of the plan]. Optionally, Café participants are encouraged to read portions of the plan that interest them and [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_Annual_Plan/2026-2027 ask questions or provide feedback on the Annual Plan talk page]. Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#May_2026_meetings_with_a_focus_on_Wikimedia_Foundation_Annual_Plan/2026-2027 tables of timestamp conversions for both sessions], [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#Agenda._This_will_be_an_approximately_1_hour_Caf%C3%A9_session,_and_is_extendible_for_an_additional_30_minutes_if_needed. the agenda], and [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 19:46, 21 May 2026 (UTC) == Vote now in the 2026 U4C election == <section begin="announcement-content" /> Eligible voters are asked to participate in the 2026 [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee]] election. More information–including an eligibility check, voting process information, candidate information, and a link to the vote–are available on Meta at the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|2026 Election information page]]. The vote closes on 2 June 2026 at [https://zonestamp.toolforge.org/1780358400 00:00 UTC]. Please vote if your account is eligible. Results will be available by 14 June 2026. -- In cooperation with the U4C,<section end="announcement-content" /> [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 17:15, 27 May 2026 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> == Create an autopatrolled user group? == 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) 2qe8hur7u3pncjn9srgx469wuobu1uw 2812034 2812033 2026-05-29T16:42:43Z Koavf 147 /* Create an autopatrolled user group? */ Reply 2812034 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] == I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&#126;2026-28640-56|&#126;2026-28640-56]] ([[User talk:&#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC) :What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:35, 12 May 2026 (UTC) : Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC) ::Sounds good -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:13, 15 May 2026 (UTC) :::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC) == Proposal to rehost Wikinews here == As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance. I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:30, 14 May 2026 (UTC) :I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) ===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) == [[Wikiversity:Deletion policy]] proposed as policy == [[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]]. This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]]. Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy. === Voting === *{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:33, 18 May 2026 (UTC) *{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC) === Comments === == May 2026 Wikimedia Café meetups regarding the Wikimedia Foundation Annual Plan == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 15px; padding-right: 15px;">[[File:Wikimedia Café logo in plain SVG format.svg|75px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of May. Both sessions will focus on the [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2026-2027 the 2026-2027 Wikimedia Foundation Annual Plan]. Participants may attend either or both sessions. #'''Saturday, 30 May 2026 at 15:00 UTC''' ([https://zonestamp.toolforge.org/1780153200 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''Sunday, 31 May 2026 at 05:00 UTC''' ([https://zonestamp.toolforge.org/1780203600 timestamp converter]), at a time friendly to Asia and the Pacific Café participants are highly encouraged to read in advance [https://en.wikipedia.org/wiki/User:Sohom_Datta/annual_plan_guide at least this summary of the plan]. Optionally, Café participants are encouraged to read portions of the plan that interest them and [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_Annual_Plan/2026-2027 ask questions or provide feedback on the Annual Plan talk page]. Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#May_2026_meetings_with_a_focus_on_Wikimedia_Foundation_Annual_Plan/2026-2027 tables of timestamp conversions for both sessions], [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#Agenda._This_will_be_an_approximately_1_hour_Caf%C3%A9_session,_and_is_extendible_for_an_additional_30_minutes_if_needed. the agenda], and [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 19:46, 21 May 2026 (UTC) == Vote now in the 2026 U4C election == <section begin="announcement-content" /> Eligible voters are asked to participate in the 2026 [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee]] election. More information–including an eligibility check, voting process information, candidate information, and a link to the vote–are available on Meta at the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|2026 Election information page]]. The vote closes on 2 June 2026 at [https://zonestamp.toolforge.org/1780358400 00:00 UTC]. Please vote if your account is eligible. Results will be available by 14 June 2026. -- In cooperation with the U4C,<section end="announcement-content" /> [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 17:15, 27 May 2026 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> == Create an autopatrolled user group? == 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) kqkk0pkksup8oukb5ey6u6bch763waq 2812093 2812034 2026-05-30T01:48:25Z Codename Noreste 2969951 /* Create an autopatrolled user group? */ reply to Koavf: We'll decide on the name later. (-) ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2812093 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] == I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&#126;2026-28640-56|&#126;2026-28640-56]] ([[User talk:&#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC) :What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:35, 12 May 2026 (UTC) : Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC) ::Sounds good -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:13, 15 May 2026 (UTC) :::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC) == Proposal to rehost Wikinews here == As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance. I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:30, 14 May 2026 (UTC) :I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) ===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) == [[Wikiversity:Deletion policy]] proposed as policy == [[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]]. This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]]. Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy. === Voting === *{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:33, 18 May 2026 (UTC) *{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC) === Comments === == May 2026 Wikimedia Café meetups regarding the Wikimedia Foundation Annual Plan == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 15px; padding-right: 15px;">[[File:Wikimedia Café logo in plain SVG format.svg|75px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of May. Both sessions will focus on the [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2026-2027 the 2026-2027 Wikimedia Foundation Annual Plan]. Participants may attend either or both sessions. #'''Saturday, 30 May 2026 at 15:00 UTC''' ([https://zonestamp.toolforge.org/1780153200 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''Sunday, 31 May 2026 at 05:00 UTC''' ([https://zonestamp.toolforge.org/1780203600 timestamp converter]), at a time friendly to Asia and the Pacific Café participants are highly encouraged to read in advance [https://en.wikipedia.org/wiki/User:Sohom_Datta/annual_plan_guide at least this summary of the plan]. Optionally, Café participants are encouraged to read portions of the plan that interest them and [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_Annual_Plan/2026-2027 ask questions or provide feedback on the Annual Plan talk page]. Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#May_2026_meetings_with_a_focus_on_Wikimedia_Foundation_Annual_Plan/2026-2027 tables of timestamp conversions for both sessions], [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#Agenda._This_will_be_an_approximately_1_hour_Caf%C3%A9_session,_and_is_extendible_for_an_additional_30_minutes_if_needed. the agenda], and [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 19:46, 21 May 2026 (UTC) == Vote now in the 2026 U4C election == <section begin="announcement-content" /> Eligible voters are asked to participate in the 2026 [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee]] election. More information–including an eligibility check, voting process information, candidate information, and a link to the vote–are available on Meta at the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|2026 Election information page]]. The vote closes on 2 June 2026 at [https://zonestamp.toolforge.org/1780358400 00:00 UTC]. Please vote if your account is eligible. Results will be available by 14 June 2026. -- In cooperation with the U4C,<section end="announcement-content" /> [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 17:15, 27 May 2026 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> == Create an autopatrolled user group? == 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) 5ltym7hcy15exbxeaiedj3el3uocjjy 2812104 2812093 2026-05-30T04:16:08Z Koavf 147 /* Create an autopatrolled user group? */ Reply 2812104 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] == I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&#126;2026-28640-56|&#126;2026-28640-56]] ([[User talk:&#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC) :What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:35, 12 May 2026 (UTC) : Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC) ::Sounds good -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:13, 15 May 2026 (UTC) :::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC) == Proposal to rehost Wikinews here == As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance. I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:30, 14 May 2026 (UTC) :I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) ===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) == [[Wikiversity:Deletion policy]] proposed as policy == [[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]]. This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]]. Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy. === Voting === *{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:33, 18 May 2026 (UTC) *{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC) === Comments === == May 2026 Wikimedia Café meetups regarding the Wikimedia Foundation Annual Plan == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 15px; padding-right: 15px;">[[File:Wikimedia Café logo in plain SVG format.svg|75px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of May. Both sessions will focus on the [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2026-2027 the 2026-2027 Wikimedia Foundation Annual Plan]. Participants may attend either or both sessions. #'''Saturday, 30 May 2026 at 15:00 UTC''' ([https://zonestamp.toolforge.org/1780153200 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''Sunday, 31 May 2026 at 05:00 UTC''' ([https://zonestamp.toolforge.org/1780203600 timestamp converter]), at a time friendly to Asia and the Pacific Café participants are highly encouraged to read in advance [https://en.wikipedia.org/wiki/User:Sohom_Datta/annual_plan_guide at least this summary of the plan]. Optionally, Café participants are encouraged to read portions of the plan that interest them and [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_Annual_Plan/2026-2027 ask questions or provide feedback on the Annual Plan talk page]. Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#May_2026_meetings_with_a_focus_on_Wikimedia_Foundation_Annual_Plan/2026-2027 tables of timestamp conversions for both sessions], [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#Agenda._This_will_be_an_approximately_1_hour_Caf%C3%A9_session,_and_is_extendible_for_an_additional_30_minutes_if_needed. the agenda], and [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 19:46, 21 May 2026 (UTC) == Vote now in the 2026 U4C election == <section begin="announcement-content" /> Eligible voters are asked to participate in the 2026 [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee]] election. More information–including an eligibility check, voting process information, candidate information, and a link to the vote–are available on Meta at the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|2026 Election information page]]. The vote closes on 2 June 2026 at [https://zonestamp.toolforge.org/1780358400 00:00 UTC]. Please vote if your account is eligible. Results will be available by 14 June 2026. -- In cooperation with the U4C,<section end="announcement-content" /> [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 17:15, 27 May 2026 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> == Create an autopatrolled user group? == 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) 4vsc17bcap7d0774oh3l6zd3wuspbw8 2812123 2812104 2026-05-30T11:28:18Z Jtneill 10242 /* Create an autopatrolled user group? */ reply ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2812123 wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == [[MediaWiki:Protectedpagetext#Protected edit request on 11 December 2025]] == I posted an edit request there 5 months ago, so I’ll be taking it to this page. [[Special:Contributions/&#126;2026-28640-56|&#126;2026-28640-56]] ([[User talk:&#126;2026-28640-56|talk]]) 23:33, 12 May 2026 (UTC) :What exactly is the problem? I don't understand what needs to change and why. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 23:35, 12 May 2026 (UTC) : Pinging @[[User:Atcovi|Atcovi]], @[[User:Jtneill|Jtneill]] and @[[User:Juandev|Juandev]] for further input. Someone is requesting a modification to [[MediaWiki:Protectedpagetext]] to use {{tlx|Protected page text}}, but we might need to discuss whether to use the template. In the meantime, I'll start a sandbox version of the protected page text template. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 23:19, 14 May 2026 (UTC) ::Sounds good -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 04:13, 15 May 2026 (UTC) :::+1 Jtneill. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 12:59, 19 May 2026 (UTC) == Proposal to rehost Wikinews here == As many of you know, and mentioned here at the Colloquium, our sister project Wikinews recently closed, with all 31 active editions made read-only. [[User:BigKrow]] has asked about the prospect of writing news stories here and I suggested that since we already have [[School:Journalism]] and some resources related to the [[:Category:Journalism|broader topic of journalism]]. I would like to propose that we have continued and indefinite space for {{w|citizen journalism}} by essentially repurposing Wikinews into a sub-project here. The only special infrastructure that Wikinews required was [[:mw:Extension:DynamicPageList]], which was deactivated and caused issues due to a lack of maintenance. I will add this proposal to the site banner, but I recognize that that may be a conflict of interest, so if anyone requests that I remove it, I will. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:30, 14 May 2026 (UTC) :I would like to see this conversation go for at least 30 days to establish a consensus. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) ===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) == [[Wikiversity:Deletion policy]] proposed as policy == [[Wikiversity:Deletions]] has been operating as a [[Wikiversity:Guidelines|guideline]]. It has been revised and moved to [[Wikiversity:Deletion policy]], consistent with naming conventions used across sister projects such as Wikipedia, Wikibooks, and Wikiquote. The speedy deletion criteria have also been updated for consistency with [[MediaWiki:Deletereason-dropdown]]. This proposal is for the page to be formally adopted as [[Wikiversity:Policies|Wikiversity policy]]. Community feedback is invited, including suggestions for further improvements that may strengthen the proposed policy. === Voting === *{{support}} Seems reasonable. If there's somehow something missed here, we can just amend it later. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:33, 18 May 2026 (UTC) *{{support}} I don't see any issues with the policy. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 16:07, 18 May 2026 (UTC) === Comments === == May 2026 Wikimedia Café meetups regarding the Wikimedia Foundation Annual Plan == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 15px; padding-right: 15px;">[[File:Wikimedia Café logo in plain SVG format.svg|75px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of May. Both sessions will focus on the [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2026-2027 the 2026-2027 Wikimedia Foundation Annual Plan]. Participants may attend either or both sessions. #'''Saturday, 30 May 2026 at 15:00 UTC''' ([https://zonestamp.toolforge.org/1780153200 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''Sunday, 31 May 2026 at 05:00 UTC''' ([https://zonestamp.toolforge.org/1780203600 timestamp converter]), at a time friendly to Asia and the Pacific Café participants are highly encouraged to read in advance [https://en.wikipedia.org/wiki/User:Sohom_Datta/annual_plan_guide at least this summary of the plan]. Optionally, Café participants are encouraged to read portions of the plan that interest them and [https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_Annual_Plan/2026-2027 ask questions or provide feedback on the Annual Plan talk page]. Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#May_2026_meetings_with_a_focus_on_Wikimedia_Foundation_Annual_Plan/2026-2027 tables of timestamp conversions for both sessions], [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#Agenda._This_will_be_an_approximately_1_hour_Caf%C3%A9_session,_and_is_extendible_for_an_additional_30_minutes_if_needed. the agenda], and [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 19:46, 21 May 2026 (UTC) == Vote now in the 2026 U4C election == <section begin="announcement-content" /> Eligible voters are asked to participate in the 2026 [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee|Universal Code of Conduct Coordinating Committee]] election. More information–including an eligibility check, voting process information, candidate information, and a link to the vote–are available on Meta at the [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2026|2026 Election information page]]. The vote closes on 2 June 2026 at [https://zonestamp.toolforge.org/1780358400 00:00 UTC]. Please vote if your account is eligible. Results will be available by 14 June 2026. -- In cooperation with the U4C,<section end="announcement-content" /> [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 17:15, 27 May 2026 (UTC) <!-- Message sent by User:Keegan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&oldid=30513860 --> == Create an autopatrolled user group? == 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) 5k21yshg7mrxmlxb0bkdujb2smulesy 0 5845 2812078 2811918 2026-05-29T23:18:03Z Evan Mercer 3071189 2812078 wikitext text/x-wiki <table class="ambox" style="background-color: #e8f0fe; border-left: 5px solid #1a73e8; border-collapse: collapse; width: 100%; margin: 1em 0; padding: 0.5em 1em; border: 1px solid #a8c7fa; border-left-width: 5px;"><tr><td style="padding: 0.5em 0 0.5em 1.2em; vertical-align: middle; width: 40px;">[[File:Exquisite-khelpcenter.png|40px]]</td><td style="padding: 0.5em 1.2em; vertical-align: middle;">'''This article discusses a legacy technology.''' ActionScript is officially deprecated and no longer actively used or supported following the **End-of-Life (EOL) of Adobe Flash Player on 31 December 2020**. Content built using ActionScript has been **blocked from running in Flash Player since 12 January 2021**, and all major web browsers have completely removed support in favour of open standards like [[TypeScript]], [[JavaScript_Programming|Javascript]] and WebAssembly.</td></tr></table> {{rightTOC}} Welcome to Adobe Flash [[w:ActionScript|ActionScript]]! I'm glad you've decided to try and learn ActionScript, it is truly a wonderful programming language with plenty of potential in the domain of interactive media. Firstly, I would like to introduce myself, as teachers do. I am Raven Storm, a young Canadian programmer who has been working in ActionScript for over three years. I have plenty of free time that I've turned into a lot of experience. As an active member of the WikiMedia community, I would like to contribute in whatever way I can by creating this basic tutorial. So let's get started, shall we? ==Prerequisites== Since ActionScript is a basic scripting language, there is no real need for any programming experience prior to this tutorial. Basic knowledge of HTML and computers in general are a plus! ==[[Portal:Learning Projects|Learning Projects]]== Learning materials and [[Portal:Learning Projects|learning projects]] are located in the main Wikiversity namespace. Simply make a [[link]] to the name of the learning project (learning projects are independent pages in the [[Wikiversity:Namespaces|main namespace]]) and start writing! * ... ===ActionScript tutorial=== This tutorial will cover everything you need to get started. There are plenty of tutorials out there that cover complex and specific engines created in ActionScript: this is not (yet) a congolomeration of these tutorials (see websites like [http://www.kirupa.com/ Kirupa.com], [http://www.flashkit.com/tutorials/ Flashkit Tutorials], [http://www.actionscript.org/resources/categories/Tutorials/ ActionScript.org Tutorials]) but an introduction to ActionScript so that you can understand those tutorials! *[[ActionScript:Introduction]] ==Enrolled== Please sign below if you are participating in this topic. Use 4 tildes (~) to sign. *[[User:Ravenstorm|Ravenstorm]] 01:20, 27 September 2006 (UTC) *[[User:88.207.191.72|88.207.191.72]] 17:55, 27 March 2007 (UTC) *--[[User:Xora K Joken|Xora]] 18:34, 9 April 2007 (UTC) *--[[User:Kortex|Kortex]] 20:54, 14 December 2007 (UTC) *--[[User:Cecil|Cecil]] 00:29, 11 May 2008 (UTC) *--[[User:Gazooks113|Gazooks113]] 01:35, 28 November 2008 (UTC) *''Eternal.hazard'' *--Thomas Bebbington[[Special:Contributions/97.118.217.119|97.118.217.119]] 06:23, 29 November 2008 (UTC) *--k.krishnakanth reddy *[[User:Hughveal|Hughveal]] ([[User talk:Hughveal|discuss]] • [[Special:Contributions/Hughveal|contribs]]) 16:28, 11 March 2014 (UTC)[[User:hughveal|hughveal]]12:27, 11 March 2014 (UTC) ==See also== *[[b:Programming:Action script|Programming:Action script]] - at Wikibooks [[Category:Flash ActionScript]] [[Category:Adobe]] sfoid9jd9vbpnea4jxb5q17micezhgp 2 64001 2812101 2291057 2026-05-30T02:48:17Z CommonsDelinker 9184 Replacing Aircraft_workers_on_lunch_break_1942.gif with [[File:Men_and_women_employees_on_the_"swing_shift"_of_North_American's_Inglewood,_Calif.,_aircraft_plant_enjoy_their_lunch_periods._-_DPLA_-_d6023e28bd8009cde3b2cbe41f490f1f.gif]] (by [[:c:User:Comm 2812101 wikitext text/x-wiki <!-- This is a comment line - the line below adds user boxes and categories --> {{SPPP2008}} = Week 1: Introduction to Social Psychology = What is Social Psychology? * Social Psychology is a branch of psychology that seeks a broad understanding of how human beings think act and feel within a group – Baumeister & Bushman. * Understanding human behaviour in a social context – James Neil. * How the thoughts feelings and behaviours of others are influenced by the actual imagined of implied presence of others – Allport. * Social psychology is the study of how people and groups interact – Wikipedia [[Image:Social Psychology Definition 1.jpg|600px|center]] My own thoughts on what social psychology is – a broad definition may be that social psychology is the study of individual behaviour within a group situation. Where as sociology is the study of society as a whole, and psychology studies the processes within the individual, social psychology studies the individuals within a society, how and why they act within, and are influenced by groups and society as a whole. * Social Psychology can help you make sense of your social world. [[Image:Social-psychology-division.gif‎|right]] A few key terms and concepts that came up during the readings and the lecture: * Social Perception: How we interpret social objects * Social Influence: Attitudes and behaviours brought about by others * Social Interaction: How we interact with others in a social world * Person vs. Situation debate: is a person influenced primarily by their biology or the situation they are in? The person/situation debate surrounds the issues of what influences behaviour? Is it the person or is it the situation? I currently sit somewhere in the middle in this debate, not because I am unable to decide, but because I have witnessed plenty of evidence for both sides, highlighting that the topic is neither black nor white, but rather some shade of grey. I find myself asking many questions regarding this - Why a person might behave one way in one situation, yet another way in a different situation? How would a different person behave in the same situation? How would have they behaved across different cultural and time dimensions? I am personally aware that my behaviour is often influenced or guided by the situation that I am in, for instances there are certain sets of acceptable behaviour that differs across settings (friends vs. grandparent or university vs. work), but is this because of my personality or that society and culture dictate how we must behave in various settings. On the other hand I know other people who’s behaviour is relatively stable across all situations, whether it be around friends, family, colleagues, at work or at home. So this is evidence for both sides, and where does this leave us? What I am hoping to learn from this course is the ways in which different factors dictate or influence how individuals behave in society. == Culture and Nature (Ch 2) == Social psychology aims to answer the fundamental question surrounding how people think, feel, and act? When investigating this question many look into the important aspects of culture and nature. Nature:[[Image:Dna-split.png‎|100px|right]] * Genes, hormones, brain structures etc. * Turn to the evolutionary theory to explain behaviour. Looks at how change occurs in nature * ''Natural selection'' demonstrates that traits that increase one’s chances of survival or reproduction will endure, and those that reduce these chances will not. * Survival means living longer than others. I take nature to mean everything that is essentially innate, where as culture could be conceived as everything that is gained from socialisation. Culture: [[Image:Aboriginal_song_and_dance.jpg‎|200px|right]] * What people learn from parents, society, and from past experience. * What people share, ranging from language, values, food styles, to styles of government. * Consists of shared ideas and meanings. Ideas are mental representations that can be expressed through language. ‘ Culture is an information based system, involving both shared understandings and praxis (practical way of doing things), that allows groups of people to live together in an organised fashion and to satisfy their biological and other needs.’ – Baumeister & Bushman Human beings are often referred to as social and cultural animals. That is we seek connections to others, we prefer to live work and play with other people (social side). But whereas many other animals can be classified as social animals, humans are distinct in the fact that we are cultural animals in the way that we can take part in this culture. ===Further information === * Gustav, J. (1986). Nature, culture and social psychology. ''European Journal of Social Psychology, 16'', 17-30[http://web.ebscohost.com.ezproxy2.canberra.edu.au/ehost/detail?vid=8&hid=113&sid=d57f2c07-fc6f-4595-b1f7-7971c1ef1def%40sessionmgr108&bdata=JnNpdGU9ZWhvc3QtbGl2ZQ%3d%3d#db=pbh&AN=12144459] == Tutorial 1 - Introductory tutorial == To be perfectly honest this was one of my last tutorials so I had already been through the usual get to know you exercises, and wasn’t to thrilled at the prospect of doing another. However, I was pleasantly surprised. For this get to know each other exercise we were required to form groups based on a certain characteristic (i.e. eye colour, number of siblings, where you were born etc). As we moved on it became slightly more challenging, as we were required to form groups based on beliefs (i.e. religious and political). The thing I think I found most interesting about this tutorial was just how quickly and easily individuals create groups. Even though people may hardly know one another, we are all willingly to form groups on the basis of some characteristic or quality. It is as if this would guarantee that everyone had at least one thing in common. Another thing I found interesting was how easily you related to the people in your groups, and excluded those in another, because in some way or anther they were different. ''What I know/what do I expect'' – Well to be perfectly honest before this course I cant say that I know too much about social psychology. After doing a brief overview of social psychology in psychology 102 last year, I became very interested in social psychology as a field of study. And not knowing as much as I do now, I enrolled in a few sociology units thinking (very naively) that it would be somewhat similar to social psychology, and became quite disappointed as it wasn’t answering all the questions I had running through my head. So upon enrolling in social psychology this year I wasn’t too excited to say the least, however, so far I have been pleasantly surprised, and look forward to learning more and more throughout the semester. ''What I want to know'' – This is a bit of a difficult question to answer, as there is just so much about human beings I want to understand. What drives people to act the way they do? Why are human’s so eager to form groups, and readily exclude others from those groups? What makes people like and dislike other people? If I primary drive of humans is to form social connections, why do we exclude so many people from these connections? Why are humans so eager to follow the latest trends and fads, is it just to feel socially connected or is there something more? Why are some people happy being law-abiding citizens and others ‘driven’/ ‘attracted’ to crime? ''Essay ideas'' – Not quite sure about essay ideas, maybe something to do with crime rates in different cultures, or social influences on crime, or other factors affecting crime rates (i.e. temperature). = Week 2 – Social Self = The human ‘self’ is a tool that we all use to help us cope with human society and satisfy our needs. The self is designed to help us relate to one another as well as helping us connect more intimately with others. Building and maintaining relationships is one of the primary needs of human beings. The purpose of the self is to gain social acceptance, as well as to play social roles. We may underestimate the need to belong but in reality it is a fundamental human need - [http://www.youtube.com/watch?v=hUbJIOEr4gM The idiot's guide to social acceptance] There are no firm or tight boundaries around the self; the self appears to be everything that we do both in the public and private areas of our lives. The concept of the self could be looked at as a result of evolution or a product of society. Which has a greater influence, our brains, neurons, hormones, and other biological processes, or the requirements of the society we are living in? This does not appear to be such a clear-cut distinction, as both appear to play an importance and crucial role. It would be difficult to imagine a ‘self’ without either nature or culture. It has been proposed that the self is made up of several parts, including * [[w:self-knowledge|Self-knowledge]] (or self-concept): this is information we possess about the self. * Interpersonal self (or public self): as the name suggests this is the self that helps connect socially with others. * Agent-self (or executive functioning): this is the self as a doer; makes decisions, active responding, self-control etc. Self-construal- is a way of thinking about the self (i.e. independent vs. interdependent self-construal), either seeing the self as separate and distinct from others, or a self that emphasizes connections with others. == Where self-knowledge comes from?== Some believe that people learn about themselves through everyday interactions with others, where individuals learn how others perceive them. * This is also known as the ''looking-glass self'', and consists of three components; # You imaging how you appear to others # You imagine how other perceive you # You develop an emotional response as a result of imaging how other may judge you. This idea was then expanded to include the notion of feedback from others. Essentially other people tell us who we are? Although I find this notion a little hard to understand, because when you meet people for the first time, they may ask you to tell them a little about yourself, not let me tell you a little bit about yourself… Other ways that we may learn about ourselves include - [[Image:Lupa.na.encyklopedii.jpg|200px|right]] * ''Introspection'': People have direct knowledge of what they are like; they only have to look inwards. * ''Social comparison'': you learn not the facts about yourself, but what value they hold, by comparing themselves to others. This can make people either feel better about themselves (through downward social comparison) or worse about themselves (through upward social comparison) * ''Self-perception'': We learn about ourselves in the same way that we learn about others, though observing behaviour and drawing conclusions (self-perception theory). * ''Phenomenal self or working self'' we are only aware of a small portion of information about ourselves at any point in time. Why do we seek self-knowledge? We need self-knowledge as to enable us to fit in better with others. Although people learn about themselves there are some things they would rather know than others. There are 3 general motives that shape peoples quest for knowledge. # The simple desire to learn knowledge about one’s self (appraisal motive) # The desire to learn flattering things about the self (self-enhancement motive) # The desire to confirm what one already believes about one’s self (consistency motive) I’m sure we can all relate to times where people have asked our opinion on something about themselves (i.e. does my bum look big in this), and no matter what you tell them, if it is not consistent with what they possibly already think they wont believe you. I wonder if this is because the desire to find out the information is driven by the consistency motive, and maybe if information doesn’t align, it is not accepted? I guess this is where the question of telling people what they want to hear comes in? ===Further Information=== Swann, W. B., & Read, S. J. (1981). Acquiring self-knowledge: The search for feedback that fits. ''Journal of Personality and Social Psychology, 41'', 1119-1128. [http://web.ebscohost.com.ezproxy2.canberra.edu.au/ehost/detail?vid=22&hid=113&sid=d57f2c07-fc6f-4595-b1f7-7971c1ef1def%40sessionmgr108&bdata=JnNpdGU9ZWhvc3QtbGl2ZQ%3d%3d#db=pdh&AN=psp-41-6-1119] Ulrich, K., Hannover, B., & Schubert, B. (2001). The semantic-procedural interface model of the self: The role of self-knowledge for context-dependent versus context--independent modes of thinking. ''Journal of Personality and Social Psychology, 80'', 397-409. [http://web.ebscohost.com.ezproxy2.canberra.edu.au/ehost/detail?vid=24&hid=113&sid=d57f2c07-fc6f-4595-b1f7-7971c1ef1def%40sessionmgr108&bdata=JnNpdGU9ZWhvc3QtbGl2ZQ%3d%3d#db=pdh&AN=psp-80-3-397] ==Self-esteem== [[w:self-esteem|Self-esteem]] could be referred to as how favorable a person evaluates himself or herself, or their global feelings of self-worth. Self-esteem is often based on the comparison of one’s self to others. In western society it is usually accepted that it is better to have high self-esteem than low self-esteem. Possibly indicating that high self-esteem is ‘good for us’, and people who have high levels of self-esteem are happier and healthier. We are often told that if we all just had higher levels of self-esteem, our lives would be perfect (this is highlighted in a lot of ‘self-help’ books or programs). [[Image:Self Esteem Shop.jpg|250px|right]] This, however, does not appear to be the case. For humans to be healthy functioning individuals it is best not to have overly high levels of self-esteem, but more a balance of positive self-esteem and realism. As people with low self-esteem are prone to depression, and individuals with overly high levels of self-esteem are considered narcissists (excessive self-love). [http://www.youtube.com/watch?v=jquAX9GId_g Narcissism Rules] [http://www.youtube.com/watch?v=J38LP8v8E_E Narcissism Lesson] [[Image: LOE3.jpg‎|right]] The textbook also mentions the fact that it is not individuals with low self-esteem who have a distorted view of reality (which is what I always thought until now), but those with high self-esteem who distort reality. I finding this interesting that high self-esteem is related to a distorted view of reality. Even though it would appear that there are no benefits to having high self-esteem besides a distortion of reality, it has been suggested that there are actually many benefits of high self-esteem - * Fosters confidence that you can do the right thing. * More willing to speak up in groups. * More likely to try again after failure. * More willing to do what they think is best. ===Further information=== Raskin, R., Novacek, J., & Hogan, R. (1991). Narcissism, self-esteem, and defensive self-enhancement. ''Journal of Personality, 59'', 19-39. [http://web.ebscohost.com.ezproxy2.canberra.edu.au/ehost/detail?vid=18&hid=113&sid=d57f2c07-fc6f-4595-b1f7-7971c1ef1def%40sessionmgr108&bdata=JnNpdGU9ZWhvc3QtbGl2ZQ%3d%3d#db=pbh&AN=9106030312] Bogart, L. M., Benotsch, E. G., & Pavlovic, J. D. (2004). Feeling superior but threatened: The relation of narcissism and social comparison. ''Basic and Applied Social Psychology, 26'', 35-44 [http://web.ebscohost.com.ezproxy2.canberra.edu.au/ehost/detail?vid=18&hid=113&sid=d57f2c07-fc6f-4595-b1f7-7971c1ef1def%40sessionmgr108&bdata=JnNpdGU9ZWhvc3QtbGl2ZQ%3d%3d#db=pbh&AN=18331801] Riketta, M., & Dauenheimer,D. (2003). Manipulating self-esteem with subliminally presented words. ''European Journal of Social Psychology, 33'', 679-700. [http://web.ebscohost.com.ezproxy2.canberra.edu.au/ehost/detail?vid=24&hid=113&sid=d57f2c07-fc6f-4595-b1f7-7971c1ef1def%40sessionmgr108&bdata=JnNpdGU9ZWhvc3QtbGl2ZQ%3d%3d#db=pbh&AN=10827243] Hannover, B., Birkner, N., & Pohlmann, C. (2006). Ideal selves and self-esteem in people with independent or interdependent self-construal. ''European Journal of Social Psychology, 36'', 119-133. [http://web.ebscohost.com.ezproxy2.canberra.edu.au/ehost/detail?vid=24&hid=113&sid=d57f2c07-fc6f-4595-b1f7-7971c1ef1def%40sessionmgr108&bdata=JnNpdGU9ZWhvc3QtbGl2ZQ%3d%3d#db=pbh&AN=19475089] = Week 3 – Social Thinking = Human beings exert a large amount of time and energy thinking about other people. This may because one of our primary needs as human beings is to be accepted and to participate in society. What this means for humans is being able to manage relationships and deal with other people. ===''Attribution Theory''=== Attribution Theory suggests how we explain someone's behaviour - by either crediting the situation or the person's disposition. When we think of the cause of our own bahaviour, we are likely to attribute the causes of our actions to the situation, whereas when we think of the causes of other people's behaviour we fall into the trap of the [[w:fundamentalattributionerror|fundamental attribution error]]. This is the tendency for observers, when analyzing another's behaviour, to underestimate the impact of the situation and to overestimate the impact of personal dispositions.Although the fundamental attribution error can be witnessed in all cultures, it is suggested that it is especially strong in individualistic Western cultures. In everyday life we struggle to explain other people's actions. For instance a jury must decided whether an action was intentional or self-defense. In a job interview, the interview has to judge whether the applicant is genuine. When we make such judgments, our attributions, either to the person or the situation, have important consequences. [[Image:Bus driver.JPG|350px|left]] I think that a lot of road rage is the result of the fundamental attribution error. We interpret others bad driving to internal dispositions - such as that they must just be a bad driver. ''Attitudes and Actions'' Attitudes are feeling, often based on our beliefs, that predispose us to respond in a particular way to objects people and events. Attitudes can affect our actions, but also our actions can affect our attitudes. This can be seen in the [[w:foot-in-the-doortechnique|foot-in-the-door phenomenon]] - the tendency for people who have first agree to a small request to comply later with a larger request. It has also been suggested that role playing may affect our attitudes. When you adopt a new role to strive to follow the social prescription. At first your behaviours may not feel natural because you are playing a role, however, before long what began as action becomes you. There is a saying that i think particularly suits this notion - 'fake it, until you make it'. When we act in ways that do not align with our attitudes we may experience cognitive tension, or cognitive dissonance, an to relieve this tension we have to being our actions inline with our attitudes. Social cognition: Thoughts about people and social relationships. == Tutorial 2: How do we undertake effective interpersonal communication? == Levels (depth of communication) * Shallow – Factual information (time, date etc.) * Clichés/scripts (common greetings, e.g. hey how’re you? An act of feeling obliged to ask). * Opinions/ attitudes/ thoughts * Deep – Feelings/emotions/affect * Self-disclosure. Channels (relative amount of importance we attribute to part of information) For example in face-to-face contact we may place 50% on verbal cues and 50% on nonverbal cues (for example body language, facial expressions, tone of voice). Whereas in other forms of communication (e-mail, phone, SMS), the weight we give to verbal and nonverbal cues may change. In e-mail you are highly reliant on written words as there is a lack on nonverbal cues (although we now have the use of emoticons). On the phone you may presume that this would be the same, but I have found that the focus drastically shifts from a focus on verbal cues, to nonverbal cues like the tone, and volume, of voice. You would find it hard to believe someone who is telling you that they are really happy and excited, while they sound sad and depressed. I don’t know if everyone places the same weight on different cues, but I know that I rely very heavily on nonverbal cues as opposed to verbal ones. [[Image:Men and women employees on the "swing shift" of North American's Inglewood, Calif., aircraft plant enjoy their lunch periods. - DPLA - d6023e28bd8009cde3b2cbe41f490f1f.gif|200px|right]] [[w:bodylanguage|<u>Body language</u>]] that tells someone that you are engaged in the conversation: smiling, eye contact, and body front on. Where as things like crossed arms, wondering eyes, looking away, and changing orientation of your body are likely to inform someone that you are attempting to withdraw from the situation. When then did an exercise on eye contact … I don’t know what to say; I thought that I was pretty good at keeping eye contact with people (I always use to win staring competitions), until today. We were required to stand close to another person (someone we didn’t really know) and maintain eye contact for between 30-60. That might not sound like a very long time, but this activity got more intense as the time went on, and it seemed to go forever. It was also noticeable that there is a certain distance for optimal eye contact, anything that it too close or far away was the hardest to maintain. The first thing that I wanted to do was to laugh, because it was getting pretty awkward. I was surprised at how weird it felt. It really made me wonder how much eye contact I actually make with other people. However, I think it would make a big difference holding eye contact with someone you know well (friend, family, or partner), than someone you don’t know at all, as there seems to be a certain level of intimacy involved. They always did say that eyes are the windows to the soul, and up until now I never believed just how powerful the eyes are. === Communication === [[Image:Communication_shannon-weaver2.svg‎|150px|right]][[Image:Communication_sender-message-reciever.png‎|150px|left]] ''The Shannon weaver model of communication''- It embodies 6 concepts including; a source, an encoder, a message, a channel, a decoder, a receiver. *The emphasis here is very much on the transmission and reception of information = Week 4 – Aggression (Ch9) + Ghosts of Rwanda = * [[w:Aggression|Aggression]] is any behaviour that is intended to or motivated to harm another person. Aggression is not a thought but an action. [[Image:The Post Office was a riot.jpg|300px|right]] There are many different ‘types’ of aggression: - Hostile aggression, which is, Impulsive (heat of the moment) behaviour intended to harm someone. - Instrumental aggression, which could be thought of as the opposite of this hot impulsive behaviour, as instrumental aggression is premeditated, calculated behaviour. This could be something like spreading a rumor intended to ruin a person’s reputation. - Verbal aggression: yelling, screaming etc. - Physical aggression: hitting, kicking, physical fighting etc. - Passive aggression, which could be described as the intentional harming of others through withholding behaviour (failing to tell someone that a person called regarding the job they applied for, and consequently the person does not get the job). - Active aggression, this is harming someone by performing some behaviour. [[Image:2004327 Fighting Elephants.png|250px|center]] Aggression is not only found in humans. How do we develop aggression-? * Some people propose that aggression is not learned, but developed through the process of evolution as a means of survival – Lorenz * Others believe that aggression is learned like any other behaviour – Bandura Like everything else I don’t think that it is as simple as explaining it through one view. Both learning and instincts may be relevant to explaining aggression. There has to be some innate aspect of aggression because it can be seen all over the world. Although I don’t think that it can purely be explained by instincts either. This may be another one of those situations where nature says go and culture say stop. As socialized beings we learn that in a lot of situations it is neither appropriate nor necessary to use aggression. Inner Causes of Aggression- ''Frustration/aggression hypotheses'': the occurrence of aggression is presupposed by the existence of frustration, and the existence of frustration generally leads to the occurrence of aggression. ''Relative deprivation theory'': the sense of having less than one is entitled to, which leads to a feeling of frustration and then aggression. ''Excitation transfer'' – arousal from one situation may carry over to another, and possibly lead to heightened aggression – Zimmerman. I wonder if this is one of the reasons why people become more aggressive when they drink alcohol? Is it because they are already in an aroused state because of the alcohol, and then they misinterpret another situation (someone accidently bumping into them), which leads to aggression? === Further Information === [http://www.csulb.edu/~tstevens/b-anger.htm anger and aggression] Bjorkqvist, K., Legerspetz, K. M. J., & Kaukiainen, A. (1991) Do girls manipulate and boys fight? Developmental trends in regard to direct and indirect aggression. ''Aggressive Behavior, 18'', 117-127. == Reflections on Ghosts of Rwanda == I’m not exactly sure where I’m supposed to start. This was quite possibly one of the most emotive films I have ever seen. I would find it hard to believe that anyone could watch this film and not find it thoroughly tragic and heartbreaking. One of the most tragic things was that this could be going on and all over the world people turned a blind eye. One could only hope that if such an event were to happen again the world would be more responsive. But I have a feeling that was what a lot of people would have been saying after the holocaust, yet look what happened only 50 odd years later. There were a few questions that were continually running through my head as I was watching the film, so I though I would just outline a few of them. * I cannot understand how people can be so cruel in their killings. Is it possible to kill people in they way they did without feeling any remorse or emotion what so ever? * How could situations get so bad as to lead one group to massacre another? What kinds of situations bread such hated for fellow human beings? * What is the point of the United Nations; if in times of conflict they cant intervene? I understand their not wanting to interfere with internal conflict, but when almost 1 million people were killed, is that really an option? * What could be the psychological mechanisms at work that possibly exaggerate the aggression? Could the drive-theory, or frustration-aggression hypothesis (''the occurrence of aggressive behaviours presupposes the existence of frustration and the existence of frustration leads to some form of aggression) ''explain it? The Hutus were already frustrated by what the Tutsis use to do to them, and used this as propaganda when rounding up support. * Did the Weapons effect come in to play? Emotions were running high and weapons were readily visible and available. * Did the fact that they acted as a ‘group’ instead of individuals lead to more violence? As this may have served as a sense of anonymity and deindividuation, which would encourage more violence and more killing. * The next striking thing about the movie was how the world turned a blind eye and failed to respond. They were all too afraid to use the ‘g-word’, but whatever you want to call it, it should not matter, when 800,000 or so innocent people massacred. * Clinton never apologized for his lack of response (no friends, only interests). Even though there was a lot of tragedy, there were some acts of kindness, and some people, who made me want to believe that good does actually exist. [[Image:Punaisen Ristin lippu.jpg|350px|center]] <u> The red cross were among the few who to decided to help. </u> * What made some people have the courage to stay, putting their own lives at risk to potentially help people that they don’t even know? * Why were some people be prepared to stay behind by themselves and possibly save hundreds, when others just completely turned their backs. * It is amazing that some people could be so brave, putting others needs before their own, while so many others turned their back stating that there was nothing they could do. It helps balance humanity, if there were only people who killed and no one who was willing to reach out and lend a hand, what would this be saying about humanity? === Further Information === Smith, D. (1998) The psychocultural roots of genocide: Legitimacy and Crisis in Rwanda. ''American Psychologist,53,'' 743-753. = Week 5 – Prejudice = == Prejudice and Intergroup relations (Baumeister & Bushmen Ch 12) == [[Image:KKK_Burn_resubmit.JPG|200px|center]] The [[w:KKK|KKK]] (Klu Klux Klan) is a right-wing fraternal organization in the United States that advocates white supremacy to the exclusion of other 'races'. [[w:Prejudice|Prejudice]] is a negative attitude of feeling held towards another individual based solely on the individual (perceived) membership to a certain group. Prejudice like other attitudes is a mixture of beliefs (stereotypes) and emotions (such as, hostility, envy or fear), and the predisposition to action (discrimination). Where as prejudice is a negative attitude, discrimination is a negative behaviour. Even though prejudice seems very irrational, it is something that occurs all the time, in areas of life. I see prejudice as arising partially from human’s desire to categorize everything. If we come across something we have not encountered before, we have no category to put this person or thing into, we may become confused and don’t know how to act towards it. Social categorization cannot only often be very ill informed, but also I see it as the root of most prejudice. We are all humans; most of us like to believe that although we may belong to some groups, they do not determine our fate. Even though we like to believe this about ourselves we often fall into the trap of the outgroup homogeneity bias – where we believe all members of the ‘other’ or outgroup are similar to one another, but we as the ingroup are not as similar to each other. [[Image:No sexism racism homophobia.jpg|300px|center]] Where does prejudice come from? As mentioned above prejudice can result from social categorisations that particularly emphasis the difference between groups. In that respect prejudice is a result of culture. We are taught that different is bad, and that other groups are different. There are many aspects of this view that I agree with, and some not so much. Much information that we know about other groups or stereotypes are learned through socialization, they are not something innate. However I also believe that humans are designed to form categories and stereotypes, but it is the content of these stereotypes or the emotions felt towards other groups that is learned through society. === Social Roots of Prejudice === Prejudice also arises through inequalities, social divisions, and emotional scapegoating. Prejudice rationalizes inequalities. Us Vs. Them - thanks to our need to belong we are a group-bound species. We define who we are in part through our groups. We associate ourselves with certain groups and contrast ourselves with others. This not only tell us who you are, but also who you are not. The desire to distinguish ourselves from others predisposes us to prejudice against strangers. Prejudice stems not only from the division of society but also from emotions such as anger and fear. When something goes wrong, we want to find someone whom we can blame. Like the worlds outlast at many innocent Arabs, following the 9/11 attacks. ''Common targets of prejudice '' *Arabs- Prejudice and discrimination increased in U.S after September 11 2001. People who are overweight- another highly visible characteristic of individuals subject to prejudicial attitudes is obesity. Unlike racist and sexist attitudes, many people will openly admit and even act upon their negative attitudes towards obese people. *Stigma by association- rejection of those who associate with stigmatized others. *Homosexuals- Although one’s sexual orientation is not as readily visible as ones race or gender or weight, anti-gay prejudices are often quite strong, leading sometimes to violence and discrimination. *Homophobia: excessive fear of homosexuals or homosexual behaviour. Both men and women are intolerant of homosexuals in their own gender. Perhaps this is because of a fear of being the target of a sexual advance from homosexuals. May fear a positive response to homosexual advances. * Other potential targets of prejudice include elderly, jews, and people with stigmas. *Stigmas: characteristics of individuals that are considered socially unacceptable (e.g. being overweight, mentally ill, sick, poor, or physically scarred). == Tutorial 3 – Prejudice – ‘The Australian Eye’ (Jane Elliot) == [[Image:Blue-green eyes.jpg|200px|left]] Building on from the lecture on Prejudice, we watch the video ‘The Australian Eye’ By Jane Elliot. This video was about an experiment conducted by Jane Elliot. Although there were many interesting things about this experiment, and many individuals appeared to learn a lot, I disagreed with the ethics of this experiment and the way it was conducted. [[Image:Menschliches auge.jpg|200px|right]] Jane Elliot played (I presume she was acting) the role of an authority figure, and she made her ‘power’ known early on. The purpose of this experiment was to demonstrate the power of ''prejudice (the negative feeling towards an individual based solely on their membership to a particular group)''. This was not done through promoting acceptance and cohesion, but through demonstrating what it is like to be treated differently on the basis of a characteristic that you have no control over. I personally didn’t agree with the way that she went about doing this. I think there are other ways to promote acceptance and demonstrate the destructive nature of prejudice, other than giving one group (the people with brown eyes- the majority of which in this case were of Aboriginal descent) power to ''discriminate (the unequal treatment of different people based on the groups to which they belong'') against another group (the people with blue eyes – the majority of which were of European decent). I think one thing that was really demonstrated, was the power of an authority figure to demand obedience. Even though you got the feeling that not everyone agreed with what was happening, they did not want to be seen as going against the group, and thus did not say anything. As far as I am aware the majority of these people didn’t know one another before this experiment. If this is the case it supported the'' minimal group effect – that is that people show favoritism towards ingroup members even when the groups are randomly determined''. However, the minimal group effect may not be so relevant as; these groups weren’t so arbitrary for the reason that, they had all in one-way or another experienced discrimination from a member or members of the other group (not necessarily the individuals who made up the group in the experiment, but in a larger sense to incorporate all White Australians). One thing that was really striking to me about this experiment was the fact that Jane Elliot presumed that all the Blue eyed members had had things fairly easily and had not experienced discrimination like the brown-eyed members had. This reflects the wrong assumption of the'' outgroup homogeneity bias – this is the assumption that outgroup members are more similar to one another than ingroup members are to one another.'' Another issue with this experiment was that as much as it could be discouraging prejudice it could be encouraging it. This exercise focused on the negative aspects of the group that confirmed what the other group already believed about the Blue-eye group. As humans we tend to accept information that aligns with our beliefs and disagree with information that doesn’t. The more negative information that is highlighted about a particular group the harder it will be to eliminate these beliefs. When such deep prejudicial beliefs are held, that are continually confirmed, the more strained intergroup relations will be. == Key terms and their applications – == During this tutorial we also discussed various terms relating to prejudice, aggression and prosocial behaviour. I discuss a few that I found particularly interesting- * ''Discontinuity effect: ''this is where groups are more extreme in their views and actions, than individuals. I have often found myself in a group situation or an observer of a group situation, and wonder if the people would act the same if they were not surrounded by a group of people. I think this may play role on group violence, if you imaging two groups ‘going to war’, would their actions be the same if it was just two individuals involved instead of groups? * ''Outgroup homogeneity bias:'' (as outlined above) can be witnessed in such statements as, ‘they’re all the same’ ‘if you’ve seen one you’ve seen them all’. I just question if individuals treated others as individuals whether prejudice would be reduced. * ''Subtypes:'' when we make up a category for individuals who do not fit into our general stereotypes. Instead of re-evaluating the validity of our initial stereotype we just place people in separate categories within that stereotype. We continue doing this until the original stereotype we held virtually holds no meaning, as there are so many subtypes. Two terms that I think help to understand the aggression in Rwanda. * ''Deindividuation:'' a sense of anonymity, when acting in a large group, that makes people more likely to engage in aggression. When people act in groups they are no longer an individual that is accountable for their actions, but a member of a group that can’t be identified. Would the aggression that occurred in Rwanda been as severe if everyone knew that they were accountable for their actions? I think not, anyone is free to comment… * ''Weapons effect:'' the increase in aggression as a result of the mere presence of a weapon. I think this is especially likely to occur when people are already acting aggressively, but may increase the intensity of that aggression. In Rwanda the Hutus were already frustrated, angry and hostile, once the killings started and weapons were readily available, the violence escalated (I know this is a generalization). == Can someone turn around his or her prejudiced views? == Around the time that we were studying prejudice I watched a program that demonstrated that prejudice can be reduce, and I thought I would share my thoughts on it here. This program was about [[w:whitesupramacy|White supremacist]]. He believed that the White race was going to become extinct if we didn’t do anything to get rid of these ‘other people’. If you think to the absolute extreme Deep South white supremacist and times that by 10 that was this guy. Now the amazing thing wasn’t that he held such beliefs (although I really cant understand it), but how he dramatically turned his beliefs around. His ‘break-though’ moment didn’t come when his sons first words were ‘White is good’ or ‘nigger’ or when he was imprisoned for bashing a couple who he believe were Jewish (even though they weren’t), but when he was at the supermarket and his 3 year old son turned around and said ‘look daddy it’s a big Black nigger’. This man was able to turn his beliefs around from such extreme hatred and prejudice to acceptance and compassion. As horrible as it is to witness such hated, it is something that surrounds our everyday existence, it is encouraging to hear a story like his. = Week 10 - Attraction, Exclusion and Close Relationships= == The Need to belong == Based on evolution, humans belonged to groups in order to ensure survival. Being a member of a group allowed members to distribute the workload and increase safety. More recently, however, most people no longer belong to tribes, but they still protect those in their groups and still have a desire to belong in groups. Why do we form and maintain relationships (even bad ones)? Humans have many ‘needs’ and the need to belong is simply one of them. [[Image:Maslow's hierarchy of needs.svg|300px|left]] [[w:maslowshierarchyofneed|Maslow's hierarchy of needs]] proposed that, needs vary significantly in terms of their importance. Some needs must be met before other needs can be addressed. Lower- level needs such as physiological or safety needs must be met before our belonging and love needs can be addressed. We all need to feel that we are needed and accepted by others. As social beings, humans need to feel as though we are rooted in communities, with ties to family and friends. It is this need for belonging that motivates us to develop relationships. [[Image:SanQuentinSP.jpg|200px|right]] This need to belong is so powerful that it can lead people to behave in ways that would otherwise seem bizarre. An example of this is the prisoners at [[w:sanquentin|San Quentin]] who were sentenced to solitary confinement resorted to taking into toilets to have some connection with other humans. However, for most people their need to belong can be satisfied by simpler means, such as acceptance by family and friends. The need to belong is not simply satisfied by human contact, there also has to be an aspect of stability and mutual concern for one another. Having one of these factors without the other will only produce partial satisfaction. I think the clearest example of this is when you start university, although people may surround you, until you form new friendships, you may feel partially isolated. Once we have formed a few close relationships, this need to belong may be satisfied and we don’t feel the need to continuously seek new relationships. Some people may desire more close friends than other, but I think in general most people would prefer 2 to 5 close friendships than a dozen acquaintances, this comes back to the fact that to fulfil your need to belong there has to be some aspect of mutual concern. So in essence humans form relationships, whether they are friendships or intimate relationships to fulfil this need to belong. == Attraction == [[Image:Laughing couple.jpg|250px|right]] What causes two people to be attracted to each other, either as friends or as romantic partners? When two people meet they may come to like each other, or they may not, what factors influence this? I general I think that we seek out friends and partners who are similar to ourselves in some way, either in interest, desires, goals, hobbies etc. But apparently it is not only these factors that guide who we seek out, the matching hypothesis states that people tend to pair up with other who are equally attractive. I find this fascinating. This could be rooted in the idea that men seek partner that are like their mothers and women seek partners that are like their fathers, I’m not sure how much I agree with this idea, but it could explain why people seek out partners who are equally attractive. It is not that surprising that people like people who are similar to themselves, but one thing that I did find surprising was that people also come to like people based on familiarity [[w:exposureeffect|mere exposure effect]]. I can see that this is targeted in marketing. Often using the idea that the more that you see something, the more you come to like it, and when faced with a range of options you will choose the one that is most familiar to you. So I guess in a way it could also work on people. But I disagree that just through exposure you will come to like someone; I think that factors such as lack of similarity or a bad experience with someone could potentially undermine the notion of mere exposure. == Rejection == I think we can all relate to the fact that at one point in our lives we have been rejected. I think this really highlights one of the key themes of this textbook, that bad is stronger than good. When someone is rejected it can take a lot longer for that person to come to feel accepted again. I don’t agree with Freud on the notion that humans are innately evil, but I do think that rejection and exclusions are two instances that show the ugly side of humanity. We are so eager to reject people sometimes for no reason other than that of what a person is wearing, without considering what the possible disastrous effects may be on the other person (including rejection sensitivity). I could go on indefinitely about the long-term effects of rejection on individuals, but I find it a little depressing, so I think I will move on to something a little more positive, love... There is no such thing as a person who doesn't want any friends [http://www.apa.org/monitor/nov02/perils.html] ==Love== [[Image:Crystal Clear app package favourite.png|250|right]] [[w:Love|Love]] no one really knows what it is, or how to get it, but it is something that it so desperately sought after in our society. Those three words, ‘I Love You’, are used in a variety of situations, and very rarely mean the same thing to everyone. I find love an intriguing and fascinating topic, yet when it came time to write about it I was lost with what to say. So I turned to my good friend Wikipedia, which told me that ‘love’ relates to any number of emotions relating to strong affection. Then I started thinking about it, are strong emotions of affection the key? People say that they love a food, or a sport, but do these kinds of love really consist of strong feelings of affection. May be it would be more accurate to say I like pears or I like tennis? However, this still does not satisfy my curiosity. I love my parents, and that definitely relates to strong feelings of affection, but it does not mean the same thing as when I say I love my partner. The range of situations in which love it used, combined with the intricacy of the feelings involved, makes love not only intriguing, but also difficult to define. ''Romantic Love -'' Hatfield distinguishes two types of love: temporary passionate love and enduring companionate love. Passionate love is an aroused state of intense positive absorption in another, usually present at the beginning of a love relationship. The theory assumes that emotions have two ingredients, physical arousal and cognitive appraisal, and that arousal from any source can enhance one emotion or another, depending on how we interpret and label that emotion. ''If the inevitable odds against eternal passionate love in a relationship were better understood, more people might choose to be satisfied with the quieter feeling of satisfaction and contentment'' (Myer, 2007). As love matures it becomes a steadier companionate love - a deep, affection attachment we feel for those with whom are lives are intertwined. One key to an enduring relationship is ''equity'': receiving in proportion to what to put in. Mutually sharing the self and possessions, giving and getting emotional support, promoting and caring about each other's welfare, are the core to every type of loving relationship. Another important ingredient is ''self-disclosure'' (revealing intimate aspects of oneself to others, our likes and dislikes, our dreams and worries, our proud and shameful moments). As one person reveals a little, the other reciprocates, the first then reveals more, and so on, as friends or lovers move to deeper intimacy. ==Tutorial 4 - Cross-cultural training == === Culture Shock === [[w:cultureshock|Culture shock]] refers to the feelings of anxiety that people experience when they find themselves in an unknown culture. This is definitely something that I can relate to first hand, from my travels through South East Asia. [[Image:A plate of cooked river snails.jpg|250px|right]] I remember my first trip to Asia, I arrived in the Bangkok airport, I walked outside, and my immediate thoughts was what the hell am I doing here? (it was hot and humid, and on top of that it was smelly and covered in smog). Coming from a reasonably relaxed country, to a city that is crazy, I immediately felt like a fish out of water. However, my regret for being there did not last very long. Immediately I was fascinated by the culture and hospitality of the people who took us into their homes, and shared with us their culture, and again unfortunately this honeymoon phase did not last either. I soon became aware of the enormous child sex trade industry, with children as young as 9 or 10 being sold so their family can afford to eat. The contrast from where I came from and where I was, had never been so apparent, and unlike the first two phases this one stuck with me for much longer, as I did not have time to enter the adjustment phase. [[Image:Khmerfood5.JPG|150px|left]] When I returned to Asia a year later all that had happened the previous time was forgotten, and I started back at the honeymoon phase. However, this time I was pushed further out of my comfort zone, and although there was still the initial honeymoon period, this again soon ended. During this visit I saw things that I never believed possibly (both amazing and utterly tragic). But fortunately this time I believe I was about to come to terms with the differences between cultures,and move into the adjustment phase. I was able to accept a certain way of doing things, even if it was not what I was accustomed to. ==Information/References== Hendrick,C., & Hendrick, S. (1986). A theory and method of love. ''Journal of personality and Social Psychology''. 50, 392-402. Myer, D. (2007). ''Psychology''. New York: Worth Publishers. Sternberg, R. (1986). A triangular theory of love. ''Psychological Review'', 93, 119-135. White, G. L., Fishbeinm S., & Rutsein, J. (1981). Passionate love and the misattribution of arousal. ''Journal of Personality and Social Psycholog''y, 41, 56-62. = Week 11 Groups = == Why do we form group? == [[Image:Fsr alw foto.jpg|250px|right]] <u>We are members of many groups.</u> From the evolution point of view we formed groups as to provide protection from wild animals, and even though we may not need protection from animals humans still have the tendency to form and maintain groups. Why do we form groups? [[Image:STA 21 Graduation, U.S. Navy · DN-SD-06-10072.JPG|250px|left]] A group is a collection of at least two people who are doing or being something together – Baumeister & Bushman. Although mostly agree with this definition, I find it hard to think of a group of only two people, i just have an idea in my head that a group must be more than 2. So i guess i will compromise, and accept that a group may consist of 3 or more people. People form groups for many different reasons, one of which may be to fulfil our need to belong. However, I think this may be a little to simple, if forming groups was purely to belong, then why do we naturally classify other people into groups? I think that the formation of groups stems form our tendency to being [[w:cognitivemiser|cognitive misers]], or our tendency to classify objects in order to make sense of our world. The Self-Categorization Theory assumes that individuals divide and understand the social world through self-categorisations. Individuals see themselves as similar to some, and distinct from others and this is the basis on which groups are formed (Turner et al., 1987). [[Image:1970sfamily3.jpg|250px|right]] According to the [[w:soaiclidentitytheory|Social Identity Theory]], humans form and maintain groups in order to achieve a sense of social identity. The Social Identity Theory suggests that when individuals perceive themselves as sharing a common group membership they become highly identified with that group and become more connected with the welfare of the group. Individuals join groups that they perceive as similar (ingroup) and distinguish from those who are seen as different (outgroup) (Tajfel & Turner, 2004). These group members seek to find positive ingroup distinctiveness, differentiating their group from other groups. This distinction leads to not only ingroup favouritism; the individual treats their group more favourably than other groups, but also deindividuation; the loss of self-awareness and of individual accountability in a group. The more an individual identifies with his or her own group, the greater this distinction, and the more the individual’s identity depends on the welfare of the group. Although groups generally serve a positive role in society, these processes of ingroup favouritism, and deindividuation can produce terrible results. == Group Action == Groups can have both positive and negative effects on individuals. [[w:socialfacilitation|Social facilitation]] demonstrates how the presence of others can increase one’s performance on a well-learnt task. The key here is that the person has to already be confident in performing the task, otherwise the presence of others can decrease performance. In some situations, our attention is divided between the task at hand and the presence of others (distraction-conflict theory). [[Image:Tug-of-war.jpg|250px|center]] ''We put in less effort hoping that other people will pick up the 'slack''' On the flip side of social facilitation, are negative effect of groups is [[w:socialloafing|social-loafing]], this is when people reduce their individual effort when working in a group. This is one of the reasons why I am no to keen on group assignments. When working as part of a group, generally everyone receives the same mark no matter what their individual input was, when individuals are not identifiable for their work they may put in less effort. When individuals being to suspect that others are putting in less effort, they also put in less effort (this is know as the bad apple effect). The effects of social-loafing can be detrimental when you are trying to achieve something, however, though making individual efforts identifiable, and increasing the group cohesiveness it may be possible to reduce the effects of social-loafing. == How Groups Think?== When individuals come together as a group, the pooling of information can have great benefits for our society, but at times groups can be incredibly stupid. We are often encouraged to brainstorm when we are working in groups, thinking that it will produce a higher standard of ideas. However, it has been demonstrated that not only it the quantity of work in group lower, but the quality of this work was also lower as well. In order to perform effectively as a group, individuals must work separately before pooling information, and in these situations pooled group judgments may be better than single judgments. [[w:groupthink|Groupthink]] is the tendency for group members to think alike. When groups are working together they run the risk of conforming to the group, and not considering alternative plans of action. Groupthink is most likely to occur when the group is cohesive, there is a strong leader, groups are isolated from others, and the group perceives itself as superior to others. Even though individual members may have differing ideas or opinions, because of the pressure to conform to, and be accepted by, the group, they don’t express their ideas, enhancing the chances that groupthink will influence their decisions. == Tutorial 5 - Australia Zeitgeist == [[w:soicalcapital|Social capital]] is a loosely defined concept; I take it to mean both the quality of human connection as well as community involvement. One concept that can be reflected in social capital is global warming, how much are people willing to put in to achieve a common goal? [[w:socialisolation|Social Disengagement]]: I take this to be the opposite of social capital; it is isolation from society, exclusion and disconnection. The question can be asked, what is happening to society, when people are disengaging rather than engaging? [[w:Zeitgeist|Zeitgeist]]: is used to describe ‘the spirit of the age and its society’, it is used to describe the intellectual, cultural, ethical, and political climate of an era. In the tutorials this week we also listened to a talk by Hugh Mackay. Hugh Mackay talked about how we are going through a time of social change, this was demonstrated through noting some startling statistics: • Around 45% of marriages end in divorce, compared to less than 10% 25 years ago. • 500,000 children regularly move from the home of the custodial parent to the non-custodial parent. • 30 years ago 75% of people were married by the age of 30, now less than 30% will be married by the age of 30. • More than 50% of household contain one or two people. [[Image:CellPhone.png|100px|left]] [[Image:IPod_5G,_nano_2G,_shuffle_2G.jpg|100px|center]] Mackay talked about how these factors are leading to disengaging from the social scene. I think that this disengagement can also be witnessed in the marked increase in media use. The ipod, and mobile phones are prime examples of factors that contribute to social disengagement.Instead of connecting with our social world, we are caught up with what's going on with ourselves. But as I am aware that most of you all listened to this talk as well, I’m going to refrain from discussing it any more. What I do want to discuss, is what I see as the Australian Zeitgeist, and the Zeitgeist of the times in general. [[Image:Barack Obama Fold.jpg|150px|left]] [[Image:Hillary Rodham Clinton.jpg|150px|right]] Zeitgeist is used to describe the spirit or flavour of the times. I think that Australian and the world in general are going through a time of dramatic change. One only has to look at the American election to see such a change, when else is in American history would you see an African-American and a woman contending leadership, and later running for president. And although all have not welcomed him, I definitely think it is a huge step towards equality. Maybe ours is a time of transition. == Information/References== Watson, G.B. (1928). Do groups think more effectively than individuals? The ''Journal of Abnormal and Social Psychology,23'', 328-336. Tajfel, H. (1978). Social categorization, social identity and social comparison. In H. Tajfel (Ed.) ''Differentiation between social groups''. (pp. 61-77). London: Academic Press. Tajfel, H., Billig, M. G., Bundy, R. P., & Flament, C. (1971). Social categorization and intergroup behaviour. E''uropean Journal of Social Psychology'', 1, 149-178. Tajfel, H. & Turner, J. (2004) An integrative theory of intergroup conflict. In M. J. Hatch & M. Schults (Eds.) ''Organizational Identity''. (pp. 56-66). London: Oxford University Press. Turner, J. C., Hoggs, M. A., Oaks, P. J., Reicher, S. D., & Wetherell, M. S. (1987). ''Rediscovering the social group: A self-categorization theory''. Oxford, UK: Basil Blackwell Ltd. = Week 12 Prosocial Behaviour = [[Prosocial behaviour]] is doing something that is good for other people or for society as a whole. It can be seen in voluntary acts that are intended to benefit others. One purpose of prosocial behaviour is to be accepted. At times prosocial behaviour can come at a cost to the individual, so why do we engage in this behaviour, as mentioned above one reason is to gain acceptance and social status, others could include; self-interest, guilt, altruism, convenience etc. When we engage in certain prosocial behaviour we have an expectation that our good deeds will be both reciprocated and fair. Reciprocation can be both direct and indirect; we may help someone with the anticipation that they can help us later, or we may help someone and later when we need help someone else may help us (this can be seen in the notion of karma – what goes around comes around). We also act on the notion of fairness. One thing that intrigued me about fairness was the fact that individual’s feel guilty for having lived when others may have died (such as in a natural disaster) – also known as ''survivor guilt''. ==Why do people help?== Humans are unique in the way that they help strangers. According to the evolution we help people who have our genes (Kin selection). Yet this does not explain why we help people we do not know. Auguste Compte described two types of helping based on different motives; Egotistic helping, and altruistic helping. In egotistic helping the helper looks for something in return for their helping. In contrast in altruistic helping the helper seeks to help another without expecting anything in return, and is motivated by empathy. === Does altruistic behaviour truly exist? === [[Image:Todd Huffman - Lattice (by).jpg|250px|left]] ''Although people assume that altruistic helping exists, it might not.'' Baumeister and Bushman [[Image:Oskar shindler factory krakow.jpg|200px|right]] Oskar Shindler's factory where he saved 1200 Jews from concentration camps [[Altruism]] is the idea of a selfless act of providing aid to others than has no benefits for oneself, this often occurs at the expense to one’s self. As humans are selfish in nature there is considerable question as to whether altruism actually exists. ''The empathy-altruism hypothesis'' states that empathy motivates people to reduce other people’s distress, such as by helping or comforting them. When empathy is low, people can reduce their own stress either by helping the person in need or by escaping the situation so they don’t have to see the person suffer any longer. If empathy is high, however, then simply shutting your eyes or leaving the situation wont work because the other person is still suffering, in that case the only solution is to help the victim feel better. We decrease our distress by decreasing others distress. What I understood altruism to be was helping others with NO benefit to us. This hypothesis states that we help others to reduce our own discomfort, so can altruism truly exist if we are motivated to help others for personal gain (even if it is just to stop us from feeling bad)? Even in the smallest act that could be classified as altruistic, we gain some benefit, for someone just to say ‘thank you’ isn’t that a benefit in itself if it makes us feel good. The Negative State Relief Theory is no better at explaining altruistic behaviour, as it states that we are motivated to help others to relieve our own distress. However such acts as that of Carl Wilkins in Rwanda that make you believe that altruism might just exist. Even after every other American had left Kigali, he alone stayed and contested the 800,000-person genocide. Despite death threats he stayed, saving lives time and time again. Such selfless goodness exemplifies altruism. ===Information=== [http://www.in-mind.org/issue-6/altruism-myth-or-reality.html Altruism, Myth or Reality?] == Bystander Helping == [[Image:AV89-26-14 600d.jpg|200px|right]] There have been many situations in which bystanders can help, and yet more often than not they don’t. Darley and Latane attributed this to an important situational factor, the presence of others. After staging emergencies under various conditions Darley and Latane came up with a decision making process for bystander intervention. We will help only if the situation first enables us to first notice in incident, then interpret it as an emergency, and then finally assume responsibility for helping. At each step the presence of others turns people away from the path that leads to helping. When strangers are in a group on the street, they are more likely than individuals to focus on what they themselves are doing, and where they are going. If they notice an unusual situation, they may infer from other people's reactions that it is not an emergency, i.e. that person must be lying in that doorway because they are drunk. But in other situations, when the emergency is not ambiguous people still fail to help. They may notice the event, perceive it as an emergency yet fail to take any responsibility (this is also know as ''diffusion of responsibility'', where we think that others will take responsibility). [[w: Bystandereffect|The bystander effect]] is the tendency for any given bystander to be less likely to give aid if other bystanders are present. It has been suggested that helping decreases by 50% when we are in the presence of others. The best odds of our helping someone occur when: * The victim appears to need and deserve help. * The victim is in some way similar to us * We have just observed someone else being helpful * We are not in a hurry * We are in a small town, or rural areas * We are feeling guilty * We are focused on others and not preoccupied. * We are in a good mood. It has been suggested that happy people are helpful people. No matter why people feel good, they become more generous and more eager to help. Bystander Calculus Model- has also been proposed to explain why bystanders help in some situations and not others. Bystanders calculate the (perceived) costs and benefits of providing help. 3 stages # ''Physiological arousal'': witnessing an emergency – physiological arousal – greater chance of helping. # ''Labelling the arousal'': is arousal labelled as a personal distress or empathetic concern? – usually labelled as personal distress. # ''Evaluating the consequence''s: weigh up costs of helping, choose the action that reduces personal distress to the lowest cost. *Costs of helping: Time and effort – less likely to help if it involves greater time and effort *Costs of not helping: empathy costs (bystander experiences distress) *Personal costs (bystander experiences blame or guilt). *The greater the similarity to the victim the more likely the bystander is to help. <u>What are you going to do next time you see someone who may need help? </u> === Information=== [http://greatergood.berkeley.edu/greatergood/archive/2006fallwinter/ Why people ignore people in need.] [http://www.abcnews.go.com/Health/story?id=5015458&page=1 News: What Would You do in a Hit and Run?] [http://www.thestar.com/News/article/196946 Act Now and Defy the Bystander Effect] = Week 14 Environmental Psychology = [[Image:Auwald leipzig.jpg|250px|left]] [[w:EnvironmentalPsychology|Environmental psychology]] studies the interactions and relations between humans and their environments. Environmental psychology looks at this in two ways; how the physical environment affects human thought, feelings, and behaviours as well as how human actions affect the environment. Ecological issues of people’s relationship to their environment, both natural and human-made, have assumed crucial importance to our quality of life, and even to the survival of humanity. [[Image:Global Warming Predictions Map.jpg|250px|right]] The environment can have both positive and negative impact on human. Crowing refers to the negative subjective feeling that there are too many people in a particular space, whereas density refers to the number of people in a given space. One example of this could be getting in to an elevator with 12 other people, although this may not be referred to as densely populated, people may have the negative experience that there are too many people in there. [[Image:07-Chevrolet-Tahoe.jpg|200px|center]] <u>Humans are having a deadly affect on the environment</u> One thing that really stood out at me in this weeks reading, was the trend in environmental concern. This reading was out of what I presume was an American textbook, it stated that in 1990 71% of Americans were concerned that there was not enough being done about the environment. All I have to say to that is WHAT (any of you who have been to the US may understand)? Okay let me explain, the US is the land of big, they want it big and they want it no matter what the cost to the environment (SUVs are a prime example of this mentality). I have never seen as much packaging and waste than I did in the US. The US has the most per capita waste. [[Image:Vuilnis bij Essent Milieu.jpg|150px|right]] However, we shouldn’t feel so good just yet, Australia is number 2. This is not to say that all states in the US are totally inactive, some states (such as California and Vermont) have made progress towards becoming environmentally friendly.But i don't think they contain the entire 70%. It just got me thinking about how attitudes really aren't a very good predictor of behaviour. ===Information=== * [http://www.vexen.co.uk/USA/pollution.html Oil, Pollution and the Kyoto Protocol] * [http://www.globalissues.org/article/233/climate-change-and-global-warming-introduction Climate Change and Global Warming] = Wrap up = Wow, is it really that time of the year again? Boy how time flies when your having "fun". Well what can i say i got out of this course? I think it has really encouraged me to look at the things i do and think and possibly re-ass them. One example would be the fundamental attribution error, i know that i can fall in to its trap more often than not. In general i think the course has taught me a great deal, not only about social psychology but also about myself. I think that the e-portfolio gave us a chance to really understand, and reflect on, the course material. What feedback would i have for this course? In general i think that this course was great, the only thing i would comment on is the timing of the assessments, i would recommend that possibly the essay should be due a bit earlier (possibly wk 12 or so). I just think it is a little rough having all our assessment for the one unit within a space of 12 days. Okay so i'm signing off hopefully for the last time, not more battles with wikiversity!!! Take care, and happy marking James. bxwuomqoe858a7kv7mvu2x6hcqwu8b1 0 64081 2812056 2777032 2026-05-29T19:33:44Z ~2026-32231-75 3085699 /* Body parts */ 2812056 wikitext text/x-wiki == Before we start: Back-pocket notes== Note: This is not all of Spanish 1! As you start learning a different language, remember to concentrate primarily on four (4) aspects: *Reading *Writing *Listening *Speaking ===Unfamiliar characters=== In Spanish, there are a few characters that, in general terms, are not used in the English language. * '''Opening marks'''. Written questions and expressions of exclamation begin with a '''¿''' or a '''¡''' (characters inverted from the ones used in English); the purpose of these is to establish the beginning of the question or exclamation and, thus, alert the reader to prepare for the according intonation. The opening marks are paired with the (closing) ending marks. * '''Accents'''. The vowels are the only letters that may present an accent: á é í ó ú Notice that only one type of accent is used: "'''´'''". Good to know: interrogative pronouns are always written with an accent: * what? ''¿qué?'' * who? ''¿quién?'' * where? ''¿dónde?'' * when? ''¿cuándo?'', * how? ''¿cómo?'' * why? ''¿por qué?'' Also, later you'll learn about [[w:Diaeresis (diacritic)|diaeresis]], which may be presented over the letter ''u'' as: '''ü''' In Spanish, ''ü'' is used in a limited number of words and in conjunction with the letter ''g'', as in: ''pingüino'' (penguin). Additionally, an important known and characteristic letter of the Spanish language is: '''ñ''' called ''eñe''. Though it looks like an ''n'', it is a different letter (its sound is similar to that of the '''''gn''''' in lasa''gn''a; thus, lasaña in Spanish). The character " '''~''' " (tilde) is formally called ''virgulilla de la eñe'', and is not used on any other letter. ===Sounds=== As in general terms, every letter of the [[Spanish alphabet|alphabet]] has its own sound, so if you learn the sounds of the letters, you will soon be able to read the words. Note that there are some particularities about the letters ''c'', ''s'', ''x'' and ''z''; the letters ''h'' and ''ch''; as well as ''y'' and ''ll''. '''H''' or '''h''' is generally silent; if it follows a '''C''' or '''c''', the digraph sounds like the initial sound in '''''ch'''eese'' [t͡ʃ]. Regarding pronunciation, the sound of the double ''r'' " '''rr''' " [r], is stronger than the sound of the single ''r'' [ɾ]. No word in Spanish starts with double-r (Rr). ---- Good luck. ''Buena suerte''. Don't be nervous, and don't be afraid of letting the ''Trigeminal nerve'' do its thing! Cheers! ¡Comencemos! == Chapter 1 (Intro to Greetings and Phrases & sounds) == Hello and welcome to this chapter of Spanish 1: Intro to Greetings and Phrases & Sounds. Please note that this is unfinished. You can check out the Greetings and Phrases down below. Thank you! ==Greetings and Phrases== *'''Buenos días.''' - Good morning. *'''Buenas noches.''' - Good night. *'''Buenas tardes.''' - Good afternoon./Good evening. *'''Hola.''' - Hello. *'''¿Cómo te llamas?''' - What is your name? *'''Me llamo...''' - My name is... *'''Encantado(a).''' - Delighted. *'''Igualmente.''' - Likewise. / Same here. *'''Mucho gusto.''' - Pleased to meet you. *'''Señor (Sr.)''' - sir, Mr. *'''Señora (Sra.)''' - madam, Mrs. *'''Señorita (Srta.)''' - miss, Ms. *'''¿Cómo estás?''' - How are you? (familiar) *'''¿Cómo está usted?''' - How are you? (formal) *'''¿Qué?''' - What? *'''¿Cómo le va?''' - How do you do? (formal) *'''¿Cómo te va?''' - How's it going? (informal) *'''¿Qué tal?''' - How are you? *'''¿Y tú?''' - And you? (familiar) *'''¿Y usted?''' - And you? (formal) *'''bien''' - well *'''más o menos''' - more or less (so, so) *'''nada''' - nothing *'''regular''' - regular, okay *'''Lo siento''' - I am sorry *'''Gracias.''' - Thank you. *'''De nada.''' - You're welcome. / Not at all. *'''A su servicio.''' - At your service. *'''Adiós.''' - Good-bye. *'''Chao.''' - Bye. *'''Hasta luego.''' - See you later. *'''Hasta mañana.''' - See you tomorrow. *'''Hasta pronto.''' - See you soon. *'''Nos vemos.''' - See you. ====Note==== Remember: * in most Spanish-speaking countries, "ll" (double "L") is pronounced as "j" [d͡ʒ~ʒ]; however the sound becomes "y" [ʎ~j] in Mexico and "sh" [ʃ] in Argentina... * in Latin America, the "Z" is pronounced as "s" [s]; in Spain, it is sounded as "th" [θ]. * J is used for the sound like in Ba'''ch''' [x~ɣ~h], g is pronounced hard [ɡ] before a, o and u, but the same as j [x~ɣ~h] before e and i. (To indicate that it is pronounced hard before these, gu- is written. Gü- simply is [ɡʷ]. * letters in parentheses (a) are feminine, while those without (o) are masculine. ====How To Use Ud and Uds (Tú/Usted [Ud.] - Vosotros/Ustedes[Uds.])==== In Castilian Spanish (Spain), there are several ways to say "you." We must differentiate between familiar/formal and singular/plural forms. ''Tú'' (singular) is used when talking to someone you know, such as family, friends, or pets. ''Usted'' (abbreviated Ud.) is used in formal events, such as talking to a teacher, someone who you don't know well, or a person who has a high title, such as a police officer, etc. The plural of tú is vosotros/as, the feminine form is used only for a wholly female group. The plural of ''usted'' is ''ustedes''. In Latinoamerica ''usted'' can also be used for a close friend, and ''tú'' is never used as a polite form. ''Tú'' may be lost altogether and another pronoun ''vós'' may be used for certain contexts. {| class=wikitable ! !Singular !Plural |- !Familiar |align=center|tú |align=center|vosotros/as (Spain)<br>ustedes (Latin America) <small>[http://lema.rae.es/dpd/srv/search?id=yhtVtQ5pQD6ONJL2Gd RAE]</small> |- !Formal |align=center|usted |align=center|ustedes |} === FANBOYS in Spanish (FANBOYS en Español) === '''Para''' - For '''Y -''' And '''Ni -''' Nor '''Pero''' '''-''' But '''O -''' Or '''Sin embargo -''' Yet '''Así que -''' So ===Time (tiempo)=== :''Please read [[b:Spanish/Lessons/¿Qué hora es?|¿Qué hora es?]]'' *'''¿Qué hora es?''' - What time is it? *'''¿A qué hora ....?''' - At what time ....? *'''Es la una.''' - It's one o' clock. *'''Son las dos.''' - It's two o' clock. *'''Son las tres y cuarto.''' - It's a quarter past three. *'''Son las cuatro y media.''' - It's half past four. *'''Son las siete menos cuarto.''' - It's a quarter to seven. ===Whole numbers=== *'''cero''' - zero (0) *'''uno''' - one (1) *'''dos''' - two (2) *'''tres''' - three (3) *'''cuatro''' - four (4) *'''cinco''' - five (5) *'''seis''' - six (6) *'''siete''' - seven (7) *'''ocho''' - eight (8) *'''nueve''' - nine (9) *'''diez''' - ten (10) *'''once''' - eleven (11) *'''doce''' - twelve (12) *'''trece''' - thirteen (13) *'''catorce''' - fourteen (14) *'''quince''' - fifteen (15) *'''dieciséis''' - sixteen (16) *'''diecisiete''' - seventeen (17) *'''dieciocho''' - eighteen (18) *'''diecinueve''' - nineteen (19) *'''veinte''' - twenty (20) *'''veintiuno''' - twenty-one (21) *'''veintidós''' - twenty-two (22) *'''veintitrés''' - twenty-three (23) *'''veinticuatro''' - twenty-four (24) *'''veinticinco''' - twenty-five (25) *'''veintiséis''' - twenty-six (26) *'''veintisiete''' - twenty-seven (27) *'''veintiocho''' - twenty-eight (28) *'''veintinueve''' - twenty-nine (29) *'''treinta''' - thirty (30) Note: '''After 30''', numbers are named using the conjunction "y" (''and''), as to say: "thirty-and-one", "thirty-and-two", etcetera... "forty-and-eight", forty-and-nine", and so forth.... *'''treinta y uno''' - thirty-one (31) *'''cuarenta''' - forty (40) *'''cincuenta''' - fifty (50) *'''sesenta''' - sixty (60) *'''setenta''' - seventy (70) *'''ochenta''' - eighty (80) *'''noventa''' - ninety (90) *'''cien''' - one hundred (100) ===Body parts === *(La) '''cabeza''' - (The) head *(La) '''oreja''' - (The) ear (outside) *(El) '''oído''' - (The) ear (inside) *(El) '''ojo''' - (The) eye *(La) '''nariz''' - (The) nose *(La) '''boca''' - (The) mouth *(La) '''lengua''' - (The) tongue *(El) '''cuello''' - (The) neck *(El) '''brazo''' - (The) arm *(El) '''pecho''' - (The) chest *(La) '''mano''' - (The) hand <<--- Notice that though the noun in Spanish ends with ''''''o'''''' the article used is '''la'''. *(El) '''dedo''' - (The) finger / toe *(El) '''estómago''' - (The) stomach *(La) '''pierna''' - (The) leg *(El) '''muslo''' - (The) thigh *(El) '''pie''' - (The) foot *(El) '''pelo''' - body hair <<--- Colloquially, "pelo" may mean head hair as well *(El) '''cabello''' - head hair *(La) '''piel''' - (The) skin *(El) '''músculo''' - (The) muscle *(El) '''corazón''' - (The) heart *(La) '''espalda''' - (The) back *(La) '''rodilla''' - (The) knee *(El) '''codo''' - (The) elbow *(El) '''hombro''' - (The) shoulder *(El) Pene - (The) penis *(El) Vagina - (The) vagina * ===Office items=== *'''bolígrafo/pluma''' - pen *'''carpeta''' - folder *'''cuaderno''' - notebook *'''estudiante''' - student *'''hoja de papel''' - sheet of paper *'''(el) lápiz''' - pencil *'''(los) lápices''' - pencils *'''(el) borrador''' - eraser *'''libro''' - book *'''profesor, profesora''' - professor (male, female) *'''maestro, maestra''' - teacher (male, female) *'''sala de clases (salón de clases) / aula''' - classroom ====Please read==== * The articles '''el''' and '''la''' both mean ''the'' in Spanish. ''El'' is the masculine form, and words that use ''el'' usually end in -o. ''La'' is the feminine form; most words that end in -a and all words that end in -ción are feminine. There are exceptions, as with the noun ''mano'' (hand) the article may not necessarily match the "o" or "a" standard: ''Las manos'' ("The hands). Some nouns can be either gender, such as ''estudiante'', which can be used as ''el estudiante'' or ''la estudiante'' to mean ''the student''. Other examples of nouns contrary to the norms of gender are "''el paraguas''" (the umbrella), "''el día''" (the day) and "''la radio''" (the radio). * Some feminine words, like ''agua'', use the form ''El'' when in singular form, as it is considered hard to say ''la agua'' due to the [[w:Glottal stop|glottal stop (hiatus)]]. * Some Spanish speakers prefer terms that may vary from other regions, for example, in central Mexico ''salón de clases'' is used instead of ''aula'', and ''maestro(a)'' instead of profesor(a). ===Calendar (el calendario)=== *(El) '''año''' - (The) year *(El) '''día''' - (The) day *(El) '''mes''' - (The) month *(La) '''semana''' - (The) week *(El) '''fin de semana''' - (The) weekend <<--- the article ''El'' is directed to ''fin'' (end); so it may be understood as: ''"'''El''' fin de '''la''' semana"'' (''The week's end'' or ''the end of the week''). *'''¿Qué día es hoy?''' - What day is today? *'''¿Cuál es la fecha?''' - What is the date? *'''Es el ... de ...''' - It's the ... of ... *'''Es el primero de ...''' - It's the first of ... *'''hoy''' - today *'''ahora''' – now *'''mañana''' - tomorrow *'''en el futuro''' – in the future *'''ayer''' - yesterday [[File:Orchard calandar 1919.jpg|thumb|200px|Calendar in English.]] [[File:Kalendaroj.jpg|thumb|200px|Calendario en español.]] ==== Days of the week ==== As in English, the roots of the names for each of the days of the week maintain a link to planets and other cosmic bodies, which in instances were ultimately named after deities; Norse mythology for English (Germanic language) and Roman mythology for Spanish (Romance language). So for example, for Monday, we find "moon" (day), which translates to ''luna'', giving us lunes. Another example is Thursday, which was named after Thor, and has as parallel Jupiter; Jupiter (which is spelled the same in Spanish) is for ''jueves''. *'''lunes''' - Monday *'''martes''' - Tuesday *'''miércoles''' - Wednesday *'''jueves''' - Thursday *'''viernes''' - Friday *'''sábado''' - Saturday *'''domingo''' - Sunday Note: Look at the calendars, and notice that the week may start on Monday instead of Sunday ==== Months ==== *'''enero''' - January *'''febrero''' - February *'''marzo''' - March *'''abril''' - April *'''mayo''' - May *'''junio''' - June *'''julio''' - July *'''agosto''' - August *'''septiembre''' - September *'''octubre''' - October *'''noviembre''' - November *'''diciembre''' - December =====Cultural note===== * The names of days in Spanish and many other European languages are always lowercase, except at the beginning of a sentence. [[Image:AztecCalendarMuseoAntropologia.JPG|thumb|200px|Aztec calendar]] * The Aztecs of Ancient Mexico developed a calendar. "The Aztec calendar stone, Mexica sun stone, Stone of the Sun (Spanish: Piedra del Sol), or Stone of the Five Eras, is a large monolithic sculpture that was excavated in the Zócalo, Mexico City's main square, on December 17, 1790.[1] It was discovered whilst Mexico City Cathedral was being repaired.[2] The stone is around 12 feet across and weighs about 24 tons.[3]."(taken from wikipedia) [1] Florescano, Enrico (2006). National Narratives in Mexico. Nancy T. Hancock (trans.), Raul Velasquez (illus.) (English-language edition of Historia de las historias de la nación mexicana, ©2002 [Mexico City:Taurus] ed.). Norman: University of Oklahoma Press. {{ISBN|0-8061-3701-0}}. OCLC 62857841 . [2] a b Aztec Civilization [3] The Aztec Sun Stone ===Other words and phrases; some questions and answers=== *'''¿Cómo se escribe ...?''' - How is ... spelled? *'''Se escribe ...''' - It's spelled ... *'''¿Cómo se dice...?''' - How does one say...? (How do you say...?) *'''Se dice...''' - One says... (You say...) *'''¿Qué quiere decir ...?''' - What does ... mean? *'''Quiere decir ...''' - It means ... *'''¿Cuántos(as)?''' - How many? *'''en''' - in / on *'''hay''' - there is, there are *'''por favor''' - please *'''Punta...''' - Tip (point or vertex) ===[[Spanish alphabet|Alphabet]]=== Showing uppercase and how the letter is proununced or named in Spanish. *'''A''' - a *'''B''' - be *'''C''' - ce *'''D''' - de *'''E''' - e *'''F''' - efe *'''G''' - ge *'''H''' - hache *'''I''' - i *'''J''' - jota *'''K''' - ka *'''L''' - ele *'''M''' - eme *'''N''' - ene *'''Ñ''' - eñe *'''O''' - o *'''P''' - pe *'''Q''' - cu *'''R''' - erre *'''S''' - ese *'''T''' - te *'''U''' - u *'''V''' - uve *'''W''' - uve doble *'''X''' - equis *'''Y''' - i griega/ye *'''Z''' - zeta ===Country focus === [[Image:Flag_of_Mexico.svg|150px|right]] '''Mexico''' (Spanish: México) is a federal constitutional republic in North America. It is bordered on the north by the United States; on the south and west by the North Pacific Ocean; on the southeast by Guatemala, Belize, and the Caribbean Sea; and on the east by the Gulf of Mexico. Mexico is a federation comprising thirty-one states and a federal district, the capital Mexico City (CDMX), whose metropolitan area is one of the world's most populous. [[Image:mx-map.png|thumb|right|200px|Map of Mexico]] Covering almost 2.3 million square kilometers, Mexico is the fifth-largest country in the Americas by total area and the 14th largest independent nation in the world. With an estimated population of 129 million (2019 estimated), it is the 10th most populous country and the most populous Spanish-speaking country in the world. As a regional power and the only Latin American member of the Organization for Economic Co-operation and Development (OECD) since 1994, Mexico is firmly established as an upper middle-income country. Mexico is the 12th largest economy in the world by GDP by purchasing power parity. The economy is strongly linked to those of its North American Free Trade Agreement (NAFTA) partners. Despite being considered an emerging world power, the uneven distribution of income and the increase in insecurity are issues of concern. Mexican culture reflects the complexity of the country's history through the blending of pre-Hispanic civilizations and the culture of Spain, imparted during Spain's 300-year colonization of Mexico. Exogenous cultural elements, mainly from the United States, have been incorporated into Mexican culture. As was the case in most Latin American countries, when Mexico became an independent nation, it had to slowly create a national identity, being an ethnically diverse country in which, for the most part, the only connecting element amongst the newly independent inhabitants was Catholicism. [[Image:Angel de la Independencia Mexico City.jpg|right|thumb|150px|Paseo de la Reforma in Mexico City at night, with the Angel of Independence.]] The Porfirian era (el Porfiriato), in the last quarter of the nineteenth century and the first decade of the twentieth century, was marked by economic progress and peace. After four decades of civil unrest and war, Mexico saw the development of philosophy and the arts, promoted by President Díaz himself. Since that time, though accentuated during the Mexican Revolution, cultural identity had its foundation in the mestizaje, of which the indigenous (i.e. Amerindian) element was the core. In light of the various ethnicities that formed the Mexican people, José Vasconcelos in his publication ''La Raza Cósmica'' (The Cosmic Race) (1925) defined Mexico to be the melting pot of all races (thus extending the definition of the mestizo) not only biologically but culturally as well. This exalting of mestizaje was a revolutionary idea that sharply contrasted with the now discredited idea of a superior, ''pure'' race, still prevalent in Europe at the time. [[Image:CH10.JPG|left|thumb|150px|The famous Guadalajara Cathedral, a symbol of Mexico]] '''Factbox''': Official Language: Spanish Other Languages: 62 other Native languages, English Capital: Mexico City (Ciudad de México, México D.F.) Government: Democracy Area: 1,972,550 sq km (761,606 sq mi) (15th) Population: 111,211,789 (July 2010) (11th) Religion: Christianity 95% (mainly Catholic), Non-religious 2.5%, Buddhism 0.1%, other 2.4% (mainly traditional Mayan or Aztec beliefs) Human Development Index [[w:Human_Development_Index|HDI]] : 0.854 (HIGH, 53rd) [[w:List_of_countries_by_Human_Development_Index|HDI List of Countries]] Independence: September 27, 1821 Currency: Mexican pesos [[Image:Palenque Ruins.jpg|thumbnail|150px|right|The ruins of Palenque, an Ancient Mayan city that was abandoned by the time Spanish explorers landed in Mexico.]] [[Category:Spanish One]] iebz72mp9wkozfnv1xanlv5fik16bpa 2812057 2812056 2026-05-29T19:34:44Z ~2026-32231-75 3085699 /* Body parts */ 2812057 wikitext text/x-wiki == Before we start: Back-pocket notes== Note: This is not all of Spanish 1! As you start learning a different language, remember to concentrate primarily on four (4) aspects: *Reading *Writing *Listening *Speaking ===Unfamiliar characters=== In Spanish, there are a few characters that, in general terms, are not used in the English language. * '''Opening marks'''. Written questions and expressions of exclamation begin with a '''¿''' or a '''¡''' (characters inverted from the ones used in English); the purpose of these is to establish the beginning of the question or exclamation and, thus, alert the reader to prepare for the according intonation. The opening marks are paired with the (closing) ending marks. * '''Accents'''. The vowels are the only letters that may present an accent: á é í ó ú Notice that only one type of accent is used: "'''´'''". Good to know: interrogative pronouns are always written with an accent: * what? ''¿qué?'' * who? ''¿quién?'' * where? ''¿dónde?'' * when? ''¿cuándo?'', * how? ''¿cómo?'' * why? ''¿por qué?'' Also, later you'll learn about [[w:Diaeresis (diacritic)|diaeresis]], which may be presented over the letter ''u'' as: '''ü''' In Spanish, ''ü'' is used in a limited number of words and in conjunction with the letter ''g'', as in: ''pingüino'' (penguin). Additionally, an important known and characteristic letter of the Spanish language is: '''ñ''' called ''eñe''. Though it looks like an ''n'', it is a different letter (its sound is similar to that of the '''''gn''''' in lasa''gn''a; thus, lasaña in Spanish). The character " '''~''' " (tilde) is formally called ''virgulilla de la eñe'', and is not used on any other letter. ===Sounds=== As in general terms, every letter of the [[Spanish alphabet|alphabet]] has its own sound, so if you learn the sounds of the letters, you will soon be able to read the words. Note that there are some particularities about the letters ''c'', ''s'', ''x'' and ''z''; the letters ''h'' and ''ch''; as well as ''y'' and ''ll''. '''H''' or '''h''' is generally silent; if it follows a '''C''' or '''c''', the digraph sounds like the initial sound in '''''ch'''eese'' [t͡ʃ]. Regarding pronunciation, the sound of the double ''r'' " '''rr''' " [r], is stronger than the sound of the single ''r'' [ɾ]. No word in Spanish starts with double-r (Rr). ---- Good luck. ''Buena suerte''. Don't be nervous, and don't be afraid of letting the ''Trigeminal nerve'' do its thing! Cheers! ¡Comencemos! == Chapter 1 (Intro to Greetings and Phrases & sounds) == Hello and welcome to this chapter of Spanish 1: Intro to Greetings and Phrases & Sounds. Please note that this is unfinished. You can check out the Greetings and Phrases down below. Thank you! ==Greetings and Phrases== *'''Buenos días.''' - Good morning. *'''Buenas noches.''' - Good night. *'''Buenas tardes.''' - Good afternoon./Good evening. *'''Hola.''' - Hello. *'''¿Cómo te llamas?''' - What is your name? *'''Me llamo...''' - My name is... *'''Encantado(a).''' - Delighted. *'''Igualmente.''' - Likewise. / Same here. *'''Mucho gusto.''' - Pleased to meet you. *'''Señor (Sr.)''' - sir, Mr. *'''Señora (Sra.)''' - madam, Mrs. *'''Señorita (Srta.)''' - miss, Ms. *'''¿Cómo estás?''' - How are you? (familiar) *'''¿Cómo está usted?''' - How are you? (formal) *'''¿Qué?''' - What? *'''¿Cómo le va?''' - How do you do? (formal) *'''¿Cómo te va?''' - How's it going? (informal) *'''¿Qué tal?''' - How are you? *'''¿Y tú?''' - And you? (familiar) *'''¿Y usted?''' - And you? (formal) *'''bien''' - well *'''más o menos''' - more or less (so, so) *'''nada''' - nothing *'''regular''' - regular, okay *'''Lo siento''' - I am sorry *'''Gracias.''' - Thank you. *'''De nada.''' - You're welcome. / Not at all. *'''A su servicio.''' - At your service. *'''Adiós.''' - Good-bye. *'''Chao.''' - Bye. *'''Hasta luego.''' - See you later. *'''Hasta mañana.''' - See you tomorrow. *'''Hasta pronto.''' - See you soon. *'''Nos vemos.''' - See you. ====Note==== Remember: * in most Spanish-speaking countries, "ll" (double "L") is pronounced as "j" [d͡ʒ~ʒ]; however the sound becomes "y" [ʎ~j] in Mexico and "sh" [ʃ] in Argentina... * in Latin America, the "Z" is pronounced as "s" [s]; in Spain, it is sounded as "th" [θ]. * J is used for the sound like in Ba'''ch''' [x~ɣ~h], g is pronounced hard [ɡ] before a, o and u, but the same as j [x~ɣ~h] before e and i. (To indicate that it is pronounced hard before these, gu- is written. Gü- simply is [ɡʷ]. * letters in parentheses (a) are feminine, while those without (o) are masculine. ====How To Use Ud and Uds (Tú/Usted [Ud.] - Vosotros/Ustedes[Uds.])==== In Castilian Spanish (Spain), there are several ways to say "you." We must differentiate between familiar/formal and singular/plural forms. ''Tú'' (singular) is used when talking to someone you know, such as family, friends, or pets. ''Usted'' (abbreviated Ud.) is used in formal events, such as talking to a teacher, someone who you don't know well, or a person who has a high title, such as a police officer, etc. The plural of tú is vosotros/as, the feminine form is used only for a wholly female group. The plural of ''usted'' is ''ustedes''. In Latinoamerica ''usted'' can also be used for a close friend, and ''tú'' is never used as a polite form. ''Tú'' may be lost altogether and another pronoun ''vós'' may be used for certain contexts. {| class=wikitable ! !Singular !Plural |- !Familiar |align=center|tú |align=center|vosotros/as (Spain)<br>ustedes (Latin America) <small>[http://lema.rae.es/dpd/srv/search?id=yhtVtQ5pQD6ONJL2Gd RAE]</small> |- !Formal |align=center|usted |align=center|ustedes |} === FANBOYS in Spanish (FANBOYS en Español) === '''Para''' - For '''Y -''' And '''Ni -''' Nor '''Pero''' '''-''' But '''O -''' Or '''Sin embargo -''' Yet '''Así que -''' So ===Time (tiempo)=== :''Please read [[b:Spanish/Lessons/¿Qué hora es?|¿Qué hora es?]]'' *'''¿Qué hora es?''' - What time is it? *'''¿A qué hora ....?''' - At what time ....? *'''Es la una.''' - It's one o' clock. *'''Son las dos.''' - It's two o' clock. *'''Son las tres y cuarto.''' - It's a quarter past three. *'''Son las cuatro y media.''' - It's half past four. *'''Son las siete menos cuarto.''' - It's a quarter to seven. ===Whole numbers=== *'''cero''' - zero (0) *'''uno''' - one (1) *'''dos''' - two (2) *'''tres''' - three (3) *'''cuatro''' - four (4) *'''cinco''' - five (5) *'''seis''' - six (6) *'''siete''' - seven (7) *'''ocho''' - eight (8) *'''nueve''' - nine (9) *'''diez''' - ten (10) *'''once''' - eleven (11) *'''doce''' - twelve (12) *'''trece''' - thirteen (13) *'''catorce''' - fourteen (14) *'''quince''' - fifteen (15) *'''dieciséis''' - sixteen (16) *'''diecisiete''' - seventeen (17) *'''dieciocho''' - eighteen (18) *'''diecinueve''' - nineteen (19) *'''veinte''' - twenty (20) *'''veintiuno''' - twenty-one (21) *'''veintidós''' - twenty-two (22) *'''veintitrés''' - twenty-three (23) *'''veinticuatro''' - twenty-four (24) *'''veinticinco''' - twenty-five (25) *'''veintiséis''' - twenty-six (26) *'''veintisiete''' - twenty-seven (27) *'''veintiocho''' - twenty-eight (28) *'''veintinueve''' - twenty-nine (29) *'''treinta''' - thirty (30) Note: '''After 30''', numbers are named using the conjunction "y" (''and''), as to say: "thirty-and-one", "thirty-and-two", etcetera... "forty-and-eight", forty-and-nine", and so forth.... *'''treinta y uno''' - thirty-one (31) *'''cuarenta''' - forty (40) *'''cincuenta''' - fifty (50) *'''sesenta''' - sixty (60) *'''setenta''' - seventy (70) *'''ochenta''' - eighty (80) *'''noventa''' - ninety (90) *'''cien''' - one hundred (100) ===Body parts === *(La) '''cabeza''' - (The) head *(La) '''oreja''' - (The) ear (outside) *(El) '''oído''' - (The) ear (inside) *(El) '''ojo''' - (The) eye *(La) '''nariz''' - (The) nose *(La) '''boca''' - (The) mouth *(La) '''lengua''' - (The) tongue *(El) '''cuello''' - (The) neck *(El) '''brazo''' - (The) arm *(El) '''pecho''' - (The) chest *(La) '''mano''' - (The) hand <<--- Notice that though the noun in Spanish ends with ''''''o'''''' the article used is '''la'''. *(El) '''dedo''' - (The) finger / toe *(El) '''estómago''' - (The) stomach *(La) '''pierna''' - (The) leg *(El) '''muslo''' - (The) thigh *(El) '''pie''' - (The) foot *(El) '''pelo''' - body hair <<--- Colloquially, "pelo" may mean head hair as well *(El) '''cabello''' - head hair *(La) '''piel''' - (The) skin *(El) '''músculo''' - (The) muscle *(El) '''corazón''' - (The) heart *(La) '''espalda''' - (The) back *(La) '''rodilla''' - (The) knee *(El) '''codo''' - (The) elbow *(El) '''hombro''' - (The) shoulder *(El) '''Pene''' - (The) penis *(El) '''Vagina''' - (The) vagina * ===Office items=== *'''bolígrafo/pluma''' - pen *'''carpeta''' - folder *'''cuaderno''' - notebook *'''estudiante''' - student *'''hoja de papel''' - sheet of paper *'''(el) lápiz''' - pencil *'''(los) lápices''' - pencils *'''(el) borrador''' - eraser *'''libro''' - book *'''profesor, profesora''' - professor (male, female) *'''maestro, maestra''' - teacher (male, female) *'''sala de clases (salón de clases) / aula''' - classroom ====Please read==== * The articles '''el''' and '''la''' both mean ''the'' in Spanish. ''El'' is the masculine form, and words that use ''el'' usually end in -o. ''La'' is the feminine form; most words that end in -a and all words that end in -ción are feminine. There are exceptions, as with the noun ''mano'' (hand) the article may not necessarily match the "o" or "a" standard: ''Las manos'' ("The hands). Some nouns can be either gender, such as ''estudiante'', which can be used as ''el estudiante'' or ''la estudiante'' to mean ''the student''. Other examples of nouns contrary to the norms of gender are "''el paraguas''" (the umbrella), "''el día''" (the day) and "''la radio''" (the radio). * Some feminine words, like ''agua'', use the form ''El'' when in singular form, as it is considered hard to say ''la agua'' due to the [[w:Glottal stop|glottal stop (hiatus)]]. * Some Spanish speakers prefer terms that may vary from other regions, for example, in central Mexico ''salón de clases'' is used instead of ''aula'', and ''maestro(a)'' instead of profesor(a). ===Calendar (el calendario)=== *(El) '''año''' - (The) year *(El) '''día''' - (The) day *(El) '''mes''' - (The) month *(La) '''semana''' - (The) week *(El) '''fin de semana''' - (The) weekend <<--- the article ''El'' is directed to ''fin'' (end); so it may be understood as: ''"'''El''' fin de '''la''' semana"'' (''The week's end'' or ''the end of the week''). *'''¿Qué día es hoy?''' - What day is today? *'''¿Cuál es la fecha?''' - What is the date? *'''Es el ... de ...''' - It's the ... of ... *'''Es el primero de ...''' - It's the first of ... *'''hoy''' - today *'''ahora''' – now *'''mañana''' - tomorrow *'''en el futuro''' – in the future *'''ayer''' - yesterday [[File:Orchard calandar 1919.jpg|thumb|200px|Calendar in English.]] [[File:Kalendaroj.jpg|thumb|200px|Calendario en español.]] ==== Days of the week ==== As in English, the roots of the names for each of the days of the week maintain a link to planets and other cosmic bodies, which in instances were ultimately named after deities; Norse mythology for English (Germanic language) and Roman mythology for Spanish (Romance language). So for example, for Monday, we find "moon" (day), which translates to ''luna'', giving us lunes. Another example is Thursday, which was named after Thor, and has as parallel Jupiter; Jupiter (which is spelled the same in Spanish) is for ''jueves''. *'''lunes''' - Monday *'''martes''' - Tuesday *'''miércoles''' - Wednesday *'''jueves''' - Thursday *'''viernes''' - Friday *'''sábado''' - Saturday *'''domingo''' - Sunday Note: Look at the calendars, and notice that the week may start on Monday instead of Sunday ==== Months ==== *'''enero''' - January *'''febrero''' - February *'''marzo''' - March *'''abril''' - April *'''mayo''' - May *'''junio''' - June *'''julio''' - July *'''agosto''' - August *'''septiembre''' - September *'''octubre''' - October *'''noviembre''' - November *'''diciembre''' - December =====Cultural note===== * The names of days in Spanish and many other European languages are always lowercase, except at the beginning of a sentence. [[Image:AztecCalendarMuseoAntropologia.JPG|thumb|200px|Aztec calendar]] * The Aztecs of Ancient Mexico developed a calendar. "The Aztec calendar stone, Mexica sun stone, Stone of the Sun (Spanish: Piedra del Sol), or Stone of the Five Eras, is a large monolithic sculpture that was excavated in the Zócalo, Mexico City's main square, on December 17, 1790.[1] It was discovered whilst Mexico City Cathedral was being repaired.[2] The stone is around 12 feet across and weighs about 24 tons.[3]."(taken from wikipedia) [1] Florescano, Enrico (2006). National Narratives in Mexico. Nancy T. Hancock (trans.), Raul Velasquez (illus.) (English-language edition of Historia de las historias de la nación mexicana, ©2002 [Mexico City:Taurus] ed.). Norman: University of Oklahoma Press. {{ISBN|0-8061-3701-0}}. OCLC 62857841 . [2] a b Aztec Civilization [3] The Aztec Sun Stone ===Other words and phrases; some questions and answers=== *'''¿Cómo se escribe ...?''' - How is ... spelled? *'''Se escribe ...''' - It's spelled ... *'''¿Cómo se dice...?''' - How does one say...? (How do you say...?) *'''Se dice...''' - One says... (You say...) *'''¿Qué quiere decir ...?''' - What does ... mean? *'''Quiere decir ...''' - It means ... *'''¿Cuántos(as)?''' - How many? *'''en''' - in / on *'''hay''' - there is, there are *'''por favor''' - please *'''Punta...''' - Tip (point or vertex) ===[[Spanish alphabet|Alphabet]]=== Showing uppercase and how the letter is proununced or named in Spanish. *'''A''' - a *'''B''' - be *'''C''' - ce *'''D''' - de *'''E''' - e *'''F''' - efe *'''G''' - ge *'''H''' - hache *'''I''' - i *'''J''' - jota *'''K''' - ka *'''L''' - ele *'''M''' - eme *'''N''' - ene *'''Ñ''' - eñe *'''O''' - o *'''P''' - pe *'''Q''' - cu *'''R''' - erre *'''S''' - ese *'''T''' - te *'''U''' - u *'''V''' - uve *'''W''' - uve doble *'''X''' - equis *'''Y''' - i griega/ye *'''Z''' - zeta ===Country focus === [[Image:Flag_of_Mexico.svg|150px|right]] '''Mexico''' (Spanish: México) is a federal constitutional republic in North America. It is bordered on the north by the United States; on the south and west by the North Pacific Ocean; on the southeast by Guatemala, Belize, and the Caribbean Sea; and on the east by the Gulf of Mexico. Mexico is a federation comprising thirty-one states and a federal district, the capital Mexico City (CDMX), whose metropolitan area is one of the world's most populous. [[Image:mx-map.png|thumb|right|200px|Map of Mexico]] Covering almost 2.3 million square kilometers, Mexico is the fifth-largest country in the Americas by total area and the 14th largest independent nation in the world. With an estimated population of 129 million (2019 estimated), it is the 10th most populous country and the most populous Spanish-speaking country in the world. As a regional power and the only Latin American member of the Organization for Economic Co-operation and Development (OECD) since 1994, Mexico is firmly established as an upper middle-income country. Mexico is the 12th largest economy in the world by GDP by purchasing power parity. The economy is strongly linked to those of its North American Free Trade Agreement (NAFTA) partners. Despite being considered an emerging world power, the uneven distribution of income and the increase in insecurity are issues of concern. Mexican culture reflects the complexity of the country's history through the blending of pre-Hispanic civilizations and the culture of Spain, imparted during Spain's 300-year colonization of Mexico. Exogenous cultural elements, mainly from the United States, have been incorporated into Mexican culture. As was the case in most Latin American countries, when Mexico became an independent nation, it had to slowly create a national identity, being an ethnically diverse country in which, for the most part, the only connecting element amongst the newly independent inhabitants was Catholicism. [[Image:Angel de la Independencia Mexico City.jpg|right|thumb|150px|Paseo de la Reforma in Mexico City at night, with the Angel of Independence.]] The Porfirian era (el Porfiriato), in the last quarter of the nineteenth century and the first decade of the twentieth century, was marked by economic progress and peace. After four decades of civil unrest and war, Mexico saw the development of philosophy and the arts, promoted by President Díaz himself. Since that time, though accentuated during the Mexican Revolution, cultural identity had its foundation in the mestizaje, of which the indigenous (i.e. Amerindian) element was the core. In light of the various ethnicities that formed the Mexican people, José Vasconcelos in his publication ''La Raza Cósmica'' (The Cosmic Race) (1925) defined Mexico to be the melting pot of all races (thus extending the definition of the mestizo) not only biologically but culturally as well. This exalting of mestizaje was a revolutionary idea that sharply contrasted with the now discredited idea of a superior, ''pure'' race, still prevalent in Europe at the time. [[Image:CH10.JPG|left|thumb|150px|The famous Guadalajara Cathedral, a symbol of Mexico]] '''Factbox''': Official Language: Spanish Other Languages: 62 other Native languages, English Capital: Mexico City (Ciudad de México, México D.F.) Government: Democracy Area: 1,972,550 sq km (761,606 sq mi) (15th) Population: 111,211,789 (July 2010) (11th) Religion: Christianity 95% (mainly Catholic), Non-religious 2.5%, Buddhism 0.1%, other 2.4% (mainly traditional Mayan or Aztec beliefs) Human Development Index [[w:Human_Development_Index|HDI]] : 0.854 (HIGH, 53rd) [[w:List_of_countries_by_Human_Development_Index|HDI List of Countries]] Independence: September 27, 1821 Currency: Mexican pesos [[Image:Palenque Ruins.jpg|thumbnail|150px|right|The ruins of Palenque, an Ancient Mayan city that was abandoned by the time Spanish explorers landed in Mexico.]] [[Category:Spanish One]] 92x5lzelixz1c4p9xi9bdurpcoua9h0 0 84801 2812088 2247458 2026-05-30T00:39:19Z MathXplore 2888076 Added {{[[Template:BookCat|BookCat]]}} using [[User:1234qwer1234qwer4/BookCat.js|BookCat.js]] 2812088 wikitext text/x-wiki {{Infobox |name = |bodystyle = |title = |image = [[File:Fragonard, The Reader.jpg|220px]] |caption = "Young Girl Reading"<br>by Jean-Honoré Fragonard (c. 1776) |headerstyle = background:#ccf; |labelstyle = background:#ddf; |datastyle = |header1 = Course Metadata |label1 = Faculty |data1 = [[Portal:Humanities|Humanities]] |label2 = School |data2 = [[School:Language and Literature|Language and Literature]] |label3 = Department |data3 = [[Topic:Literary Studies|Literary Studies]] |label4 = Course Number |data4 = LS100 |label5 = Course Section |data5 = A |label6 = Course Instructor |data6 = [[User:Yangjustinc|Justin Yang]] |label7 = Prerequisites |data7 = High School English }} == Introduction == Through the study of selected examples of [[w:poetry|poetry]], [[w:fiction|fiction]], and [[w:drama|drama]], this course will introduce students to the fundamentals of the university-level literary study, and furnish them with the skills to think and write critically about literature. Students will be taught the basic concepts of genre and form in literature, and methods of literary analysis, to enable them to continue in more specialized English courses at the second year or beyond. This section of [[LS100A - Introduction to Literary Studies | Introduction to Literary Studies]] will emphasize an exploration of wit and satire, beginning with English literature of the Middle Ages and ending sometime near the end of the eighteenth century. ===Evaluation=== *10% Participation<br><small>Participation is counted through contributions to this course wiki. All contributions should be signed.</small> *20% Reading Questions<br><small>Only the top five reading question packages submitted will be counted.</small> *40% Research Paper<br><small>1500-2000 word research paper on a pre-selected topic or, with instructor permission, a topic of a student's own choosing.</small> *30% Final Examination<br><small>Open-book examination comprised of 5 passage recognition questions and 2 essay questions.</small> ==Course== {|width=100% |valign=top width=50%| {{robelbox|icon=Epiphany-bookmarks.png|iconwidth=64px|title=Syllabus and Learning Materials}} <div style="{{Robelbox/pad}}"> === Syllabus and Learning Materials === # Introduction to Literary Study ## [[/History|History of Literature]] ## [[/Genre|Genre]] ## [[/Literary Criticism|Literary Criticism]] ## [[/Humour|Theories of Humour]] # from ''The Canterbury Tales'' ## [[/Introduction to the Canterbury Tales|Introductory Notes]] ## [[/The General Prologue|The General Prologue]] ## [[/The Miller's Tale|The Miller's Tale]] # ''The Country Wife'' ## [[/Introduction to The Country Wife|Introductory Notes]] ## [[/Prologue to The Country Wife|Prologue]] ## [[/Act I of The Country Wife|Act I]] ## [[/Act II of The Country Wife|Act II]] ## [[/Act III of The Country Wife|Act III]] ## [[/Act IV of The Country Wife|Act IV]] ## [[/Act V of The Country Wife|Act V]] ## [[/Epilogue to The Country Wife|Epilogue]] # from ''Gulliver's Travels'' ## [[/Introduction to Gulliver's Travels|Introductory Notes]] ## [[/A Voyage to Lilliput|A Voyage to Lilliput]] ## [[/A Voyage to Brobdingnag|A Voyage to Brobdingnag]] ## [[/A Voyage to the Country of the Houyhnhnms|A Voyage to the Country of the Houyhnhnms]] # ''The Rape of the Lock'' ## [[/Introduction to The Rape of the Lock|Introductory Notes]] ## [[/Canto I of The Rape of the Lock|Canto I]] ## [[/Canto II of The Rape of the Lock|Canto II]] ## [[/Canto III of The Rape of the Lock|Canto III]] ## [[/Canto IV of The Rape of the Lock|Canto IV]] ## [[/Canto V of The Rape of the Lock|Canto V]] # ''Northanger Abbey'' ## [[/Introduction to Northanger Abbey|Introductory Notes]] ## [[/Chapter 1|Chapter 1]] ## [[/Chapter 2|Chapter 2]] ## [[/Chapter 3|Chapter 3]] ## [[/Chapter 4|Chapter 4]] ## [[/Chapter 5|Chapter 5]] ## [[/Chapter 6|Chapter 6]] ## [[/Chapter 7|Chapter 7]] ## [[/Chapter 8|Chapter 8]] ## [[/Chapter 9|Chapter 9]] ## [[/Chapter 10|Chapter 10]] ## [[/Chapter 11|Chapter 11]] ## [[/Chapter 12|Chapter 12]] ## [[/Chapter 13|Chapter 13]] ## [[/Chapter 14|Chapter 14]] ## [[/Chapter 15|Chapter 15]] ## [[/Chapter 16|Chapter 16]] ## [[/Chapter 17|Chapter 17]] ## [[/Chapter 18|Chapter 18]] ## [[/Chapter 19|Chapter 19]] ## [[/Chapter 20|Chapter 20]] ## [[/Chapter 21|Chapter 21]] ## [[/Chapter 22|Chapter 22]] ## [[/Chapter 23|Chapter 23]] ## [[/Chapter 24|Chapter 24]] ## [[/Chapter 25|Chapter 25]] ## [[/Chapter 26|Chapter 26]] ## [[/Chapter 27|Chapter 27]] ## [[/Chapter 28|Chapter 28]] ## [[/Chapter 29|Chapter 29]] ## [[/Chapter 30|Chapter 30]] ## [[/Chapter 31|Chapter 31]] # ''Alice's Adventures in Wonderland'' ## [[/Introduction to Alice's Adventures in Wonderland|Introductory Notes]] ## [[/Down the Rabbit-Hole|Down the Rabbit Hole]] ## [[/The Pool of Tears|The Pool of Tears]] ## [[/A Caucus-Race and A Long Tale|A Caucus-Race and A Long Tale]] ## [[/The Rabbit Sends in a Little Bill|The Rabbit Sends in a Little Bill]] ## [[/Advice from a Caterpillar|Advice from a Caterpillar]] ## [[/Pig and Pepper|Pig and Pepper]] ## [[/A Mad Tea-Party|A Mad Tea-Party]] ## [[/The Queen's Croquet-Ground|The Queen's Croquet-Ground]] ## [[/The Mock Turtle's Story|The Mock Turtle's Story]] ## [[/The Lobster-Quadrille|The Lobster-Quadrille]] ## [[/Who Stole the Tarts|Who Stole the Tarts?]] ## [[/Alice's Evidence|Alice's Evidence]] </div> {{Robelbox/close}} |valign=top width=50%| {{robelbox|icon=Crystal 128 desktop.png|iconwidth=64px|title=Assignments & Examinations}} <div style="{{Robelbox/pad}}"> ===Assignments=== * [[/Reading Questions 1|Reading Questions 1]] <small>for ''The Canterbury Tales''</small> * [[/Reading Questions 2|Reading Questions 2]] <small>for ''The Country Wife''</small> * [[/Reading Questions 3|Reading Questions 3]] <small>for ''Gulliver's Travels''</small> * [[/Reading Questions 4|Reading Questions 4]] <small>for ''The Rape of the Lock''</small> * [[/Reading Questions 5|Reading Questions 5]] <small>for ''Northanger Abbey''</small> * [[/Reading Questions 6|Reading Questions 6]] <small>for ''Alice's Adventures in Wonderland''</small> * [[/Research Paper| Research Paper]] ===Examination=== *[[/Final Examination | Final Examination]] </div> {{Robelbox/close}} |} ==Required Readings== ===Primary Texts=== Any standard, unabridged edition of the following works will be acceptable. * [[s:The Canterbury Tales |''The Canterbury Tales'']] by [[w:Geoffrey Chaucer | Geoffrey Chaucer]] * [[s:The Country Wife | ''The Country Wife'']] by [[w:William Wycherley | William Wycherley]] * [[s:Gulliver's Travels | ''Gulliver's Travels'']] by [[w:Jonathan Swift | Jonathan Swift]] * [[s:Rape of the Lock | ''The Rape of the Lock'']] by [[w:Alexander Pope | Alexander Pope]] * [[s:Northanger Abbey | ''Northanger Abbey'']] by [[w:Jane Austen | Jane Austen]] * [[s:Alice's Adventures in Wonderland | ''Alice's Adventures in Wonderland'']] by [[w:Lewis Carroll | Lewis Carroll]] == See Also == * [[Portal: Literary Studies]] [[Category:Introductions]] [[Category:Courses]] {{BookCat}} rdchdcdtfb9rgvzm835jcsg8z7545mk 0 85954 2811984 2811983 2026-05-29T12:01:47Z Jtneill 10242 Reduce left bias 2811984 wikitext text/x-wiki <div style="max-width:900px; margin:0 auto; padding:0 1em; background:transparent;"> {| style="width:100%; border:0; background:transparent; background-color:transparent; margin:0 auto; " |- | style="width:72px; padding:0 0 10px 0;" | [[File:Start.svg|68px|link=Motivation and emotion/About/Welcome and getting started]] | style="padding:0 0 10px 0;" | [[/About|About]] | [[/About/Welcome and getting started|Start]] | style="width:2em;" | | style="width:72px; padding:0 0 10px 0;" | [[File:WikiProject Brazil assessment icon.svg|68px|link=Motivation and emotion/Assessment]] | style="padding:0 0 10px 0;" | [[/Assessment|Assessment]] |- | style="width:72px; padding:0 0 10px 0;" | [[File:Eo circle indigo white info.svg|68px|link=Motivation and emotion/About/Outline]] | style="padding:0 0 10px 0;" | [[/About/Outline|Unit outline]] | | style="width:72px; padding:0 0 10px 0;" | [[File:Emoji u1f989.svg|68px|link=Motivation and emotion/About/Outline/About/Staff]] | style="padding:0 0 10px 0;" | [[/About/Staff|Teaching team]] |- | style="width:72px; padding:0 0 10px 0;" | [[File:Toicon-icon-avocado-schedule.svg|68px|link=Motivation and emotion/About/Schedule|class=skin-invert]] | style="padding:0 0 10px 0;" | [[/About/Schedule|Schedule]] | | style="width:72px; padding:0 0 10px 0;" | [[File:Icon Clock.svg|68px|link=Motivation and emotion/About/Timetable]] | style="padding:0 0 10px 0;" | [[Motivation and emotion/About/Timetable|Timetable]] |- | style="width:72px; padding:0 0 10px 0;" | [[File:Lecture icon.png|68px|link=Motivation and emotion/Lectures]] | style="padding:0 0 10px 0;" | [[/Lectures|Lectures]] | | style="width:72px; padding:0 0 10px 0;" | [[File:Discussion icon.png|68px|link=Motivation and emotion/Tutorials]] | style="padding:0 0 10px 0;" | [[/Tutorials|Tutorials]] |- | style="width:72px; padding:0 0 20px 0;" | [[File:Uc-logo.png|68px|link=https://unicanberra.instructure.com/courses/17386]] | style="padding:0 0 20px 0;" | {{Motivation and emotion/Canvas}} | | style="width:72px; padding:0 0 20px 0;" | [[File:Antu devhelp.svg|68px|link=Motivation and emotion/About/Help]] | style="padding:0 0 20px 0;" | [[/About/Help|Help]] |- | style="width:72px; padding:0 0 20px 0;" | [[File:Book stub img.svg|68px|link=Motivation and emotion/Book/2026]] | style="padding:0 0 20px 0;" | [[/Book|Book]] -<br>[[/Book/2026|2026]] | | colspan="2" style="padding:0 0 20px 0;" | <div style="float:left; width:13em"> <inputbox> type=search width=25 namespaces=Main** prefix=Motivation and emotion searchbuttonlabel=Search break=no bgcolor=transparent </inputbox> </div> |} </div> [[Category:{{PAGENAME}}| ]] [[Category:Featured resources]] </noinclude> caom3subkpme7401ajrwojzmzy6mqho 0 98508 2812102 2811504 2026-05-30T02:53:59Z CommonsDelinker 9184 Replacing Man_eats_at_Volunteers_of_America_soup_kitchen_Washington_DC_1936.gif with [[File:Farm_Security_Administration-_Works_Progress_Administration-_Unemployed_men_eating_in_Volunteers_of_America_Soup_Kitchen_in_Washington,_D.C._-_DPLA_-_6dee28bfef6e3 2812102 wikitext text/x-wiki <noinclude> {{title|Motivation and emotion image gallery}} * This gallery features images related to [[motivation and emotion]] from [[commons:|Wikimedia Commons]]. * These images may be useful for embedding in [[Motivation and emotion/Book|motivation and emotion book chapter]]s to help illustrate key points and to provide examples. * To find 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Franklin D. Roosevelt Presidential Library and Museum]] '''A health education and development facilitator, working with people one-to-one, in small groups, or in community or institutional settings, will engage salient teaching and learning practices that serve to empower people to improve their health outcomes.''' Through assignments, tutorials and lectorials you will work in teams, using a range of [[experiential learning]] formats to develop knowledge, skills and attitudes needed to facilitate the planning, implementation and evaluation of health education and development in diverse circumstances. '''The Right of Deaf and Hard-of-Hearing Children to Access Sign Language During the Critical Period of Language Acquisition''' The film “And Your Name is Jonah” tells the heart-wrenching story of a young Deaf boy whose misdiagnosis as intellectually disabled leads to years of neglect in terms of language and communication. Jonah’s parents, who are hearing, struggle to understand his needs, delaying his access to sign language and creating barriers to his cognitive and emotional development. His journey resonates deeply with me, not only because of my professional work as a Certified ASL/ English Interpreter but also because I am a Child of Deaf Adults (CODA). My Deaf parents were raised in hearing families who did not sign, which is a reality for 90% of Deaf children born into hearing families, according to the CDC. As someone now experiencing hearing loss, I recognize the profound gift of already knowing ASL. This dual perspective drives my passion for advocating that all Deaf and hard-of-hearing children should have access to sign language during their critical period of language acquisition. The right to language is ultimate. If Deaf children are not exposed to ASL, problems related to social security, growth, and learning occur to Deaf children. These ideas will be developed on the basis of the film, “And Your Name is Jonah” and complemented by an analysis of works by other authors and scholars. It is a human right as well as the need to take sign language in early childhood, especially for children from hearing families.   '''“And Your Name is Jonah” Film Summary''' This film was made known in the year 1979 under the name “And Your Name is Jonah”. It focuses on the biography of Jonah Corelli who is a Deaf boy but has been assumed to have low intelligence. In this case, due to that misunderstanding of the kind of disability he has, it takes time for his parents to realize what he requires. It also prevents Jonah from getting the right type of education resources and mode of communication such as ASL. The movie illustrates Jonah as more aggravated and isolated as a result of his failure to fit into society, which does not accommodate his kind. Jonah’s difficulties are increased by Jonah’s parents’ ignorance of the Deaf culture and sign language which resulted in his difficulties with his intellectual and educational progress. Jonah’s parents are one of the many hearing families who do mean well and could have established the linguistic that their Deaf child requires. Jonah’s delayed access to ASL matches the experiences of numerous Deaf children who face language deprivation, the central theme of discussion in this paper. '''The Science of Language Acquisition''' Andrews and Baker's (2019) research underscores that early language exposure is vital for intellectual and social development during children’s early growth periods. Andrews and Baker (2019) clearly point out that the first five years of a child is a critical period for language acquisition. During this time, a child's mind is highly receptive to linguistic input, whether spoken or signed. Deaf children risk irreversible cognitive delays when they are limited to early exposure to a fully accessible language like ASL. Andrews and Baker's (2019) research on the analysis of the emergence of sign language, shows that Mesoamerican backgrounds portray that enough support is vital in language access. Due to its unique linguistic and cultural diversity, Mesoamerica is perceived as a significant example of a context emergence of sign language. Andrews and Baker (2019) point out that numerous sign languages have emerged in Mayan, with others appearing in communities with different spoken language families. Based on comparative analyses, the cultural and linguistic environments highly affect similarities and differences in sign language. For instance, facets, including multimodal communication in hearing communities (featured by vital dependence on conventional gestures and non-verbal behaviors) highly influence these sign languages. Likewise, affirmative attitudes toward hearing impairment and communicative practices lead to the emergence of sign language in these backgrounds. In addition, the variation in sign language use also replicates differences in speech communities. For example, as Andrews and Baker (2019) outline, individuals often develop home sign systems with smaller vocabularies and lower levels of conventionalization than village sign languages. This arises more in communities with many deaf people and shared cultural contexts. Evans (2004) further highlights that Deaf children with early access to sign language achieve literacy milestones comparable to hearing peers. Literacy, closely tied to language acquisition, forms the foundation for academic success. Based on Jonah's case of delayed access to ASL as pointed out in the film, it reflects this article's findings. This is because his frustrations with communication are evident as behavioral and emotional struggles every moment in cases of language deprivation. Scott and Dostal (2019) provide more information for this argument, showing that bilingual approaches incorporating ASL and written English nurture language acquisition. This approach is vital for literacy development and underscores the significance of embracing ASL as a valid and primary language for Deaf children, especially during their early development periods of language acquisition. Scott & Dostal (2019) point out clearly that the primary language of ASL provides Deaf children with a firm linguistic foundation, especially when acknowledged and embraced as a substantial language. This formed foundation is important to cognitive and emotional growth and development. It also supports the argument that ASL should be used as a proper language to teach the Deaf. Moreover, Evans (2004); Scott & Dostal (2019) explain the objective adverse effects of early sign language experience on developmental aspects in children with hearing impairments. Early access to ASL enhances linguistic skills, self-identity, and emotional well-being as well. The language and ASL-based instruction outlined by (Evans, 2004; Scott & Dostal, 2019) shows cognitive advantages granted by early exposure to language improve literacy and academic achievement. As a result, the importance of ensuring that children with hearing impairments have the chance to learn sign language should be embraced significantly since it is important during the language development period of a child. Despite the use of sign language being crucial for the child to be able to interact in society, think, and even develop emotionally and intellectually, sign language thus becomes a central right for deaf or hard-of-hearing children. Finton et al. (2014) supported their findings with the study which resulted in the conclusion that early exposure to ASL is beneficial, in that it enhances deaf children’s language and education development by the appropriate age. In purely technical terms, denying deaf children early language profoundly results in catastrophic risks of linguistic, cognitive, and academic delays. Apart from that, the article emphasizes the importance of early mediation for children with hearing loss having hearing caregivers. In a study conducted by Finton et al. (2024), they observed that learners who enrolled in ASL-focused bilingual education before three years were able to perform better academically as compared to those learners with deaf parents. In addition, the evidence of the article suggests a positive perspective on hearing caregivers’ effectiveness in enhancing ASL development. Similarly, it describes how and when language matters for mitigating the impact of language delay. For this case, Jonah’s delayed access to ASL has also been evidenced by the discovery of the Finton et al. (2024) study. This is because the effects of the delayed language learning impacted him to the extent of folding up and struggling to express himself; the major points re-echoed by the article for individuals who were never exposed to early formal language learning. Without the ASL-focused bilingual intervention, many deaf children risk being limited in terms of linguistic eloquence, mental development, academic success, and social and emotional comfort, as in the case of Jonah. Besides, a lack of linguistic fluency during the early growth years of a child due to language deprivation hinders the development of critical thinking skills, literacy, and the ability to socialize. Under social context, language-deprived children struggle with frustration due to isolation, which can lead to behavior challenges (Finton et al., 2024). Thus, early exposure to ASL and hearing caregivers is the only meaningful solution to these outcomes among Deaf children. Finton et al. (2024) point out that bilingual education is essential and effective in ensuring enduring academic and cognitive growth and development. Therefore, deaf children with hearing caregivers (provided with early access to bilingual and ASL education) portray affirmative outcomes in their behavior and academic journey. '''Challenges Faced by Deaf Children in Hearing Families''' Crume (2013) identifies systemic barriers within educational systems that prioritize oralism or speech therapy over sign language. Like many hearing families, Jonah's parents are unaware of ASL's potential to bridge communication gaps. Barriers like these often emanate from beliefs that spoken language is fundamentally superior for developing language and literacy skills. This, in turn, disadvantages deaf children since they lack full access to spoken language. The oralism approach limits the linguistic options available to deaf children (Crume, 2013), perpetuating the fallacy that sign language is incompatible with academic success. Consequently, many deaf children with hearing backgrounds face rampant communication challenges. Also, the challenges are compounded by their caregivers, who lack awareness regarding the potential of ASL as a fully accessible language, which is vital in bridging these gaps. This aspect reflects Jonah's situation as depicted in the film. Furthermore, Crume (2013) stresses the relevance of promoting sign language phonological awareness as an intervention to literacy for deaf children. Instructors in ASL or English bilingual programs recognize that structural knowledge of signs can play the same role as spoken language phonological awareness in supporting literacy development (Crume, 2013). The systematic prioritization of oralism in many educational backgrounds deprives deaf children of prospects for knowledge and linguistic development. Jonah's family, among many other hearing families, should focus on speech therapy. In addition, limited resources and support for hearing children and their families cause more challenges. For instance, Jonah's family lacked enough guidance to help them recognize the complexities and better ways to raise him with all the support he needed. Lack of guidance and resources forms an environment where deaf children are linguistically deprived, and their primary communication mode is devalued (Crume, 2013). In this case, the issue created an environment where Jonah was not only linguistically deprived but also socially isolated. Thus, Crume's article outlines the need for a systematic change to prioritize sign language and provide enough resources to deaf children and hearing families. Edwards (2005) notes that the prevalence of cochlear implants has further complicated this issue, making many families and communities divided. Hearing parents face complicated decision-making processes when defining language and communication pathways for their deaf children. While implants can provide auditory input, they do not guarantee speech proficiency or cognitive development unless paired with a robust foundation in sign language. Edwards (2005) highlights how this issue is reflected in ''Sound and Fury (2000)'', where the Artinian family struggles with the choice to undergo CI surgery for their deaf children. This choice nearly affected their extended family. The deaf brother rejects the implant for his child together with the hearing brother who supports the implants, creating a conflicting decision (Edwards, 2005). This shows the reflective cultural and emotional aspects involved in the main challenge. Likewise, the hearing parents of the brothers accuse the deaf sibling of child abuse for his implant refusal (Edwards, 2005), outlining societal pressure on deaf families to follow hearing norms and beliefs. The deaf community perceives choosing cochlear implants as a rejection of deaf culture. Reflective on Jonah's situation, many hearing families need to be more informed about the significance of early language acquisition, mainly the role of ASL in fostering intellectual and social development. Due to a lack of proper guidance, hearing parents may perceive CI as an objective solution. They should realize that their children need access to spoken or ASL language to develop cognitive and linguistic skills. Thus, according to Edwards's (2005) analysis, despite Cochlear implants being beneficial for few deaf children, they must be incorporated with early exposure to a fully accessible language. It is important to embrace these for effective cognitive and language development among deaf children. The majority of Deaf children are born into hearing families unfamiliar with Deaf culture or sign language. This leads to significant challenges for their language and cognitive development. As the study by Finton et al. (2024) reports, Deaf children in hearing families regularly experience delays in both language acquisition and academic achievement compared to those in Deaf families. This variation is as a result of the limited first exposure to an open language, particularly the ASL. Parents of children with hearing impairment need advice from various healthcare practitioners who advocate for oral-only interventions. This, in turn, expresses the important role of ASL in the cognitive and linguistic development of a child. Similarly, Finton et al. (2024) have described the role of bilingual education and early practice regarding the experiences that deaf children and hearing families come across. Finton et al., (2024) revealed that deaf children particularly those who enrolled for bilingual education at the age of 3 years and were raised by hearing have a superior favorable association of academic accomplishment related to those raised by deaf caregivers. This shows the role of early exposure to ASL in enhancing language development and positively influencing Deaf children's academic performance. In addition, many hearing families go through a lack of awareness and knowledge just like Jonah’s parents. Because of this privation of responsiveness and support resources as well, hearing parents may delay sign language from their deaf children Finton et al., 2024. As a result, this leads to language deprivation and has long-lasting impacts, mainly on intellectual, social, and academic development. Jonah’s experience mirrors these realities. His parents’ reliance on spoken communication isolates him, delaying his ability to express his thoughts and emotions. This isolation is preventable through early ASL exposure and education. Therefore, there is a need for educational programs and support that enlighten hearing families about the benefits of early exposure of deaf children to ASL and bilingual education, especially in the language acquisition period. '''Role of Education and Deaf Role Models''' Bilingual ASL/English programs enhance language acquisition and academic outcomes for Deaf children. Crume (2013) highlights that these programs provide phonological awareness in ASL, translating to stronger literacy skills. Unlike traditional approaches focusing on spoken language PA, English/ASL bilingual programs prioritize the structural knowledge of signs, which helps deaf children bridge language and literacy. Jonah’s struggles in the film demonstrate the consequences of being excluded from such programs. Instructors in these initiatives use different teaching methods, enhancing surroundings where deaf people can advance their cognitive and linguistic skills more effectively (Crume, 2013). Also, Crume's (2013) analysis shows how teachers in these contexts believe in the value of ASL as a linguistic tool that supports literacy development among deaf children. These teachers aid deaf children in building refined skills that are useful for decoding and understanding written English through supporting sign language PA. This bilingual approach also empowers deaf children to explore academic settings with greater competence and confidence (Crume, 2013). Jonah's struggles portray the impact of excluding deaf children from such initiatives well, as the film depicts. Jonah's ability to acquire bilingual ASL/English education led to delayed language acquisition. Consequently, this led to his low social integration and poor academic progress. Thus, it necessitates embracing bilingual programs as a significant support resource for deaf children, especially those born in hearing families lacking sign language skills and knowledge. It is important to shift educational programs for deaf students from reliance on spoken language PA to bilingual education approaches embracing ASL (Crume, 2013). These education programs extend their benefits to deaf students, from literacy development to affirming cultural and linguistic identity. Shantie and Hoffmeister (2000) emphasize the importance of hiring Deaf educators in schools to support the cultural and linguistic development of deaf children. Deaf teachers provide linguistic models and help Deaf students develop a positive self-identity. According to Shantie and Hoffmeister (2000), 90 to 97 percent of deaf children are born to hearing families who are not aware of ASL, and schools offer their first exposure to a fully accessible language. Teachers are cultural and linguistic role models, which helps Deaf students acquire ASL more effectively and develop affirmative self-identity. Jonah’s story could have been different if he had been exposed to role models who shared his linguistic and cultural background. Most hearing teachers lack the skills and proficiency needed to serve as effective language role models. As Shantie and Hoffmeister (2000) point out, 33 percent of hearing teachers claim to comprehend their leaner's signing on par with their English understanding, and only 45 percent account for signing as well as their students. This shows the created gap, compounded by the use of manually coded English language which is less effective than ASL in fostering literacy and second language acquisition. A firm basis in ASL is vital for the academic and linguistic success of deaf students. With the lack of many language models, they will be subjected to an underprivileged educational experience that fails to address their needs (Shantie & Hoffmeister, 2000). In contrast, deaf educators are distinctly positioned to provide effective bilingual education. Their experience and skills in ASL help them to form a supportive surrounding that enhances cultural superiority and language acquisition at the same time. Besides, these role models assist students in recognizing their deaf condition, enabling them to excel academically and socially. To align this with the film, Jonah's story shows the potential effect of these teachers. If Jonah had been exposed to deaf role models at early stages, who shared his cultural and linguistic background, his language development and self-esteem would have improved. Shantie & Hoffmeister's (2000) study advocates for a logical change to ensure that many deaf teachers and Children of Deaf Adults are present in classroom settings, especially during the vital preschool years, to guarantee the future thriving of deaf students. Scott and Dostal (2019) add that culturally competent education fosters social-emotional development. Deaf children who see their language and culture reflected in their learning environments feel more valued, confident, and connected. This form of a sense of affirmation nurtures self-esteem and forms a base for affirmative social interactions and emotional comfort. Jonah’s initial frustrations stem from a lack of these affirming experiences. As highlighted by Scott and Dostal (2019) study findings, Deaf children exposed to natural languages, like ASL in learning settings benefit from a surrounding that validates their cultural identity and communication essentials. According to the film, Jonah's frustrations are a valid example of the mere impacts of deprivation of both ethnic and linguistic assertion. For example, Jonah fought with a detached and isolated state of mind. Consequently, this barred him from engaging in learning development and crafting a significant association with peers or educators. An ethnically competent education mitigates social implications, which may include sinking stigma and nurturing inclusivity, as Scott & Dostal (2019) findings outline. Through learning in educational settings that respect and assimilate language and ethos, deaf children benefit from mutual respect from their hearing peers/friends. This creates a school culture that values all students, fostering language understanding and empathy. Scott and Dostal's (2019) research accentuates that social attachment and language fulfillment support literacy and language development. It builds a child's ability to navigate social situations and build a firm emotional resilience. This is what Jonah lacked in enhancing his linguistic and communication skills. '''Socio-Cultural and Emotional Impacts'''            Language deprivation has profound emotional consequences. Baker (2023) discusses how children without access to a natural language often experience anxiety, depression, and behavioral issues. These emotional struggles are evident in Jonah's difficulties in connecting with his peers and family. Depression, anxiety, and behavioral issues are common aspects of emotional challenges for people facing language deprivation during early child development ages (Baker, 2023). The inability to form a meaningful relationship or a sensitive sense of isolation from their families and peers is one of the ways through which emotional difficulties can manifest. The case of Jonah portrays one of the emotional escalations of language deprivation. Primarily, the main reason why he struggled to establish some kind of connection with the people around him is a limitation to his ability to communicate and profound emotional issues of feeling a lack of support and being misunderstood. This disconnection produces a sense of loneliness and promotes negative emotive states, most especially to every deaf child. Baker (2023) notes that, unlike in childhood alone, these challenges persist into adulthood, and they are more likely to affect an individual’s self-esteem, learning achievement, and occupational prospects. As Baker (2023) points out, the communities that either deny language differences or provide insufficient support services make these emotional challenges worse. This in turn increases feelings of incompetence and rejection, making their emotional distress and isolation even worse. To address this, Baker's (2023) study suggests early and inclusive strategies that aim at addressing linguistic access and emotional support tools, and strive to build a more understanding society. Unfortunately, Deaf education or sign language classes are out of reach for the majority of the families with little resources, especially the minority ones. Systematic discrimination, differential access to resources, and the inadequacy of support services that families require for effective communication through sign language or deaf education present barriers to these families (Bourgois & Hirsch, 2012). This depicts Jonah’s parent’s orientation to circumnavigate the unfamiliarity of the system. Further, these difficulties are compounded by a lack of policymaker understanding and support. As a result, the families who have little cash resources as well as the discriminated groups are the most affected. Achievement of professional training and community networks also become easier for them to obtain, which makes it challenging for the affected deaf children to overcome the communication barrier (Bourgois et al., 2012). These societal hurdles, particularly the incapability to socialize, made Jonah's parents face tough challenges steering the unfamiliar system of bringing him into his condition. For example, with a lack of clear direction and ample resources, Jonah’s parents became overwhelmed by the difficulties of securing suitable communication devices and instruction tools for their Deaf child. Their experience resonates with the link between fundamental factors and economic challenges to collective segregation and language deprivation succession. The role of social elements such as social stigma and communal attitudes as pointed out by Bourgois & Hirsch (2012), deject open communication and resource allocation for marginalized populations. These underlying forces have a far-reaching impact on dividing families. It goes further to leave them struggling with responsive and social encounters. Thus, a systematic change is essential to surge public awareness regarding the effective ways to develop deaf children and support hearing families. This shift may also necessitate equitable resource allocation and the formation of an inclusive education system, reducing disparities that disseminate harm. Shantie and Hoffmeister (2000) also argue that Deaf children benefit from environments that celebrate their language and identity. This fosters psychological and emotional growth and comfort among deaf children. When schools and communities embrace ASL, Deaf children develop stronger self-esteem and resilience (Shantie & Hoffmeister, 2000). Students are surrounded by peers and role models who share the same experiences, which creates a supportive and inclusive environment where they feel valued and well-understood. Jonah’s eventual exposure to ASL and Deaf culture offers him a pathway to self-expression and belonging. This cultural affirmation boosts deaf children's self-esteem and forms their resilience to face challenges in their hearing community (Shantie & Hoffmeister, 2000). For instance, taking Jonah's case as a good illustrator for this aspect, by gaining access to a mode of communication that matches his natural linguistic skills, he would experience a new sense of self-language. This would serve as a solution to his social disconnection and challenges in self-expression and identity. Shantie and Hoffmeister (2000) also focus on the key role of Deaf instructors in forming an inclusive learning environment. Although teachers are role models to their students, those who share the same cultural and linguistic backgrounds serve as powerful models (Shantie & Hoffmeister, 2000). For deaf children/students, this demonstrates possibilities for academic success and self-realization within the deaf community. For instance, this kind of teacher would help learners like Jonah realize an affirmative perspective of their deaf identity, promote psychological comfort, and prevent societal stigma. Thus, integrating deaf culture and language into education contexts enhances academic outcomes and nurtures emotional and psychological well-being, hence equipping deaf children with vital tools to succeed in life. '''Conclusion''' The video “And Your Name is Jonah" is a powerful reminder of the importance of language access for Deaf children. The film's narrative, supported by extensive research, underscores that ASL is not just a communication tool but also a right. Early exposure to sign language during the critical period of language acquisition ensures cognitive, social, and emotional development. As a CODA and ASL/English interpreter, I see daily how ASL empowers Deaf individuals. For hearing families, embracing ASL is an act of love and advocacy. Policymakers, educators, and medical professionals must prioritize access to sign language to prevent the lifelong consequences of language deprivation. Jonah’s story may be fictional, but its lessons are real and urgent. Language is not a privilege but a right. This subject draws on the [[w:Ottawa Charter for Health Promotion|Ottawa Charter for Health Promotion]] (WHO 1986), the [[w:Sundsvall Statement|Sundsvall Statement]] (WHO 1991), the [[w:Jakarta Declaration|Jakarta Declaration]] (WHO 1997) and the [[w:Bangkok Charter for Health Promotion|Bangkok Charter for Health Promotion]] (2005), to define its scope. ==Acknowledgement of Country== This material was originally developed by the staff of a university located on land of which the [[w:Wurundjeri People|Wurundjeri]] (Woiwurrung language) of the Kulin Nation or Alliance are the traditional custodians. For this reason, we pay our respects to their elders, past and present, and we rejoice in the rising generations. ==Assignments== This subject is driven by assignments. Your submitted assignments are used to assess your [[/Intended learning outcomes/]]. Use the topics and tutorials to guide and inform your assignment work. You will need to devote up to 150 hours to study and assignment work in this subject (for example: 15 hours per week for 10 weeks). #[[/Facilitate an activity/]] - team assignment #[[/Critical essay/]] - cooperative learning theory #[[/Explain an activity/]] - program logic #[[/Funding submission brief/]] - strength-based approach; vulnerable groups ==Topics and schedule== #[[/Introduction to the unit/]] and [[/How to get good marks/]] #[[/What is Health Education?/]] #[[/Cooperative Learning Theory/]] #[[/Group Dynamics and Group Processes/]] #[[/Settings or Environments/]] #[[/Strengths-based approach/]] #[[/Planning a group-based lesson/]] #[[/Implementing a group-based lesson/]] #[[/Evaluating a group-based lesson/]] #[[/Funding submissions/]] #[[/Funding ethnic diversity/]] #[[/Funding gender equity/]] #[[/Funding class equality/]] ==Topics and schedule 2016== #[[/What is Health Education?/]] ==Bibliography== *[http://www.businessballs.com/bloomstaxonomyoflearningdomains.htm Bloom's Taxonomy of Learning Domains] (essential resource) *[http://applications.emro.who.int/dsaf/EMRPUB_2012_EN_1362.pdf Health education: theoretical concepts, effective strategies and core competencies (WHO 2012)] *[http://sjp.sagepub.com/content/39/6_suppl/85.full.pdf+html From Health Education to Healthy Learning] *[http://www.who.int/healthpromotion/Milestones_Health_Promotion_05022010.pdf?ua=1 Milestones in Health Promotion] *[http://www.usc.edu/hsc/ebnet/Cc/awareness/Johari%20windowexplain.pdf Johari Window--A model for self-awareness, personal development, group development] *[http://www.massedpartnership.org/wp-content/uploads/2013/08/Feedback-Johari-Window.pdf Understanding Feedback: The Johari Window] *[http://www1.umn.edu/ohr/prod/groups/ohr/@pub/@ohr/documents/asset/ohr_89185.pdf Cooperative Learning Group Activities for College Courses - a Guide for Instructors] - Prepared by Alice Macpherson, Kwantlen University College *[http://trove.nla.gov.au/work/30362925?q=isbn%3A9780195183436&c=book&versionId=36849131 Communicating in groups : building relationships for group effectiveness] - Joann Keyton If a link is not working, please put the title of the document in a search engine and see if you can find it that way. ==Communications== * [http://lms.latrobe.edu.au/course/view.php?id=32479 Learning Management System website] for enrolled students (http://lms.latrobe.edu.au) * The hashtag is CHD-HED [[Category:Health Education Development|*]] [[Category:La Trobe Health Sciences]] [[Category:Public health]] bzab7qam8dgdr8ttn15wpufqgjng6og 0 239713 2812095 2207361 2026-05-30T02:41:11Z CommonsDelinker 9184 Replacing WPA_Volunteers_of_America_soup_kitchen_Washington_DC_1936.gif with [[File:Photograph_of_a_Soup_Kitchen_during_the_Depression_-_DPLA_-_97b2f78434e423409df1de0cc93f960d.gif]] (by [[:c:User:CommonsDelinker|CommonsDelinker]] because: [[:c:COM:FR|Fil 2812095 wikitext text/x-wiki {| class="wikitable" |- ! Developed countries !! Developing countries |- | US, Canada, Japan, Australia, NZ, Europe (generally) || Most are in Africa, Asia and Latin America |- | Pop.: 1.4 billion, 18% of the WP || Pop.: 6 billion, 82% of the WP |- | 85% of the world's wealth || 15% of the world's wealth |- | 88% of natural resource use || 12% of natural resource use |- | Generate 75% of pollution/waste || Generate 25% of pollution/waste |- | Growth rate: .1% ([[AP Environmental Science/CH. 1. and 2. Introduction to Environmental Science/Rule of 70|rule of 70]]: 700 years) || Growth rate: 1.5% (rule of 70: 47 years) |} ==Population Growth (World)== {| class="wikitable" |- ! Years !! Pop. Growth !! Years (Count) |- | 0-1927 || 2 billion || 18 centuries |- | 1928-1974 || 2 billion || 25 years |- | 1975-1999 || 2 billion || 25 years |- | 2000-2013 || 2 billion || 13 years |} ==What's the Differences?== [[File:Maddison GDP per capita 1500-1950.svg|thumb|right|Gross domestic product (at purchasing power parity) per capita between 1500 and 1950 in 1990 International Dollars for selected nations (https://commons.wikimedia.org/wiki/File:Maddison_GDP_per_capita_1500-1950.svg)]] [[File:Photograph of a Soup Kitchen during the Depression - DPLA - 97b2f78434e423409df1de0cc93f960d.gif|thumb|right|If a recession lasts for too long, then it eventually turns into a depression]] To judge whether a country is developed or developing, we have to take a look at its economic development. Countries like the US and Japan are considered '''developed countries''' due to their high levels of industrialization and (avg) per capita income (38,000 dollars) while '''developing countries''' like Sri Lanka, Togo, and Uzbekistan are less industrialized and have a lower (avg) per capita income (4,000 dollars). You also have to look at the '''GDP per capita''', which is the number of people living in the country divided by the country's GDP (Gross Domestic Product). The GDP is the total annual value of goods and services made by the whole population, including its people and companies, both domestic and international, within its border. The GDP shows how big the economy of the country is. If the growth rate shows a bigger growth rate than the quarter before, this means the economy is growing and is doing well for the country. If the growth rate is slower than the quarter before, then it causes a recession, a huge decline in economic activity. If the recession lasts for a while, it eventually turns into a depression (The Great Depression). Although we don't want the economic activity to decline, we also don't want it to ridiculously increase, causing inflation (fall of the general purchasing power of money). People use the GDP's of countries to compare--to see which economy is thriving more. A 100 year gap exists between developed and developing countries in terms of education. Many students in developing countries don't get their necessary education due to various issues, such as political issues or war (see https://edlab.tc.columbia.edu/blog/12261-Developed-Countries-VS-Developing-Countries). In developed countries, the population is usually more stable (steady/moderate growth rate '''of .1%''') and less infant mortality rates (number of infant deaths per 1,000 births) and total fertility rates (number of births expected by a woman through her childbearing years) than developing countries (growth rate of '''1.5%'''), who see a higher rate of infant mortality rates and total fertility rates. Poor medical care, widespread diseases, cultural practices are to blame for these high rates. Wealth is unequally distributed in developing, while developed countries have equal utilization of wealth. Developed countries have 85% of the world's wealth while developing countries have 15% of the world's wealth. In Africa, high oil prices and the absence of money made by foreign trade has led to energy poverty. Developed countries use 88% of the world's natural resource while developing countries only use 12% of the world's natural resource. [[File:Bin.JPG|thumb|left|Trash cans in developed countries have became a powerful weapon in the war against waste]] With poor work in the medical field and a lack of awareness on the topic of properly handling waste, the effect of pollution (environmental pollution, for example) has definitely taken a toll in several developing countries. In Botswana, the 11,000 kg of waste created per day caused a serious danger to community health (disease-carrying insects) and the environment (destruction of ecosystems). Seemingly small things in developed countries, such as the formation of trash cans and the law against littering, has allowed the levels of pollution to be well-managed and the waste to be under control. Practices such as source reduction (designing products to limit waste), recycling (using useful items, such as paper, to make them into new items) and composting (collecting organic waste and restricting it under conditions that allow it to break down naturally), all regular practices in the USA, have prevented risks to community health from arising. [[Category:AP Environmental Science]] [[Category:Developing Countries]] a0r1lzi1tbvrd8t9jbws3wep49w2ttx 0 263508 2811988 2811513 2026-05-29T14:31:55Z Scogdill 1331941 2811988 wikitext text/x-wiki == Overview == Prince Arthur, Duke of Connaught and Strathearn was Queen Victoria and Prince Albert's son. His is a royal rather than an inherited dukedom. == Also Known As == *Family name: he, Saxe-Coburg and Gotha (Albert's patronymic) *Arthur William Patrick Albert Connaught *Prince Arthur *Duke of Connaught and Strathearn **Arthur William Patrick Albert Connaught, 1st Duke of Connaught and Strathearn (24 May 1874 - 16 January 1942)<ref name=":0">"Arthur William Patrick Albert Saxe-Coburg and Gotha, 1st Duke of Connaught and Strathearn." "Person Page — 10066." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe'' <nowiki>http://www.thepeerage.com/p10066.htm#i100656</nowiki> (accessed May 2019).</ref> == Acquaintances, Friends and Enemies == === Friends === * Leonie, Lady Leslie (sister of Jennie Churchill) was Arthur's mistress.<ref>{{Cite journal|date=2020-09-06|title=Prince Arthur, Duke of Connaught and Strathearn|url=https://en.wikipedia.org/w/index.php?title=Prince_Arthur,_Duke_of_Connaught_and_Strathearn&oldid=977081968|journal=Wikipedia|language=en}}</ref> * [[Social Victorians/People/Adele Grant Capell|Adele Grant Capell]] after 1916 == Organizations == *[[Social Victorians/People/Royal Mob|The Royal Mob]] == Timeline == '''1874 May 24''', Prince Arthur was "created a royal peer," the Duke of Connaught and Strathearn and Earl of Sussex.<ref name=":0" /> '''1879 March 13''', Prince Arthur and Princess Louise Margaret of Prussia married.<ref name=":1">{{Cite journal|date=2020-09-06|title=Princess Louise Margaret of Prussia|url=https://en.wikipedia.org/w/index.php?title=Princess_Louise_Margaret_of_Prussia&oldid=977036048|journal=Wikipedia|language=en}}</ref> '''1897 July 2, Friday''', Prince Arthur, Duke of Connaught and Duchess of Connaught attended the Duchess of Devonshire's [[Social Victorians/1897 Fancy Dress Ball|fancy-dress ball at Devonshire House]]. (Prince Arthur, Duke of Connaught is #369 in the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of attendees]]; Princess Louise, Duchess of Connaught is #9.) == Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball == === Prince Arthur, Duke of Connaught === The portrait by Lafayette of Prince Arthur, Duke of Connaught and Strathearn at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 2 July 1897 fancy-dress ball]], as Effingham, is photograph #12 in the [[Social Victorians/1897 Fancy Dress Ball/Photographs#Album of Photographs|album presented to the Duchess of Devonshire]] and now in the National Portrait Gallery.<ref name=":6">"Devonshire House Fancy Dress Ball (1897): photogravures by Walker & Boutall after various photographers." 1899. National Portrait Gallery <nowiki>https://www.npg.org.uk/collections/search/portrait-list.php?set=515</nowiki> (accessed March 2020).</ref> The printing on the portrait says, "H.R.H. The Duke of Connaught as Effingham."<ref>"Prince Arthur, Duke of Connaught and Strathearn as Effingham." ''Devonshire House Fancy Dress Ball Album''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158364/Prince-Arthur-1st-Duke-of-Connaught-and-Strathearn-as-Effingham (accessed May 2019).</ref> This is a Lafayette portrait, but no other image of Prince Arthur in costume is in the [http://lafayette.org.uk/dhblist.html Lafayette Archive of photographs of this event].[[File:Prince-Arthur-1st-Duke-of-Connaught-and-Strathearn-as-Effingham.jpg|thumb|alt=Black-and-white photograph of a standing man richly dressed in armor and a historical costume|Prince Arthur, 1st Duke of Connaught and Strathearn as Effingham. © National Portrait Gallery, London.]]Prince Arthur had "sought the aid of Alias"<ref name=":7" />{{rp|41, Col. 1a}} — [[Social Victorians/People/Dressmakers and Costumiers#Mr. Charles Alias|Charles Alias]], the costumier. ==== Newspaper Reports of the Duke of Connaught's Costume ==== *The Duke of Connaught was "an Elizabethan General, wore a steel cuirass inlaid with gold. The dark grey velvet trunks, sleeves, and cap were slashed with grey satin embroidered in gold."<ref>"Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses." London ''Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and http://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.</ref> *"The Duke of Connaught, as a Military Commander. (Elizabethan period.) Doublet of gray velvet, with slashed sleeves of same, the puffs of gray silk, beaded with steel cut beads. Trunks of gray velvet, with slashing of gray silk embroidered gold and studded with cabochons and steel. Mantle of gray velvet, with embroidered gold bands. Cuirasse of steel damascened with gorget and ruff attached. Trunk hose gray silk. High boots of gray leather turned back. Toque of black velvet, with gray puffs and gray feathers. Orders, Riband and Badge of the Garter. Crispin gloves of gray leather. Sword belt, gray velvet with steel mountings. Sword, black velvet scabbard, steel hilt and blade."<ref name=":2">"Ball at Devonshire House." ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.</ref> *"The Duke of Connaught appeared in the uniform of a Military Commander of the Elizabethan period. The doublet and sleeves were of grey velvet, beaded with steel; the trunks of grey velvet, embroidered with gold, and a mantle of similar material with gold bands. The cuirass was of steel damascened, with gorget and ruff attached; the sword-belt of grey velvet, with steel mountings. His Royal Highness also wore the Riband and Badge of the Garter."<ref name=":8">“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.</ref>{{rp|p. 3, Col. 2c}} *"The Duke of Connaught, as a military commander of the same [Elizabethan] period, was robed in grey and black velvet."<ref name=":10">“Devonshire House Ball.” ''St. James’s Gazette'' 3 July 1897, Saturday: 8 [of 16], Col. 2a – 9, Col. 2b [of 2]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/BL/0001485/18970703/032/0008.</ref>{{rp|p. 8, Col. 2c}} *"THE DUKE OF CONNAUGHT as a Military Commander. (Elizabethan period). Doublet of grey velvet, with [?] sleeves of same, the puffs of grey silk, with steel cut beads. Trunks of grey velvet with slashing of grey silk embroidered gold and studded with cabochons and steel. Mantle of grey velvet with embroidered gold bands. Cuirasse [sic] of steel, damascened with gorget, and ruff attached. Trunk hose grey silk. High boots of leather turned back. Toque of black velvet with grey puffs and grey feathers. Orders — Ribbon and badge of the Garter. Crispin gloves of grey leather. Sword-belt — Grey velvet with steel mountings. Sword, black velvet scabbard, steel hilt and blade."<ref name=":9">“A Jubilee Ball. Brilliant Scene at Devonshire House. Some of the Costumes Worn.” The London ''Echo'' 3 July 1897, Saturday: 2 [of 4], Cols. 6a – 7a [of 7]. ''British Newspaper Archiv''e https://www.britishnewspaperarchive.co.uk/viewer/bl/0004596/18970703/027/0002.</ref>{{rp|2, Col. 6b}} *"The Duke of Connaught personated the Commander of the Forces in the time of Queen Elizabeth, wearing a steel and gold cuirass, with doublets and trunks of grey velvet and satin, a mantle to match, a jewelled rapier, and a grey velvet toque finished with jewels and white plumes."<ref name=":5">"The Duchess of Devonshire's Fancy Dress Ball. Special Telegram." ''Belfast News-Letter'' Saturday 03 July 1897: 5 [of 8], Col. 9 [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0000038/18970703/015/0005.</ref>{{rp|p. 5, Col. 9a}} *"[T]he Duke of Connaught as an Elizabethan General, looked extremely well in his steel cuirass, inlaid with gold, and dark grey velvet doublet, and trunks slashed with gold-embroidered grey satin."<ref name=":11">“Girls’ Gossip.” ''Truth'' 8 July 1897, Thursday: 41 [of 70], Col. 1b – 42, Col. 2c. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0002961/18970708/089/0041.</ref>{{rp|41, Col. 2a}} *"H.R.H. the Duke of Connaught was a Military Commander (Elizabethan period), and wore a doublet of grey velvet, with slashed sleeves of same; trunks of grey velvet, with slashing of grey silk; mantle of grey velvet, with embroidered gold bands; cuirass of steel, damascened with gorget, and ruff attached; high boots of grey leather, turned back. Orders, ribbon and badge of the Garter."<ref name=":7" />{{rp|p. 41, Col. 1b}} *This description accompanies a flattering line drawing of the Duke in costume (Top middle drawing, with sword, signed “Rook”): "T<small>HE</small> D<small>UKE OF</small> C<small>ONNAUGHT</small>, as a Military Commander (Elizabethan period). — Doublet of grey velvet with slashed sleeves of same, the puffs of grey silk, beaded with steel cut beads; trunks of grey velvet with slashing of grey silk embroidered gold, and studded with cabochons and steel; mantle of grey velvet with embroidered gold bands; cuirasse of steel damascened with gorget and ruff attached; trunk hose grey silk; high boots of grey leather turned back; toque of black velvet with grey puffs and grey feathers; Orders, ribbon and badge of the Garter; Crispin gloves of grey leather; sword belt, grey velvet with steel mountings; sword, black velvet scabbard, steel hilt and blade. Made by Alias, 36, Soho-square."<ref>“Dress at Devonshire House on July 2.” The ''Queen, The Lady’s Newspaper'' 10 July 1897, Saturday: 39 [of 98 in BNA; p. 65 on print page], full page [3 of 3 cols.]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/BL/0002627/18970710/137/0039.</ref>{{rp|Col. 2a–b}} Prince Arthur is not wearing boots in the drawing, although the description says he wore "high boots." *"The Duke of Connaught as a military commander — Elizabethan period — wore an effective dress of grey velvet and satin, with a steel breast-plate."<ref name=":3">“The Devonshire House Ball.” ''The Man of Ross'' 10 July 1897, Saturday: 2 [of 8], Col. 4B. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0001463/18970710/033/0002.</ref> [[File:Order of the Garter badge (United Kingdom) - Tallinn Museum of Orders.jpg|thumb|alt=Gold carved pendant representing the Badge of the Order of the Garter|Badge of the Order of the Garter (United Kingdom)]] ==== Commentary on His Costume ==== * Hanging on what is probably a royal-blue ribbon around his neck is the Badge of the Order of the Garter (right). * Prince Arthur's cuirass is damascened, that is, decorated with a brocade-type floral pattern made by gold inlaid into the steel. * His gorget merges in the photograph with the points of the collar on his mantle or cloak. It lies more or less flat on his chest and shoulders, covering the top of his cuirass. * Prince Arthur's pumpkin breeches are slashed with his shirt showing between the ribbons of the slashing. The shirt would be pulled through the gaps made by the slashing if the doublet sleeves were actually slashed. With their many, very small "slashes," trim and stripes going in 2 directions, the doublet sleeves are only made to look as if they are slashed. The hat also looks as if it were slashed, but the effect is made by fabric pulled through loops of trimmed ribbon. * While the newspapers say he is wearing a mantle (possibly copied from the description in the ''Times''), this term is likely a Victorian anachronism. Although the terms were (and are) often used generically, a short outer wrap would be a cape, a longer one would be a cloak and, after the 17th century, a full-length one possibly buttoned down the front would be a mantle. *His "high boots" come to mid thigh, with a turned down cuff. The leather wrinkles, suggesting that it is a soft leather. The bands around his instep are probably for holding spurs. *We're not certain what the newspapers mean by "Crispin gloves". The St. Crispin brothers were the patron saints of leather workers. *The newspaper say he is wearing a ruff and a gorget, but the ruff is really a ruffle. === Princess Louise, Duchess of Connaught === [[File:Princess-Louise-Duchess-of-Connaught-ne-Princess-of-Prussia-as-Anne-of-Austria.jpg|thumb|alt=Black-and-white photograph of a standing woman richly dressed in a historical costume|Princess Louise, Duchess of Connaught as Anne of Austria. © National Portrait Gallery, London.]]Princess Louise, Duchess of Connaught (at 9) sat at Table 8 in the first seating for supper, escorted to the table by [[Social Victorians/People/Mensdorff|Count Mensdorff]]. She was born Princess of Prussia. The portrait by Lafayette (right) of Princess Louise, Duchess of Connaught (née Princess of Prussia) as Anne of Austria is photograph #11 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.<ref name=":6" /> The printing on the portrait says, "H.R.H. The Duchess of Connaught as Anne of Austria," with a Long S in ''Duchess''.<ref>"Princess Louise, Duchess of Connaught (née Princess of Prussia) as Anne of Austria." ''Devonshire House Fancy Dress Ball Album''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158363/Princess-Louise-Duchess-of-Connaught-ne-Princess-of-Prussia-as-Anne-of-Austria (accessed May 2019).</ref> ==== Newspaper Descriptions of Her Costume ==== *"Ann of Austria. Robe of old ciselé velvet, havana colour, the turnback of skirt of rose colour silk velvet embroidered silver. Panel of havana colour silk velvet embroidered silver. Front of dress of white satin with embroidery of gold fleur-de-lys and beautiful bordered collar and cuffs of old guipure lace with sémé of pearls. Very simple headdress. Bandeau, pearl and gold and plume Ecran of feathers in hand. Handsome jewelled necklace and earrings."<ref name=":2" /> *She was dressed as a "Lady of the Court of Charles II. Flowered brocade in shaded apricot velvet on a deep cream satin ground, the fronts turned back with pink silk, richly embroidered in gold and a deeper shade of pink, opening to show an under-dress of white satin embroidered in gold fleur-de-lis. The bodice of brocade had a white lace collar over satin; a white satin stomacher with four pink rosettes down the front; and sleeves formed of two puffs of white satin trimmed with bands of embroidery edged with silver."<ref name=":4">"Fancy Dress Ball at Devonshire House." ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.</ref>{{rp|p. 7, Col. 6c}} *Her costume was "supplied by Mr. Alias, of Soho-square."<ref name=":4" />{{rp|p. 8, Col. 2a}} *Mr. W. Clarkson "supplied the wigs and headdresses for the Royal Family."<ref name=":4" />{{rp|p. 8, Col. 2a}} *The Duchess of Connaught was "wearing a lovely gown of brown velvet embroidered with silver and old lace."<ref name=":10" />{{rp|p. 8, Col. 2c}} *"The Duchess of Connaught came as Anne of Austria, Queen of France, in a robe of ciselé velvet, havana-coloured, the turnback of the skirt being of rose-coloured silk velvet, embroidered with silver. The front of the dress was of white satin, embroidered with gold fleur de lys, and the collar and cuffs were of old guipure lace gown [sic] with pearls. Her Royal Highness wore a handsome jewelled necklace and ear-rings."<ref name=":8" />{{rp|p. 3, Col. 2c}} *"DUCHESS OF CONNAUGHT as Ann of Austria. Robe of old ciselé velvet, havana colour, the turnback of skirt of rose-colour silk velvet embroidered silver. Panel of havana colour silk velvet embroidered silver. Front of dress of white satin, with embroidery of gold fleur de lys and beautiful bordered collar and cuffs of old guipure lace with sémé of [/] pearls. Very simple headdress, Bandeau, pearl and gold, and plume Ecran [sic] of feathers in band. Handsome jewelled necklace and earrings."<ref name=":9" />{{rp|p. 2, Col. 6c–Col. 7a}} *"Anne of Austria, in a superb cream satin gown, brocaded in raised velvet leaves in delicate hues of fawn and green. This was turned back with rich gold embroidery to show a petticoat of white satin, worked over in gold fleur-de-lys. The stomacher was encrusted with jewels, and there was a high lace collar and a low crown in the hair."<ref name=":5" />{{rp|p. 5, Col. 9b}} The first sentence of this ''Belfast News-Letter'' article is identical to the story in the ''Carlisle Patriot''<ref>"Fancy Dress Ball: Unparalleled Splendour." ''Carlisle Patriot'' Friday 9 July 1897: 7 [of 8], Col. 4a–b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000365/18970709/084/0007.</ref>). *"The Duchess of Connaught was Anne of Austria, in amber velvet brocade, her hair arranged in short ringlets under a jewelled cap. The high Vandyck collar was thickly sewn with pearls and gold."<ref name=":11" />{{rp|41, Col. 2a}} *This description accompanies a flattering line drawing of the Duchess of Connaught in costume (Numeral 5 below the drawing, bottom right drawing, head turned to her right, body slightly turned to the right, dress differs in a few ways from the photograph portrait, feather fan): "Made by Alias, 36, Soho Square. ... No. 5. T<small>HE</small> D<small>UCHESS OF</small> C<small>ONNAUGHT</small>, Anne of Austria. — Robe of old [col. 2c / 3c ] ciselé velvet, Havana colour turned back with rose silk velvet, embroidered with silver. Panels of Havana velvet, embroidered with silver. The front of the dress of white satin, embroidered with gold fleur de lys. Collar and cuffs of old guipure, with sémé of pearls. Headdress, pearl and gold plumes."<ref>“Dress at Devonshire House on July 2.” The ''Queen, The Lady’s Newspaper'' 10 July 1897, Saturday: 39 [of 98 in BNA; p. 65 on print page], full page [3 of 3 cols.]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/BL/0002627/18970710/137/0039.</ref>{{rp|Cols. 2–3b–c}} *"The Duchess of Connaught personated Anne of Austria, in a very handsome gown of brocaded velvet, the fronts turned back with rose velvet, embroidered in silver. The white satin front was enriched with gold fleur de lys, and there were cuffs and a collar of beautiful lace."<ref name=":3" /> *"H.R.H. the Duchess of Connaught, as Anne of Austria, ... looked [her character] very well."<ref name=":7">“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.</ref>{{rp|p. 34, Col. 1a}} == Demographics == *Nationality: he, English; she, Prussian === Residences === == Family == *Arthur William Patrick Albert (1 May 1850 – 16 January 1942)<ref name=":0" /> *Princess Louise Margaret Alexandra Victoria Agnes of Prussia (25 July 1860 – 14 March 1917)<ref name=":1" /> #Princess Margaret Victoria Charlotte Augusta Norah (15 January 1882 – 1 May 1920) #Prince Arthur Frederick Patrick Albert (13 January 1883 – 12 September 1938) #Princess Victoria Patricia Helena Elizabeth (17 March 1886 – 12 January 1974) === Relations === *[[Social Victorians/People/Queen Victoria | Queen Victoria]] and Albert's 7th child, 3rd son == Notes and Questions == # Prince Arthur and Princess Louise Margaret of Prussia married on 13 March 1879. Her bridesmaids were Lady Cecilia Hay (daughter of the Earl of Erroll, married Captain Webbe), Lady Victoria Edgecumbe (daughter of the Earl of Mount Edgcumbe, married Lord Algernon Percy), Lady Ela Russell (daughter of the Duke of Bedford), Lady Georgiana Churchill (daughter of the Duke of Marlborough, married Lord Curzon), Lady Blanche Conyngham (daughter of Marquis Conyngham), Lady Adelaide Taylour (daughter of the Marquis of Headfort), Lady Louisa Bruce (daughter of the Earl of Elgin), Lady Mabel Bridgeman (daughter of the Earl of Bradford, married Lieutenant-Colonel Kenyon Slaney).<ref>"Bridesmaids at Royal Weddings." ''Gentlewoman'' 08 July 1893 Saturday: 20 [of 56], Col. 2c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18930708/117/0020.</ref> == Footnotes == {{reflist}} 6rvsccxz49r2kj85a7bc0trcr8nq4ti 2812055 2811988 2026-05-29T18:27:56Z Scogdill 1331941 2812055 wikitext text/x-wiki == Overview == Prince Arthur, Duke of Connaught and Strathearn was Queen Victoria and Prince Albert's son. His is a royal rather than an inherited dukedom. == Also Known As == *Family name: he, Saxe-Coburg and Gotha (Albert's patronymic) *Arthur William Patrick Albert Connaught *Prince Arthur *Duke of Connaught and Strathearn **Arthur William Patrick Albert Connaught, 1st Duke of Connaught and Strathearn (24 May 1874 - 16 January 1942)<ref name=":0">"Arthur William Patrick Albert Saxe-Coburg and Gotha, 1st Duke of Connaught and Strathearn." "Person Page — 10066." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe'' <nowiki>http://www.thepeerage.com/p10066.htm#i100656</nowiki> (accessed May 2019).</ref> == Acquaintances, Friends and Enemies == === Friends === * Leonie, Lady Leslie (sister of Jennie Churchill) was Arthur's mistress.<ref>{{Cite journal|date=2020-09-06|title=Prince Arthur, Duke of Connaught and Strathearn|url=https://en.wikipedia.org/w/index.php?title=Prince_Arthur,_Duke_of_Connaught_and_Strathearn&oldid=977081968|journal=Wikipedia|language=en}}</ref> * [[Social Victorians/People/Adele Grant Capell|Adele Grant Capell]] after 1916 == Organizations == *[[Social Victorians/People/Royal Mob|The Royal Mob]] == Timeline == '''1874 May 24''', Prince Arthur was "created a royal peer," the Duke of Connaught and Strathearn and Earl of Sussex.<ref name=":0" /> '''1879 March 13''', Prince Arthur and Princess Louise Margaret of Prussia married.<ref name=":1">{{Cite journal|date=2020-09-06|title=Princess Louise Margaret of Prussia|url=https://en.wikipedia.org/w/index.php?title=Princess_Louise_Margaret_of_Prussia&oldid=977036048|journal=Wikipedia|language=en}}</ref> '''1897 July 2, Friday''', Prince Arthur, Duke of Connaught and Duchess of Connaught attended the Duchess of Devonshire's [[Social Victorians/1897 Fancy Dress Ball|fancy-dress ball at Devonshire House]]. (Prince Arthur, Duke of Connaught is #369 in the [[Social Victorians/1897 Fancy Dress Ball#List of People Who Attended|list of attendees]]; Princess Louise, Duchess of Connaught is #9.) == Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball == === Prince Arthur, Duke of Connaught === The portrait by Lafayette of Prince Arthur, Duke of Connaught and Strathearn at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 2 July 1897 fancy-dress ball]], as Effingham, is photograph #12 in the [[Social Victorians/1897 Fancy Dress Ball/Photographs#Album of Photographs|album presented to the Duchess of Devonshire]] and now in the National Portrait Gallery.<ref name=":6">"Devonshire House Fancy Dress Ball (1897): photogravures by Walker & Boutall after various photographers." 1899. National Portrait Gallery <nowiki>https://www.npg.org.uk/collections/search/portrait-list.php?set=515</nowiki> (accessed March 2020).</ref> The printing on the portrait says, "H.R.H. The Duke of Connaught as Effingham."<ref>"Prince Arthur, Duke of Connaught and Strathearn as Effingham." ''Devonshire House Fancy Dress Ball Album''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158364/Prince-Arthur-1st-Duke-of-Connaught-and-Strathearn-as-Effingham (accessed May 2019).</ref> This is a Lafayette portrait, but no other image of Prince Arthur in costume is in the [http://lafayette.org.uk/dhblist.html Lafayette Archive of photographs of this event].[[File:Prince-Arthur-1st-Duke-of-Connaught-and-Strathearn-as-Effingham.jpg|thumb|alt=Black-and-white photograph of a standing man richly dressed in armor and a historical costume|Prince Arthur, 1st Duke of Connaught and Strathearn as Effingham. © National Portrait Gallery, London.]]Prince Arthur had "sought the aid of Alias"<ref name=":7" />{{rp|41, Col. 1a}} — [[Social Victorians/People/Dressmakers and Costumiers#Mr. Charles Alias|Charles Alias]], the costumier. ==== Newspaper Reports of the Duke of Connaught's Costume ==== *The Duke of Connaught was "an Elizabethan General, wore a steel cuirass inlaid with gold. The dark grey velvet trunks, sleeves, and cap were slashed with grey satin embroidered in gold."<ref>"Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses." London ''Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and http://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.</ref> *"The Duke of Connaught, as a Military Commander. (Elizabethan period.) Doublet of gray velvet, with slashed sleeves of same, the puffs of gray silk, beaded with steel cut beads. Trunks of gray velvet, with slashing of gray silk embroidered gold and studded with cabochons and steel. Mantle of gray velvet, with embroidered gold bands. Cuirasse of steel damascened with gorget and ruff attached. Trunk hose gray silk. High boots of gray leather turned back. Toque of black velvet, with gray puffs and gray feathers. Orders, Riband and Badge of the Garter. Crispin gloves of gray leather. Sword belt, gray velvet with steel mountings. Sword, black velvet scabbard, steel hilt and blade."<ref name=":2">"Ball at Devonshire House." ''Times'' Saturday 3 July 1897: 12, Cols. 1a–4c ''The Times Digital Archive''. Web. 28 Nov. 2015.</ref> *"The Duke of Connaught appeared in the uniform of a Military Commander of the Elizabethan period. The doublet and sleeves were of grey velvet, beaded with steel; the trunks of grey velvet, embroidered with gold, and a mantle of similar material with gold bands. The cuirass was of steel damascened, with gorget and ruff attached; the sword-belt of grey velvet, with steel mountings. His Royal Highness also wore the Riband and Badge of the Garter."<ref name=":8">“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.</ref>{{rp|p. 3, Col. 2c}} *"The Duke of Connaught, as a military commander of the same [Elizabethan] period, was robed in grey and black velvet."<ref name=":10">“Devonshire House Ball.” ''St. James’s Gazette'' 3 July 1897, Saturday: 8 [of 16], Col. 2a – 9, Col. 2b [of 2]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/BL/0001485/18970703/032/0008.</ref>{{rp|p. 8, Col. 2c}} *"THE DUKE OF CONNAUGHT as a Military Commander. (Elizabethan period). Doublet of grey velvet, with [?] sleeves of same, the puffs of grey silk, with steel cut beads. Trunks of grey velvet with slashing of grey silk embroidered gold and studded with cabochons and steel. Mantle of grey velvet with embroidered gold bands. Cuirasse [sic] of steel, damascened with gorget, and ruff attached. Trunk hose grey silk. High boots of leather turned back. Toque of black velvet with grey puffs and grey feathers. Orders — Ribbon and badge of the Garter. Crispin gloves of grey leather. Sword-belt — Grey velvet with steel mountings. Sword, black velvet scabbard, steel hilt and blade."<ref name=":9">“A Jubilee Ball. Brilliant Scene at Devonshire House. Some of the Costumes Worn.” The London ''Echo'' 3 July 1897, Saturday: 2 [of 4], Cols. 6a – 7a [of 7]. ''British Newspaper Archiv''e https://www.britishnewspaperarchive.co.uk/viewer/bl/0004596/18970703/027/0002.</ref>{{rp|2, Col. 6b}} *"The Duke of Connaught personated the Commander of the Forces in the time of Queen Elizabeth, wearing a steel and gold cuirass, with doublets and trunks of grey velvet and satin, a mantle to match, a jewelled rapier, and a grey velvet toque finished with jewels and white plumes."<ref name=":5">"The Duchess of Devonshire's Fancy Dress Ball. Special Telegram." ''Belfast News-Letter'' Saturday 03 July 1897: 5 [of 8], Col. 9 [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0000038/18970703/015/0005.</ref>{{rp|p. 5, Col. 9a}} *"[T]he Duke of Connaught as an Elizabethan General, looked extremely well in his steel cuirass, inlaid with gold, and dark grey velvet doublet, and trunks slashed with gold-embroidered grey satin."<ref name=":11">“Girls’ Gossip.” ''Truth'' 8 July 1897, Thursday: 41 [of 70], Col. 1b – 42, Col. 2c. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0002961/18970708/089/0041.</ref>{{rp|41, Col. 2a}} *"H.R.H. the Duke of Connaught was a Military Commander (Elizabethan period), and wore a doublet of grey velvet, with slashed sleeves of same; trunks of grey velvet, with slashing of grey silk; mantle of grey velvet, with embroidered gold bands; cuirass of steel, damascened with gorget, and ruff attached; high boots of grey leather, turned back. Orders, ribbon and badge of the Garter."<ref name=":7" />{{rp|p. 41, Col. 1b}} *This description accompanies a flattering line drawing of the Duke in costume (Top middle drawing, with sword, signed “Rook”): "T<small>HE</small> D<small>UKE OF</small> C<small>ONNAUGHT</small>, as a Military Commander (Elizabethan period). — Doublet of grey velvet with slashed sleeves of same, the puffs of grey silk, beaded with steel cut beads; trunks of grey velvet with slashing of grey silk embroidered gold, and studded with cabochons and steel; mantle of grey velvet with embroidered gold bands; cuirasse of steel damascened with gorget and ruff attached; trunk hose grey silk; high boots of grey leather turned back; toque of black velvet with grey puffs and grey feathers; Orders, ribbon and badge of the Garter; Crispin gloves of grey leather; sword belt, grey velvet with steel mountings; sword, black velvet scabbard, steel hilt and blade. Made by Alias, 36, Soho-square."<ref>“Dress at Devonshire House on July 2.” The ''Queen, The Lady’s Newspaper'' 10 July 1897, Saturday: 39 [of 98 in BNA; p. 65 on print page], full page [3 of 3 cols.]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/BL/0002627/18970710/137/0039.</ref>{{rp|Col. 2a–b}} Prince Arthur is not wearing boots in the drawing, although the description says he wore "high boots." *"The Duke of Connaught as a military commander — Elizabethan period — wore an effective dress of grey velvet and satin, with a steel breast-plate."<ref name=":3">“The Devonshire House Ball.” ''The Man of Ross'' 10 July 1897, Saturday: 2 [of 8], Col. 4B. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0001463/18970710/033/0002.</ref> [[File:Order of the Garter badge (United Kingdom) - Tallinn Museum of Orders.jpg|thumb|alt=Gold carved pendant representing the Badge of the Order of the Garter|Badge of the Order of the Garter (United Kingdom)]] ==== Commentary on His Costume ==== * Hanging on what is probably a royal-blue ribbon around his neck is the Badge of the Order of the Garter (right). * Prince Arthur's cuirass is damascened, that is, decorated with a brocade-type floral pattern made by gold inlaid into the steel. * His gorget merges in the photograph with the points of the collar on his mantle or cloak. It lies more or less flat on his chest and shoulders, covering the top of his cuirass. * Prince Arthur's pumpkin breeches are slashed with his shirt showing between the ribbons of the slashing. The shirt would be pulled through the gaps made by the slashing if the doublet sleeves were actually slashed. With their many, very small "slashes," trim and stripes going in 2 directions, the doublet sleeves are only made to look as if they are slashed. The hat also looks as if it were slashed, but the effect is made by fabric pulled through loops of trimmed ribbon. * While the newspapers say he is wearing a mantle (possibly copied from the description in the ''Times''), this term is likely a Victorian anachronism. Although the terms were (and are) often used generically, a short outer wrap would be a cape, a longer one would be a cloak and, after the 17th century, a full-length one possibly buttoned down the front would be a mantle. *His "high boots" come to mid thigh, with a turned down cuff. The leather wrinkles, suggesting that it is a soft leather. The bands around his instep are probably for holding spurs. *We're not certain what the newspapers mean by "Crispin gloves." The St. Crispin brothers were the patron saints of leather workers. *The newspaper say he is wearing a ruff and a gorget, but the ruff is really a ruffle. === Princess Louise, Duchess of Connaught === [[File:Princess-Louise-Duchess-of-Connaught-ne-Princess-of-Prussia-as-Anne-of-Austria.jpg|thumb|alt=Black-and-white photograph of a standing woman richly dressed in a historical costume|Princess Louise, Duchess of Connaught as Anne of Austria. © National Portrait Gallery, London.]]Princess Louise, Duchess of Connaught (at 9) sat at Table 8 in the first seating for supper, escorted to the table by [[Social Victorians/People/Mensdorff|Count Mensdorff]]. She was born Princess of Prussia. The portrait by Lafayette (right) of Princess Louise, Duchess of Connaught (née Princess of Prussia) as Anne of Austria is photograph #11 in the [[Social Victorians/1897 Fancy Dress Ball/Photographs#The Album of Photographs|album presented to the Duchess of Devonshire]] and now in the National Portrait Gallery.<ref name=":6" /> The printing on the portrait says, "H.R.H. The Duchess of Connaught as Anne of Austria," with a Long S in ''Duchess''.<ref>"Princess Louise, Duchess of Connaught (née Princess of Prussia) as Anne of Austria." ''Devonshire House Fancy Dress Ball Album''. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158363/Princess-Louise-Duchess-of-Connaught-ne-Princess-of-Prussia-as-Anne-of-Austria (accessed May 2019).</ref> Even though this portrait was taken by Lafayette, no copy of it or another pose exists in the [http://lafayette.org.uk/dhblist.html Lafayette Negative Archive]. As part of the party of Royals, the Duchess probably did not walk in a procession. ==== Newspaper Descriptions of Her Costume ==== *"Ann of Austria. Robe of old ciselé velvet, havana colour, the turnback of skirt of rose colour silk velvet embroidered silver. Panel of havana colour silk velvet embroidered silver. Front of dress of white satin with embroidery of gold fleur-de-lys and beautiful bordered collar and cuffs of old guipure lace with sémé of pearls. Very simple headdress. Bandeau, pearl and gold and plume Ecran of feathers in hand. Handsome jewelled necklace and earrings."<ref name=":2" /> *She was dressed as a "Lady of the Court of Charles II. Flowered brocade in shaded apricot velvet on a deep cream satin ground, the fronts turned back with pink silk, richly embroidered in gold and a deeper shade of pink, opening to show an under-dress of white satin embroidered in gold fleur-de-lis. The bodice of brocade had a white lace collar over satin; a white satin stomacher with four pink rosettes down the front; and sleeves formed of two puffs of white satin trimmed with bands of embroidery edged with silver."<ref name=":4">"Fancy Dress Ball at Devonshire House." ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.</ref>{{rp|p. 7, Col. 6c}} *Her costume was "supplied by Mr. Alias, of Soho-square."<ref name=":4" />{{rp|p. 8, Col. 2a}} *Mr. W. Clarkson "supplied the wigs and headdresses for the Royal Family."<ref name=":4" />{{rp|p. 8, Col. 2a}} *The Duchess of Connaught was "wearing a lovely gown of brown velvet embroidered with silver and old lace."<ref name=":10" />{{rp|p. 8, Col. 2c}} *"The Duchess of Connaught came as Anne of Austria, Queen of France, in a robe of ciselé velvet, havana-coloured, the turnback of the skirt being of rose-coloured silk velvet, embroidered with silver. The front of the dress was of white satin, embroidered with gold fleur de lys, and the collar and cuffs were of old guipure lace gown [sic] with pearls. Her Royal Highness wore a handsome jewelled necklace and ear-rings."<ref name=":8" />{{rp|p. 3, Col. 2c}} *"DUCHESS OF CONNAUGHT as Ann of Austria. Robe of old ciselé velvet, havana colour, the turnback of skirt of rose-colour silk velvet embroidered silver. Panel of havana colour silk velvet embroidered silver. Front of dress of white satin, with embroidery of gold fleur de lys and beautiful bordered collar and cuffs of old guipure lace with sémé of [Col. 6c–7a] pearls. Very simple headdress, Bandeau, pearl and gold, and plume Ecran [sic] of feathers in band. Handsome jewelled necklace and earrings."<ref name=":9" />{{rp|p. 2, Col. 6c–Col. 7a}} *"Anne of Austria, in a superb cream satin gown, brocaded in raised velvet leaves in delicate hues of fawn and green. This was turned back with rich gold embroidery to show a petticoat of white satin, worked over in gold fleur-de-lys. The stomacher was encrusted with jewels, and there was a high lace collar and a low crown in the hair."<ref name=":5" />{{rp|p. 5, Col. 9b}} (The first sentence of this ''Belfast News-Letter'' article is identical to the story in the ''Carlisle Patriot''<ref>"Fancy Dress Ball: Unparalleled Splendour." ''Carlisle Patriot'' Friday 9 July 1897: 7 [of 8], Col. 4a–b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000365/18970709/084/0007.</ref>, so copy-and-paste journalism is at work in this desription as well.) *"The Duchess of Connaught was Anne of Austria, in amber velvet brocade, her hair arranged in short ringlets under a jewelled cap. The high Vandyck collar was thickly sewn with pearls and gold."<ref name=":11" />{{rp|41, Col. 2a}} *This description accompanies a flattering line drawing of the Duchess of Connaught in costume (Numeral 5 below the drawing, bottom right drawing, head turned to her right, body slightly turned to the right, dress differs in a few ways from the photograph portrait, feather fan): "Made by Alias, 36, Soho Square. ... No. 5. T<small>HE</small> D<small>UCHESS OF</small> C<small>ONNAUGHT</small>, Anne of Austria. — Robe of old [Col. 2c–3c ] ciselé velvet, Havana colour turned back with rose silk velvet, embroidered with silver. Panels of Havana velvet, embroidered with silver. The front of the dress of white satin, embroidered with gold fleur de lys. Collar and cuffs of old guipure, with sémé of pearls. Headdress, pearl and gold plumes."<ref>“Dress at Devonshire House on July 2.” The ''Queen, The Lady’s Newspaper'' 10 July 1897, Saturday: 39 [of 98 in BNA; p. 65 on print page], full page [3 of 3 cols.]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/BL/0002627/18970710/137/0039.</ref>{{rp|Cols. 2–3b–c}} *"The Duchess of Connaught personated Anne of Austria, in a very handsome gown of brocaded velvet, the fronts turned back with rose velvet, embroidered in silver. The white satin front was enriched with gold fleur de lys, and there were cuffs and a collar of beautiful lace."<ref name=":3" /> *"H.R.H. the Duchess of Connaught, as Anne of Austria, ... looked [her character] very well."<ref name=":7">“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.</ref>{{rp|p. 34, Col. 1a}} ==== Commentary on Her Costume ==== The newspaper accounts do not always agree, and the ones that do agree are copies — rather than confirmation — of each other. The artist of the line drawing in the ''Queen'' may have seen the costume in the costumier's shop, and other drawings from this issue of the ''Queen'' suggest that the accessories were not complete at the time of the drawing. We know that Charles Alias, who made this costume, provided descriptions about costumes to the newspapers, which may have standardized the language. The repeated description of the Duchess's costume includes technical terms, suggesting further that the source may be Charles Alias: [[Social Victorians/Terminology#Ciselé|ciselé]], [[Social Victorians/Terminology#Guipure|guipure]], havana color, sémé, bandeau, Ecran. ==== The Historical Anne of Austria ==== Anne of Austria (1601–1666) was queen consort of Louis XIII and mother and regent of Louis XIV until he came of age. == Demographics == *Nationality: he, English; she, Prussian === Residences === == Family == *Arthur William Patrick Albert (1 May 1850 – 16 January 1942)<ref name=":0" /> *Princess Louise Margaret Alexandra Victoria Agnes of Prussia (25 July 1860 – 14 March 1917)<ref name=":1" /> #Princess Margaret Victoria Charlotte Augusta Norah (15 January 1882 – 1 May 1920) #Prince Arthur Frederick Patrick Albert (13 January 1883 – 12 September 1938) #Princess Victoria Patricia Helena Elizabeth (17 March 1886 – 12 January 1974) === Relations === *[[Social Victorians/People/Queen Victoria | Queen Victoria]] and Albert's 7th child, 3rd son == Notes and Questions == # Prince Arthur and Princess Louise Margaret of Prussia married on 13 March 1879. Her bridesmaids were Lady Cecilia Hay (daughter of the Earl of Erroll, married Captain Webbe), Lady Victoria Edgecumbe (daughter of the Earl of Mount Edgcumbe, married Lord Algernon Percy), Lady Ela Russell (daughter of the Duke of Bedford), Lady Georgiana Churchill (daughter of the Duke of Marlborough, married Lord Curzon), Lady Blanche Conyngham (daughter of Marquis Conyngham), Lady Adelaide Taylour (daughter of the Marquis of Headfort), Lady Louisa Bruce (daughter of the Earl of Elgin), Lady Mabel Bridgeman (daughter of the Earl of Bradford, married Lieutenant-Colonel Kenyon Slaney).<ref>"Bridesmaids at Royal Weddings." ''Gentlewoman'' 08 July 1893 Saturday: 20 [of 56], Col. 2c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18930708/117/0020.</ref> == Footnotes == {{reflist}} d3njmgnradazcqk9e669g8omqfefmmb 0 265123 2812065 2790510 2026-05-29T21:31:35Z Scogdill 1331941 2812065 wikitext text/x-wiki ==Overview== David Cannadine says,<blockquote>The most successful patrician courtiers established an abiding dynastic connection, so that generation after generation, their families enjoyed royal favour and preferment. From the late nineteenth century until the Second World War, the Ponsonbys were the pre-eminent courtly dynasty. They were very aristocratic but not very rich, being a cadet branch of the Earls of Bessborough. Sir Henry Ponsonby, grandson of the third earl, became the Queen's private secretary in 1870, largely because his predecessor, General Grey, was his wife's uncle. He held the position until his death in 1895, by which time the Queen had already appointed his second son, Frederick, as an Equerry.<ref name=":2" />{{rp|247}}</blockquote>Cannadine says, "Sir Henry Ponsonby effectively created the post of Private Secretary to the Sovereign in its modern guise."<ref name=":2">Cannadine, David. ''The Decline and Fall of the British Aristocracy''. New York: Yale University Press, 1990.</ref>{{rp|245}} Queen Victoria's replacement for General Grey as her private secretary<blockquote>was Colonel Henry Ponsonby, who had been at Court since 1857. Ponsonby, himself from an aristocratic Whig family, was married to Mary Bulteel, a niece of General Grey. The Ponsonbys were, on superficial levels, surprising choices for the Queen. While she was increasingly Tory, they were unabashed Gladstonian Liberals. In religion, they were High Church, like Gladstone, whereas the Queen was her own distinctive brand of Broad Church Pantheist/Presbyterian. Mary Ponsonby was highly educated, and a feminist; Queen Victoria deplored feminism.<ref>Wilson, A. N. ''Victoria: A Life''. Penguin, 2014. Apple Books: https://books.apple.com/us/book/victoria/id828766078.</ref> (596 of 1204)</blockquote> == Also Known As == * Family name: Ponsonby *Baron Sysonby **Fritz (Frederick Edward Grey) Ponsonby, 1st Baron Sysonby ()<ref>{{Cite journal|date=2020-11-21|title=Frederick Ponsonby, 1st Baron Sysonby|url=https://en.wikipedia.org/w/index.php?title=Frederick_Ponsonby,_1st_Baron_Sysonby&oldid=989930883|journal=Wikipedia|language=en}}</ref> *Cannadine says, "very aristocratic but not very rich, being a cadet branch of the Earls of Bessborough."<ref name=":2" />{{rp|247}} == Acquaintances, Friends and Enemies == == Organizations == === Fritz (Frederick Edward Grey) Ponsonby === ==== Queen Victoria's Household ==== * Equerry-in-Ordinary (1894–1901) * Assistant Keeper of the Privy Purse and Assistant Private Secretary to Queen Victoria (1897–1901) ==== King Edward VII's Household ==== * Assistant Keeper of the Privy Purse and Assistant Private Secretary to King Edward VII (1901–1910) ==== King George V's Household ==== * Assistant Keeper of the Privy Purse and Assistant Private Secretary to King George V (1910–1914) * Keeper of the Privy Purse to King George V (1914–1935) * Lieutenant Governor of Windsor Castle (1928–1935) ==Timeline== '''1870 April 8''', Henry Frederick Ponsonby was appointed Queen Victoria's Private Secretary and Keeper of the Privy Purse.<ref name=":0">{{Cite journal|date=2020-12-05|title=Henry Ponsonby|url=https://en.wikipedia.org/w/index.php?title=Henry_Ponsonby&oldid=992528585|journal=Wikipedia|language=en}}</ref> == Demographics == * Nationality: British ==Family== * Hon. Sir Frederick Cavendish Ponsonby (6 July 1783 – 11 January 1837)<ref>{{Cite journal|date=2020-10-18|title=Frederick Cavendish Ponsonby|url=https://en.wikipedia.org/w/index.php?title=Frederick_Cavendish_Ponsonby&oldid=984182186|journal=Wikipedia|language=en}}</ref> * Lady Emily Charlotte Bathurst ( – 1877) *# '''Sir Henry Frederick Ponsonby''' (10 December 1825 – 21 November 1895) *# Lieutenant Colonel Arthur Edward Valette (3 December 1827 – 16 June 1868) *# Georgina Melita Maria Ponsonby (16 February 1829 – 18 February 1895) *# Harriet Julia Frances Ponsonby (27 October 1830 – 30 June 1906) *# Selina Barbara Wilhelmina Ponsonby (20 January 1835 – 22 July 1919) *# Frederick John Ponsonby (21 March 1837 – 3 February 1894) * Sir Henry Frederick Ponsonby (10 December 1825 – 21 November 1895)<ref name=":0" /> * Hon. Mary Elizabeth Bulteel (21 Sep 1832 – 1916)<ref>{{Cite journal|date=2020-11-07|title=John Crocker Bulteel|url=https://en.wikipedia.org/w/index.php?title=John_Crocker_Bulteel&oldid=987566588|journal=Wikipedia|language=en}}</ref> *# Alberta Victoria Ponsonby (6 May 1862 – 15 October 1945) *# Cecil William Davidge Ponsonby (28 March 1863 – 16 January 1936) *# Magdalen Ponsonby (24 June 1864 – 1 July 1934) *# John Ponsonby (25 March 1866 – 26 March 1952) *# '''Frederick Edward Grey Ponsonby''' (16 September 1867 – 20 October 1935) *# Arthur Augustus William Harry Ponsonby (16 February 1871 – 24 March 1946) * Fritz (Frederick Edward Grey) Ponsonby (16 September 1867 – 20 October 1935) ===Relations=== * Mr. and Mrs. Edward Ponsonby * Mr. Fritz Ponsonby * Mr. John Ponsonby * Miss M. Ponsonby * Miss Melita and Miss Julia Ponsonby * Ponsonby Fane *Hon. Sir Frederick Cavendish Ponsonby's sister was Lady Caroline Lamb, who married Viscount Melbourne, Prime Minister early in Queen Victoria's reign. == Notes and Questions == === Frederick Ponsonby's ''Recollections of Three Reigns'' === [[Social Victorians/Royals Amateur Theatricals|Amateur theatricals]] among the royals and household:<blockquote>Once or twice whole plays came down [to Windsor, Osborne, or Balmoral, though he probably means Windsor? Someone says Osborne somewhere, though?], but this was expensive, and amateur actors in the Household were usually asked to produce something. There were two outstanding actors, Arthur Collins [fn 1 [[Social Victorians/People/Arthur Collins|Lieutenant-Colonel Arthur Collins]], Gentleman Usher to Queen Victoria.] and Alec Yorke, and these two alternately drilled the members of the Household and produced plays. [50/51] The first time I acted was in ''She Stoops to Conquer'', stage-managed by Arthur Collins, who did the part of the Innkeeper. This was before I joined the Household [in 1894]. My brother Arthur [fn 1 Later 1^st Lord Ponsonby of Shulbrede. For many years Socialist M.P. for Brightside Division of Sheffield.] was Tony Lumpkin, while Arthur Bigge and I did the two lovers, and Princess Louise and Princess Beatrice the two principal ladies’ parts. The Queen came to the rehearsals, which frightened us all very much, and when she saw me chucking Princess Louise under the chin (I was supposed to mistake her for the barmaid) she thought this was overdone. I received a message that I had better not indulge in any chucking under the chin. The next day I went through my part but never came within touching distance of Princess Louise, and again received a message to say I was overdoing it the other way. I consulted Princess Louise herself, who roared with laughter at my dilemma, and we finally hit off a happy medium. Both Princess Louise and Princess Beatrice were quite good in their parts, but very sketchy with the words. I therefore learnt their parts as well as my own so that I could either say their words or prompt them. Everyone else did the same, but there was one small bit when they were both on together and of course they stuck, each one thinking it was the other’s fault. After an awkward pause the servants gave a round of applause, which I thought was a very intelligent way of helping them, but although the prompter was able to start them again, they could not get going and the stage carpenter solved the problem by letting the curtain down. Another time I acted in ''A Scrap of Paper'' at Balmoral, this time under Alec Yorke as stage manager. The Queen thought the performances so good that she invited Hare and his company, who were playing at Aberdeen, to come and witness the performance. They must have been amused as they all knew the play well, but of course they were loud in their praises. There were also tableaux in which all the members of the Royal Family took part. They must have been very wearying for the audience, who had to sit for two and a half hours with very long intervals between the tableaux. The only person who thoroughly enjoyed them was Clarkson, the wig-maker from London. As he hob-nobbed with the Royal Family and as he supplied all the dresses, he probably made a very good thing out of it.<ref name=":1">Ponsonby, Frederick, Sir, first Lord Sysonby. ''Recollections of Three Reigns''. Introd. and notes, Colin Welch. London, Eyre & Spottiswoode, 1951.</ref>{{rp|50–51}} </blockquote>On the transition between Victoria’s and Edward’s administrations and households:<blockquote>The King made a clean sweep of the Grooms-in-Waiting and decided to appoint in future soldiers and sailors of distinction to be Gentlemen Ushers, a position similar to that of Chamberlains abroad. In Queen Victoria’s reign these appointments were held by any type of man and there was a certain amount of nepotism. The King’s plans were not at first understood and he was hurt at the refusal of several old friends to serve as Ushers. Later many soldiers and sailors accepted but it is doubtful whether they were really suited for managing large crowds of society people at Buckingham Palace. [101/102] When the King and Queen went to London, Buckingham Palace was undergoing great changes and they therefore had to remain at Marlborough House, which had been the King’s London home since his marriage. Having been appointed Assistant Private Secretary I imagined that I should have a great deal of work to do, but when I came the next morning at 10 a.m. I was told by Francis Knollys that it was quite unnecessary for me to stay as he wanted no help. This surprised me considerably as I knew there must be a lot of work to be done, but as there is nothing more tiresome than a person hanging about idle when everyone else is busy, I made myself scarce. Even the Equerry had so much to do that he had no time to speak to me. I learnt later that while there was no friction of any sort between the new and the old Household, Marlborough House was firmly convinced that Buckingham Palace was hopelessly out of date and that none of Queen Victoria’s Household were any good. There was a great deal of truth in this, but while the other Departments were certainly out of date, the Private Secretary’s office under Sir Arthur Bigge was not only up to date, but far better managed than the equivalent office in the Prince of Wales’ Household. Bigge had organized the office on business lines, and since I had joined the Household typewriting and shorthand had been introduced, and the filing of papers brought up to date. The new appointment of myself as an Assistant Private Secretary was not altogether popular. In the first place Francis Knollys considered it quite unnecessary, and in the second place a member of Queen Victoria’s old-fashioned Household had been nominated, which probably meant incompetence. There were clerks at Buckingham Palace, but they were probably hopeless, therefore quite obviously the Marlborough House staff had better carry on as before, and the Assistant Private Secretary, who was a fifth wheel to a coach, could join the ceremonial Household.<ref name=":1" />{{rp|101–102}}</blockquote> More on the households: <blockquote> The Lord Chamberlain’s office under Sir Arthur Ellis, which was concerned with ceremonial issues, was also reorganized and all sorts of innovations and improvements were introduced: the most effective of these was the ‘Drawing-rooms’, as they were then called, being held in the evening instead of the afternoon. They were in future called ‘Courts’ and were held in the ballroom instead of the throne-room. Ellis, who had visited every Court in Europe with King Edward when he was Prince of Wales, had an unrivalled knowledge of the way things were done abroad and was able to adapt the best features of the Continental receptions. The King went into every detail and between them they stage-managed perfectly a new piece of pageantry. When I attended a ‘Court’ I was always struck by the incongruous music the band played, and determined to do what I could to have this remedied. The majority of the Household, being quite unmusical, clamoured for popular airs, and Sir Walter Parratt, the Master of the Music, who cared only for classical music and looked down on any other sort of music, complied with the demand. I argued that these popular airs robbed the ceremony of all dignity. A presentation at Court was often a great event in a lady’s life, but if she went past the King and Queen to the tune of ‘His nose was redder than it was’, the whole impression was spoilt. I maintained that minuets and old-fashioned airs, operatic music with a ‘mysterious’ touch, were what was wanted. I wrote to Sir Walter Parratt, who welcomed my opposition, to counter the pressure for popular airs; not that he carried out my proposals, but he played music that he liked.<ref name=":1" />{{rp|125}} </blockquote> On the Drawing-Rooms: <blockquote>King Edward liked French and Viennese light operas, whereas Queen Alexandra preferred grand opera, particularly Wagner. One morning the bandmaster received a message from the King to play Offenbach, and one from the Queen to play Wagner. Finding himself unable to comply with both, he thought he would hit upon the happy medium and selected Gilbert and Sullivan operas, and as always with people who compromise, he got into trouble with both the King and the Queen.<ref name=":1" />{{rp|125}}</blockquote> == Bibliography == * Ponsonby, Arthur [Arthur Augustus William Harry]. ''Henry Ponsonby, Queen Victoria's Private Secretary: His Life from His Letters''. London: Macmillan, 1943. * Ponsonby, Frederick, Sir, first Lord Sysonby. ''Recollections of Three Reigns''. Introd. and notes, Colin Welch. London, Eyre & Spottiswoode, 1951. ==Footnotes== {{reflist}} 75o35hryjqfypkpevwf9mwwf9ciapnz 2812066 2812065 2026-05-29T21:31:59Z Scogdill 1331941 /* Overview */ 2812066 wikitext text/x-wiki ==Overview== David Cannadine says,<blockquote>The most successful patrician courtiers established an abiding dynastic connection, so that generation after generation, their families enjoyed royal favour and preferment. From the late nineteenth century until the Second World War, the Ponsonbys were the pre-eminent courtly dynasty. They were very aristocratic but not very rich, being a cadet branch of the Earls of Bessborough. Sir Henry Ponsonby, grandson of the third earl, became the Queen's private secretary in 1870, largely because his predecessor, General Grey, was his wife's uncle. He held the position until his death in 1895, by which time the Queen had already appointed his second son, Frederick, as an Equerry.<ref name=":2" />{{rp|247}}</blockquote>Cannadine says, "Sir Henry Ponsonby effectively created the post of Private Secretary to the Sovereign in its modern guise."<ref name=":2">Cannadine, David. ''The Decline and Fall of the British Aristocracy''. New York: Yale University Press, 1990.</ref>{{rp|245}}<p> Queen Victoria's replacement for General Grey as her private secretary<blockquote>was Colonel Henry Ponsonby, who had been at Court since 1857. Ponsonby, himself from an aristocratic Whig family, was married to Mary Bulteel, a niece of General Grey.<p> The Ponsonbys were, on superficial levels, surprising choices for the Queen. While she was increasingly Tory, they were unabashed Gladstonian Liberals. In religion, they were High Church, like Gladstone, whereas the Queen was her own distinctive brand of Broad Church Pantheist/Presbyterian. Mary Ponsonby was highly educated, and a feminist; Queen Victoria deplored feminism.<ref>Wilson, A. N. ''Victoria: A Life''. Penguin, 2014. Apple Books: https://books.apple.com/us/book/victoria/id828766078.</ref> (596 of 1204)</blockquote> == Also Known As == * Family name: Ponsonby *Baron Sysonby **Fritz (Frederick Edward Grey) Ponsonby, 1st Baron Sysonby ()<ref>{{Cite journal|date=2020-11-21|title=Frederick Ponsonby, 1st Baron Sysonby|url=https://en.wikipedia.org/w/index.php?title=Frederick_Ponsonby,_1st_Baron_Sysonby&oldid=989930883|journal=Wikipedia|language=en}}</ref> *Cannadine says, "very aristocratic but not very rich, being a cadet branch of the Earls of Bessborough."<ref name=":2" />{{rp|247}} == Acquaintances, Friends and Enemies == == Organizations == === Fritz (Frederick Edward Grey) Ponsonby === ==== Queen Victoria's Household ==== * Equerry-in-Ordinary (1894–1901) * Assistant Keeper of the Privy Purse and Assistant Private Secretary to Queen Victoria (1897–1901) ==== King Edward VII's Household ==== * Assistant Keeper of the Privy Purse and Assistant Private Secretary to King Edward VII (1901–1910) ==== King George V's Household ==== * Assistant Keeper of the Privy Purse and Assistant Private Secretary to King George V (1910–1914) * Keeper of the Privy Purse to King George V (1914–1935) * Lieutenant Governor of Windsor Castle (1928–1935) ==Timeline== '''1870 April 8''', Henry Frederick Ponsonby was appointed Queen Victoria's Private Secretary and Keeper of the Privy Purse.<ref name=":0">{{Cite journal|date=2020-12-05|title=Henry Ponsonby|url=https://en.wikipedia.org/w/index.php?title=Henry_Ponsonby&oldid=992528585|journal=Wikipedia|language=en}}</ref> == Demographics == * Nationality: British ==Family== * Hon. Sir Frederick Cavendish Ponsonby (6 July 1783 – 11 January 1837)<ref>{{Cite journal|date=2020-10-18|title=Frederick Cavendish Ponsonby|url=https://en.wikipedia.org/w/index.php?title=Frederick_Cavendish_Ponsonby&oldid=984182186|journal=Wikipedia|language=en}}</ref> * Lady Emily Charlotte Bathurst ( – 1877) *# '''Sir Henry Frederick Ponsonby''' (10 December 1825 – 21 November 1895) *# Lieutenant Colonel Arthur Edward Valette (3 December 1827 – 16 June 1868) *# Georgina Melita Maria Ponsonby (16 February 1829 – 18 February 1895) *# Harriet Julia Frances Ponsonby (27 October 1830 – 30 June 1906) *# Selina Barbara Wilhelmina Ponsonby (20 January 1835 – 22 July 1919) *# Frederick John Ponsonby (21 March 1837 – 3 February 1894) * Sir Henry Frederick Ponsonby (10 December 1825 – 21 November 1895)<ref name=":0" /> * Hon. Mary Elizabeth Bulteel (21 Sep 1832 – 1916)<ref>{{Cite journal|date=2020-11-07|title=John Crocker Bulteel|url=https://en.wikipedia.org/w/index.php?title=John_Crocker_Bulteel&oldid=987566588|journal=Wikipedia|language=en}}</ref> *# Alberta Victoria Ponsonby (6 May 1862 – 15 October 1945) *# Cecil William Davidge Ponsonby (28 March 1863 – 16 January 1936) *# Magdalen Ponsonby (24 June 1864 – 1 July 1934) *# John Ponsonby (25 March 1866 – 26 March 1952) *# '''Frederick Edward Grey Ponsonby''' (16 September 1867 – 20 October 1935) *# Arthur Augustus William Harry Ponsonby (16 February 1871 – 24 March 1946) * Fritz (Frederick Edward Grey) Ponsonby (16 September 1867 – 20 October 1935) ===Relations=== * Mr. and Mrs. Edward Ponsonby * Mr. Fritz Ponsonby * Mr. John Ponsonby * Miss M. Ponsonby * Miss Melita and Miss Julia Ponsonby * Ponsonby Fane *Hon. Sir Frederick Cavendish Ponsonby's sister was Lady Caroline Lamb, who married Viscount Melbourne, Prime Minister early in Queen Victoria's reign. == Notes and Questions == === Frederick Ponsonby's ''Recollections of Three Reigns'' === [[Social Victorians/Royals Amateur Theatricals|Amateur theatricals]] among the royals and household:<blockquote>Once or twice whole plays came down [to Windsor, Osborne, or Balmoral, though he probably means Windsor? Someone says Osborne somewhere, though?], but this was expensive, and amateur actors in the Household were usually asked to produce something. There were two outstanding actors, Arthur Collins [fn 1 [[Social Victorians/People/Arthur Collins|Lieutenant-Colonel Arthur Collins]], Gentleman Usher to Queen Victoria.] and Alec Yorke, and these two alternately drilled the members of the Household and produced plays. [50/51] The first time I acted was in ''She Stoops to Conquer'', stage-managed by Arthur Collins, who did the part of the Innkeeper. This was before I joined the Household [in 1894]. My brother Arthur [fn 1 Later 1^st Lord Ponsonby of Shulbrede. For many years Socialist M.P. for Brightside Division of Sheffield.] was Tony Lumpkin, while Arthur Bigge and I did the two lovers, and Princess Louise and Princess Beatrice the two principal ladies’ parts. The Queen came to the rehearsals, which frightened us all very much, and when she saw me chucking Princess Louise under the chin (I was supposed to mistake her for the barmaid) she thought this was overdone. I received a message that I had better not indulge in any chucking under the chin. The next day I went through my part but never came within touching distance of Princess Louise, and again received a message to say I was overdoing it the other way. I consulted Princess Louise herself, who roared with laughter at my dilemma, and we finally hit off a happy medium. Both Princess Louise and Princess Beatrice were quite good in their parts, but very sketchy with the words. I therefore learnt their parts as well as my own so that I could either say their words or prompt them. Everyone else did the same, but there was one small bit when they were both on together and of course they stuck, each one thinking it was the other’s fault. After an awkward pause the servants gave a round of applause, which I thought was a very intelligent way of helping them, but although the prompter was able to start them again, they could not get going and the stage carpenter solved the problem by letting the curtain down. Another time I acted in ''A Scrap of Paper'' at Balmoral, this time under Alec Yorke as stage manager. The Queen thought the performances so good that she invited Hare and his company, who were playing at Aberdeen, to come and witness the performance. They must have been amused as they all knew the play well, but of course they were loud in their praises. There were also tableaux in which all the members of the Royal Family took part. They must have been very wearying for the audience, who had to sit for two and a half hours with very long intervals between the tableaux. The only person who thoroughly enjoyed them was Clarkson, the wig-maker from London. As he hob-nobbed with the Royal Family and as he supplied all the dresses, he probably made a very good thing out of it.<ref name=":1">Ponsonby, Frederick, Sir, first Lord Sysonby. ''Recollections of Three Reigns''. Introd. and notes, Colin Welch. London, Eyre & Spottiswoode, 1951.</ref>{{rp|50–51}} </blockquote>On the transition between Victoria’s and Edward’s administrations and households:<blockquote>The King made a clean sweep of the Grooms-in-Waiting and decided to appoint in future soldiers and sailors of distinction to be Gentlemen Ushers, a position similar to that of Chamberlains abroad. In Queen Victoria’s reign these appointments were held by any type of man and there was a certain amount of nepotism. The King’s plans were not at first understood and he was hurt at the refusal of several old friends to serve as Ushers. Later many soldiers and sailors accepted but it is doubtful whether they were really suited for managing large crowds of society people at Buckingham Palace. [101/102] When the King and Queen went to London, Buckingham Palace was undergoing great changes and they therefore had to remain at Marlborough House, which had been the King’s London home since his marriage. Having been appointed Assistant Private Secretary I imagined that I should have a great deal of work to do, but when I came the next morning at 10 a.m. I was told by Francis Knollys that it was quite unnecessary for me to stay as he wanted no help. This surprised me considerably as I knew there must be a lot of work to be done, but as there is nothing more tiresome than a person hanging about idle when everyone else is busy, I made myself scarce. Even the Equerry had so much to do that he had no time to speak to me. I learnt later that while there was no friction of any sort between the new and the old Household, Marlborough House was firmly convinced that Buckingham Palace was hopelessly out of date and that none of Queen Victoria’s Household were any good. There was a great deal of truth in this, but while the other Departments were certainly out of date, the Private Secretary’s office under Sir Arthur Bigge was not only up to date, but far better managed than the equivalent office in the Prince of Wales’ Household. Bigge had organized the office on business lines, and since I had joined the Household typewriting and shorthand had been introduced, and the filing of papers brought up to date. The new appointment of myself as an Assistant Private Secretary was not altogether popular. In the first place Francis Knollys considered it quite unnecessary, and in the second place a member of Queen Victoria’s old-fashioned Household had been nominated, which probably meant incompetence. There were clerks at Buckingham Palace, but they were probably hopeless, therefore quite obviously the Marlborough House staff had better carry on as before, and the Assistant Private Secretary, who was a fifth wheel to a coach, could join the ceremonial Household.<ref name=":1" />{{rp|101–102}}</blockquote> More on the households: <blockquote> The Lord Chamberlain’s office under Sir Arthur Ellis, which was concerned with ceremonial issues, was also reorganized and all sorts of innovations and improvements were introduced: the most effective of these was the ‘Drawing-rooms’, as they were then called, being held in the evening instead of the afternoon. They were in future called ‘Courts’ and were held in the ballroom instead of the throne-room. Ellis, who had visited every Court in Europe with King Edward when he was Prince of Wales, had an unrivalled knowledge of the way things were done abroad and was able to adapt the best features of the Continental receptions. The King went into every detail and between them they stage-managed perfectly a new piece of pageantry. When I attended a ‘Court’ I was always struck by the incongruous music the band played, and determined to do what I could to have this remedied. The majority of the Household, being quite unmusical, clamoured for popular airs, and Sir Walter Parratt, the Master of the Music, who cared only for classical music and looked down on any other sort of music, complied with the demand. I argued that these popular airs robbed the ceremony of all dignity. A presentation at Court was often a great event in a lady’s life, but if she went past the King and Queen to the tune of ‘His nose was redder than it was’, the whole impression was spoilt. I maintained that minuets and old-fashioned airs, operatic music with a ‘mysterious’ touch, were what was wanted. I wrote to Sir Walter Parratt, who welcomed my opposition, to counter the pressure for popular airs; not that he carried out my proposals, but he played music that he liked.<ref name=":1" />{{rp|125}} </blockquote> On the Drawing-Rooms: <blockquote>King Edward liked French and Viennese light operas, whereas Queen Alexandra preferred grand opera, particularly Wagner. One morning the bandmaster received a message from the King to play Offenbach, and one from the Queen to play Wagner. Finding himself unable to comply with both, he thought he would hit upon the happy medium and selected Gilbert and Sullivan operas, and as always with people who compromise, he got into trouble with both the King and the Queen.<ref name=":1" />{{rp|125}}</blockquote> == Bibliography == * Ponsonby, Arthur [Arthur Augustus William Harry]. ''Henry Ponsonby, Queen Victoria's Private Secretary: His Life from His Letters''. London: Macmillan, 1943. * Ponsonby, Frederick, Sir, first Lord Sysonby. ''Recollections of Three Reigns''. Introd. and notes, Colin Welch. London, Eyre & Spottiswoode, 1951. ==Footnotes== {{reflist}} kbi4o5s9f5ivvu3iibqgvkyqtjt92kn 2812067 2812066 2026-05-29T21:34:40Z Scogdill 1331941 2812067 wikitext text/x-wiki ==Overview== David Cannadine says,<blockquote>The most successful patrician courtiers established an abiding dynastic connection, so that generation after generation, their families enjoyed royal favour and preferment. From the late nineteenth century until the Second World War, the Ponsonbys were the pre-eminent courtly dynasty. They were very aristocratic but not very rich, being a cadet branch of the Earls of Bessborough. Sir Henry Ponsonby, grandson of the third earl, became the Queen's private secretary in 1870, largely because his predecessor, General Grey, was his wife's uncle. He held the position until his death in 1895, by which time the Queen had already appointed his second son, Frederick, as an Equerry.<ref name=":2" />{{rp|247}}</blockquote>Cannadine says, "Sir Henry Ponsonby effectively created the post of Private Secretary to the Sovereign in its modern guise."<ref name=":2">Cannadine, David. ''The Decline and Fall of the British Aristocracy''. New York: Yale University Press, 1990.</ref>{{rp|245}}<p> Queen Victoria's replacement for General Grey as her private secretary<blockquote>was Colonel Henry Ponsonby, who had been at Court since 1857. Ponsonby, himself from an aristocratic Whig family, was married to Mary Bulteel, a niece of General Grey.<p> The Ponsonbys were, on superficial levels, surprising choices for the Queen. While she was increasingly Tory, they were unabashed Gladstonian Liberals. In religion, they were High Church, like Gladstone, whereas the Queen was her own distinctive brand of Broad Church Pantheist/Presbyterian. Mary Ponsonby was highly educated, and a feminist; Queen Victoria deplored feminism.<ref>Wilson, A. N. ''Victoria: A Life''. Penguin, 2014. Apple Books: https://books.apple.com/us/book/victoria/id828766078.</ref> (333–334)</blockquote> == Also Known As == * Family name: Ponsonby *Baron Sysonby **Fritz (Frederick Edward Grey) Ponsonby, 1st Baron Sysonby ()<ref>{{Cite journal|date=2020-11-21|title=Frederick Ponsonby, 1st Baron Sysonby|url=https://en.wikipedia.org/w/index.php?title=Frederick_Ponsonby,_1st_Baron_Sysonby&oldid=989930883|journal=Wikipedia|language=en}}</ref> *Cannadine says, "very aristocratic but not very rich, being a cadet branch of the Earls of Bessborough."<ref name=":2" />{{rp|247}} == Acquaintances, Friends and Enemies == == Organizations == === Fritz (Frederick Edward Grey) Ponsonby === ==== Queen Victoria's Household ==== * Equerry-in-Ordinary (1894–1901) * Assistant Keeper of the Privy Purse and Assistant Private Secretary to Queen Victoria (1897–1901) ==== King Edward VII's Household ==== * Assistant Keeper of the Privy Purse and Assistant Private Secretary to King Edward VII (1901–1910) ==== King George V's Household ==== * Assistant Keeper of the Privy Purse and Assistant Private Secretary to King George V (1910–1914) * Keeper of the Privy Purse to King George V (1914–1935) * Lieutenant Governor of Windsor Castle (1928–1935) ==Timeline== '''1870 April 8''', Henry Frederick Ponsonby was appointed Queen Victoria's Private Secretary and Keeper of the Privy Purse.<ref name=":0">{{Cite journal|date=2020-12-05|title=Henry Ponsonby|url=https://en.wikipedia.org/w/index.php?title=Henry_Ponsonby&oldid=992528585|journal=Wikipedia|language=en}}</ref> == Demographics == * Nationality: British ==Family== * Hon. Sir Frederick Cavendish Ponsonby (6 July 1783 – 11 January 1837)<ref>{{Cite journal|date=2020-10-18|title=Frederick Cavendish Ponsonby|url=https://en.wikipedia.org/w/index.php?title=Frederick_Cavendish_Ponsonby&oldid=984182186|journal=Wikipedia|language=en}}</ref> * Lady Emily Charlotte Bathurst ( – 1877) *# '''Sir Henry Frederick Ponsonby''' (10 December 1825 – 21 November 1895) *# Lieutenant Colonel Arthur Edward Valette (3 December 1827 – 16 June 1868) *# Georgina Melita Maria Ponsonby (16 February 1829 – 18 February 1895) *# Harriet Julia Frances Ponsonby (27 October 1830 – 30 June 1906) *# Selina Barbara Wilhelmina Ponsonby (20 January 1835 – 22 July 1919) *# Frederick John Ponsonby (21 March 1837 – 3 February 1894) * Sir Henry Frederick Ponsonby (10 December 1825 – 21 November 1895)<ref name=":0" /> * Hon. Mary Elizabeth Bulteel (21 Sep 1832 – 1916)<ref>{{Cite journal|date=2020-11-07|title=John Crocker Bulteel|url=https://en.wikipedia.org/w/index.php?title=John_Crocker_Bulteel&oldid=987566588|journal=Wikipedia|language=en}}</ref> *# Alberta Victoria Ponsonby (6 May 1862 – 15 October 1945) *# Cecil William Davidge Ponsonby (28 March 1863 – 16 January 1936) *# Magdalen Ponsonby (24 June 1864 – 1 July 1934) *# John Ponsonby (25 March 1866 – 26 March 1952) *# '''Frederick Edward Grey Ponsonby''' (16 September 1867 – 20 October 1935) *# Arthur Augustus William Harry Ponsonby (16 February 1871 – 24 March 1946) * Fritz (Frederick Edward Grey) Ponsonby (16 September 1867 – 20 October 1935) ===Relations=== * Mr. and Mrs. Edward Ponsonby * Mr. Fritz Ponsonby * Mr. John Ponsonby * Miss M. Ponsonby * Miss Melita and Miss Julia Ponsonby * Ponsonby Fane *Hon. Sir Frederick Cavendish Ponsonby's sister was Lady Caroline Lamb, who married Viscount Melbourne, Prime Minister early in Queen Victoria's reign. == Notes and Questions == === Frederick Ponsonby's ''Recollections of Three Reigns'' === [[Social Victorians/Royals Amateur Theatricals|Amateur theatricals]] among the royals and household:<blockquote>Once or twice whole plays came down [to Windsor, Osborne, or Balmoral, though he probably means Windsor? Someone says Osborne somewhere, though?], but this was expensive, and amateur actors in the Household were usually asked to produce something. There were two outstanding actors, Arthur Collins [fn 1 [[Social Victorians/People/Arthur Collins|Lieutenant-Colonel Arthur Collins]], Gentleman Usher to Queen Victoria.] and Alec Yorke, and these two alternately drilled the members of the Household and produced plays. [50/51] The first time I acted was in ''She Stoops to Conquer'', stage-managed by Arthur Collins, who did the part of the Innkeeper. This was before I joined the Household [in 1894]. My brother Arthur [fn 1 Later 1^st Lord Ponsonby of Shulbrede. For many years Socialist M.P. for Brightside Division of Sheffield.] was Tony Lumpkin, while Arthur Bigge and I did the two lovers, and Princess Louise and Princess Beatrice the two principal ladies’ parts. The Queen came to the rehearsals, which frightened us all very much, and when she saw me chucking Princess Louise under the chin (I was supposed to mistake her for the barmaid) she thought this was overdone. I received a message that I had better not indulge in any chucking under the chin. The next day I went through my part but never came within touching distance of Princess Louise, and again received a message to say I was overdoing it the other way. I consulted Princess Louise herself, who roared with laughter at my dilemma, and we finally hit off a happy medium. Both Princess Louise and Princess Beatrice were quite good in their parts, but very sketchy with the words. I therefore learnt their parts as well as my own so that I could either say their words or prompt them. Everyone else did the same, but there was one small bit when they were both on together and of course they stuck, each one thinking it was the other’s fault. After an awkward pause the servants gave a round of applause, which I thought was a very intelligent way of helping them, but although the prompter was able to start them again, they could not get going and the stage carpenter solved the problem by letting the curtain down. Another time I acted in ''A Scrap of Paper'' at Balmoral, this time under Alec Yorke as stage manager. The Queen thought the performances so good that she invited Hare and his company, who were playing at Aberdeen, to come and witness the performance. They must have been amused as they all knew the play well, but of course they were loud in their praises. There were also tableaux in which all the members of the Royal Family took part. They must have been very wearying for the audience, who had to sit for two and a half hours with very long intervals between the tableaux. The only person who thoroughly enjoyed them was Clarkson, the wig-maker from London. As he hob-nobbed with the Royal Family and as he supplied all the dresses, he probably made a very good thing out of it.<ref name=":1">Ponsonby, Frederick, Sir, first Lord Sysonby. ''Recollections of Three Reigns''. Introd. and notes, Colin Welch. London, Eyre & Spottiswoode, 1951.</ref>{{rp|50–51}} </blockquote>On the transition between Victoria’s and Edward’s administrations and households:<blockquote>The King made a clean sweep of the Grooms-in-Waiting and decided to appoint in future soldiers and sailors of distinction to be Gentlemen Ushers, a position similar to that of Chamberlains abroad. In Queen Victoria’s reign these appointments were held by any type of man and there was a certain amount of nepotism. The King’s plans were not at first understood and he was hurt at the refusal of several old friends to serve as Ushers. Later many soldiers and sailors accepted but it is doubtful whether they were really suited for managing large crowds of society people at Buckingham Palace. [101/102] When the King and Queen went to London, Buckingham Palace was undergoing great changes and they therefore had to remain at Marlborough House, which had been the King’s London home since his marriage. Having been appointed Assistant Private Secretary I imagined that I should have a great deal of work to do, but when I came the next morning at 10 a.m. I was told by Francis Knollys that it was quite unnecessary for me to stay as he wanted no help. This surprised me considerably as I knew there must be a lot of work to be done, but as there is nothing more tiresome than a person hanging about idle when everyone else is busy, I made myself scarce. Even the Equerry had so much to do that he had no time to speak to me. I learnt later that while there was no friction of any sort between the new and the old Household, Marlborough House was firmly convinced that Buckingham Palace was hopelessly out of date and that none of Queen Victoria’s Household were any good. There was a great deal of truth in this, but while the other Departments were certainly out of date, the Private Secretary’s office under Sir Arthur Bigge was not only up to date, but far better managed than the equivalent office in the Prince of Wales’ Household. Bigge had organized the office on business lines, and since I had joined the Household typewriting and shorthand had been introduced, and the filing of papers brought up to date. The new appointment of myself as an Assistant Private Secretary was not altogether popular. In the first place Francis Knollys considered it quite unnecessary, and in the second place a member of Queen Victoria’s old-fashioned Household had been nominated, which probably meant incompetence. There were clerks at Buckingham Palace, but they were probably hopeless, therefore quite obviously the Marlborough House staff had better carry on as before, and the Assistant Private Secretary, who was a fifth wheel to a coach, could join the ceremonial Household.<ref name=":1" />{{rp|101–102}}</blockquote> More on the households: <blockquote> The Lord Chamberlain’s office under Sir Arthur Ellis, which was concerned with ceremonial issues, was also reorganized and all sorts of innovations and improvements were introduced: the most effective of these was the ‘Drawing-rooms’, as they were then called, being held in the evening instead of the afternoon. They were in future called ‘Courts’ and were held in the ballroom instead of the throne-room. Ellis, who had visited every Court in Europe with King Edward when he was Prince of Wales, had an unrivalled knowledge of the way things were done abroad and was able to adapt the best features of the Continental receptions. The King went into every detail and between them they stage-managed perfectly a new piece of pageantry. When I attended a ‘Court’ I was always struck by the incongruous music the band played, and determined to do what I could to have this remedied. The majority of the Household, being quite unmusical, clamoured for popular airs, and Sir Walter Parratt, the Master of the Music, who cared only for classical music and looked down on any other sort of music, complied with the demand. I argued that these popular airs robbed the ceremony of all dignity. A presentation at Court was often a great event in a lady’s life, but if she went past the King and Queen to the tune of ‘His nose was redder than it was’, the whole impression was spoilt. I maintained that minuets and old-fashioned airs, operatic music with a ‘mysterious’ touch, were what was wanted. I wrote to Sir Walter Parratt, who welcomed my opposition, to counter the pressure for popular airs; not that he carried out my proposals, but he played music that he liked.<ref name=":1" />{{rp|125}} </blockquote> On the Drawing-Rooms: <blockquote>King Edward liked French and Viennese light operas, whereas Queen Alexandra preferred grand opera, particularly Wagner. One morning the bandmaster received a message from the King to play Offenbach, and one from the Queen to play Wagner. Finding himself unable to comply with both, he thought he would hit upon the happy medium and selected Gilbert and Sullivan operas, and as always with people who compromise, he got into trouble with both the King and the Queen.<ref name=":1" />{{rp|125}}</blockquote> == Bibliography == * Ponsonby, Arthur [Arthur Augustus William Harry]. ''Henry Ponsonby, Queen Victoria's Private Secretary: His Life from His Letters''. London: Macmillan, 1943. * Ponsonby, Frederick, Sir, first Lord Sysonby. ''Recollections of Three Reigns''. Introd. and notes, Colin Welch. London, Eyre & Spottiswoode, 1951. ==Footnotes== {{reflist}} i33qml8ab2tusbvbnbm0infpvxc8jkq 2812068 2812067 2026-05-29T21:34:56Z Scogdill 1331941 /* Overview */ 2812068 wikitext text/x-wiki ==Overview== David Cannadine says,<blockquote>The most successful patrician courtiers established an abiding dynastic connection, so that generation after generation, their families enjoyed royal favour and preferment. From the late nineteenth century until the Second World War, the Ponsonbys were the pre-eminent courtly dynasty. They were very aristocratic but not very rich, being a cadet branch of the Earls of Bessborough. Sir Henry Ponsonby, grandson of the third earl, became the Queen's private secretary in 1870, largely because his predecessor, General Grey, was his wife's uncle. He held the position until his death in 1895, by which time the Queen had already appointed his second son, Frederick, as an Equerry.<ref name=":2" />{{rp|247}}</blockquote>Cannadine says, "Sir Henry Ponsonby effectively created the post of Private Secretary to the Sovereign in its modern guise."<ref name=":2">Cannadine, David. ''The Decline and Fall of the British Aristocracy''. New York: Yale University Press, 1990.</ref>{{rp|245}}<p> Queen Victoria's replacement for General Grey as her private secretary<blockquote>was Colonel Henry Ponsonby, who had been at Court since 1857. Ponsonby, himself from an aristocratic Whig family, was married to Mary Bulteel, a niece of General Grey.<p> The Ponsonbys were, on superficial levels, surprising choices for the Queen. While she was increasingly Tory, they were unabashed Gladstonian Liberals. In religion, they were High Church, like Gladstone, whereas the Queen was her own distinctive brand of Broad Church Pantheist/Presbyterian. Mary Ponsonby was highly educated, and a feminist; Queen Victoria deplored feminism.<ref>Wilson, A. N. ''Victoria: A Life''. Penguin, 2014. Apple Books: https://books.apple.com/us/book/victoria/id828766078.</ref>{{rp|333–334}}</blockquote> == Also Known As == * Family name: Ponsonby *Baron Sysonby **Fritz (Frederick Edward Grey) Ponsonby, 1st Baron Sysonby ()<ref>{{Cite journal|date=2020-11-21|title=Frederick Ponsonby, 1st Baron Sysonby|url=https://en.wikipedia.org/w/index.php?title=Frederick_Ponsonby,_1st_Baron_Sysonby&oldid=989930883|journal=Wikipedia|language=en}}</ref> *Cannadine says, "very aristocratic but not very rich, being a cadet branch of the Earls of Bessborough."<ref name=":2" />{{rp|247}} == Acquaintances, Friends and Enemies == == Organizations == === Fritz (Frederick Edward Grey) Ponsonby === ==== Queen Victoria's Household ==== * Equerry-in-Ordinary (1894–1901) * Assistant Keeper of the Privy Purse and Assistant Private Secretary to Queen Victoria (1897–1901) ==== King Edward VII's Household ==== * Assistant Keeper of the Privy Purse and Assistant Private Secretary to King Edward VII (1901–1910) ==== King George V's Household ==== * Assistant Keeper of the Privy Purse and Assistant Private Secretary to King George V (1910–1914) * Keeper of the Privy Purse to King George V (1914–1935) * Lieutenant Governor of Windsor Castle (1928–1935) ==Timeline== '''1870 April 8''', Henry Frederick Ponsonby was appointed Queen Victoria's Private Secretary and Keeper of the Privy Purse.<ref name=":0">{{Cite journal|date=2020-12-05|title=Henry Ponsonby|url=https://en.wikipedia.org/w/index.php?title=Henry_Ponsonby&oldid=992528585|journal=Wikipedia|language=en}}</ref> == Demographics == * Nationality: British ==Family== * Hon. Sir Frederick Cavendish Ponsonby (6 July 1783 – 11 January 1837)<ref>{{Cite journal|date=2020-10-18|title=Frederick Cavendish Ponsonby|url=https://en.wikipedia.org/w/index.php?title=Frederick_Cavendish_Ponsonby&oldid=984182186|journal=Wikipedia|language=en}}</ref> * Lady Emily Charlotte Bathurst ( – 1877) *# '''Sir Henry Frederick Ponsonby''' (10 December 1825 – 21 November 1895) *# Lieutenant Colonel Arthur Edward Valette (3 December 1827 – 16 June 1868) *# Georgina Melita Maria Ponsonby (16 February 1829 – 18 February 1895) *# Harriet Julia Frances Ponsonby (27 October 1830 – 30 June 1906) *# Selina Barbara Wilhelmina Ponsonby (20 January 1835 – 22 July 1919) *# Frederick John Ponsonby (21 March 1837 – 3 February 1894) * Sir Henry Frederick Ponsonby (10 December 1825 – 21 November 1895)<ref name=":0" /> * Hon. Mary Elizabeth Bulteel (21 Sep 1832 – 1916)<ref>{{Cite journal|date=2020-11-07|title=John Crocker Bulteel|url=https://en.wikipedia.org/w/index.php?title=John_Crocker_Bulteel&oldid=987566588|journal=Wikipedia|language=en}}</ref> *# Alberta Victoria Ponsonby (6 May 1862 – 15 October 1945) *# Cecil William Davidge Ponsonby (28 March 1863 – 16 January 1936) *# Magdalen Ponsonby (24 June 1864 – 1 July 1934) *# John Ponsonby (25 March 1866 – 26 March 1952) *# '''Frederick Edward Grey Ponsonby''' (16 September 1867 – 20 October 1935) *# Arthur Augustus William Harry Ponsonby (16 February 1871 – 24 March 1946) * Fritz (Frederick Edward Grey) Ponsonby (16 September 1867 – 20 October 1935) ===Relations=== * Mr. and Mrs. Edward Ponsonby * Mr. Fritz Ponsonby * Mr. John Ponsonby * Miss M. Ponsonby * Miss Melita and Miss Julia Ponsonby * Ponsonby Fane *Hon. Sir Frederick Cavendish Ponsonby's sister was Lady Caroline Lamb, who married Viscount Melbourne, Prime Minister early in Queen Victoria's reign. == Notes and Questions == === Frederick Ponsonby's ''Recollections of Three Reigns'' === [[Social Victorians/Royals Amateur Theatricals|Amateur theatricals]] among the royals and household:<blockquote>Once or twice whole plays came down [to Windsor, Osborne, or Balmoral, though he probably means Windsor? Someone says Osborne somewhere, though?], but this was expensive, and amateur actors in the Household were usually asked to produce something. There were two outstanding actors, Arthur Collins [fn 1 [[Social Victorians/People/Arthur Collins|Lieutenant-Colonel Arthur Collins]], Gentleman Usher to Queen Victoria.] and Alec Yorke, and these two alternately drilled the members of the Household and produced plays. [50/51] The first time I acted was in ''She Stoops to Conquer'', stage-managed by Arthur Collins, who did the part of the Innkeeper. This was before I joined the Household [in 1894]. My brother Arthur [fn 1 Later 1^st Lord Ponsonby of Shulbrede. For many years Socialist M.P. for Brightside Division of Sheffield.] was Tony Lumpkin, while Arthur Bigge and I did the two lovers, and Princess Louise and Princess Beatrice the two principal ladies’ parts. The Queen came to the rehearsals, which frightened us all very much, and when she saw me chucking Princess Louise under the chin (I was supposed to mistake her for the barmaid) she thought this was overdone. I received a message that I had better not indulge in any chucking under the chin. The next day I went through my part but never came within touching distance of Princess Louise, and again received a message to say I was overdoing it the other way. I consulted Princess Louise herself, who roared with laughter at my dilemma, and we finally hit off a happy medium. Both Princess Louise and Princess Beatrice were quite good in their parts, but very sketchy with the words. I therefore learnt their parts as well as my own so that I could either say their words or prompt them. Everyone else did the same, but there was one small bit when they were both on together and of course they stuck, each one thinking it was the other’s fault. After an awkward pause the servants gave a round of applause, which I thought was a very intelligent way of helping them, but although the prompter was able to start them again, they could not get going and the stage carpenter solved the problem by letting the curtain down. Another time I acted in ''A Scrap of Paper'' at Balmoral, this time under Alec Yorke as stage manager. The Queen thought the performances so good that she invited Hare and his company, who were playing at Aberdeen, to come and witness the performance. They must have been amused as they all knew the play well, but of course they were loud in their praises. There were also tableaux in which all the members of the Royal Family took part. They must have been very wearying for the audience, who had to sit for two and a half hours with very long intervals between the tableaux. The only person who thoroughly enjoyed them was Clarkson, the wig-maker from London. As he hob-nobbed with the Royal Family and as he supplied all the dresses, he probably made a very good thing out of it.<ref name=":1">Ponsonby, Frederick, Sir, first Lord Sysonby. ''Recollections of Three Reigns''. Introd. and notes, Colin Welch. London, Eyre & Spottiswoode, 1951.</ref>{{rp|50–51}} </blockquote>On the transition between Victoria’s and Edward’s administrations and households:<blockquote>The King made a clean sweep of the Grooms-in-Waiting and decided to appoint in future soldiers and sailors of distinction to be Gentlemen Ushers, a position similar to that of Chamberlains abroad. In Queen Victoria’s reign these appointments were held by any type of man and there was a certain amount of nepotism. The King’s plans were not at first understood and he was hurt at the refusal of several old friends to serve as Ushers. Later many soldiers and sailors accepted but it is doubtful whether they were really suited for managing large crowds of society people at Buckingham Palace. [101/102] When the King and Queen went to London, Buckingham Palace was undergoing great changes and they therefore had to remain at Marlborough House, which had been the King’s London home since his marriage. Having been appointed Assistant Private Secretary I imagined that I should have a great deal of work to do, but when I came the next morning at 10 a.m. I was told by Francis Knollys that it was quite unnecessary for me to stay as he wanted no help. This surprised me considerably as I knew there must be a lot of work to be done, but as there is nothing more tiresome than a person hanging about idle when everyone else is busy, I made myself scarce. Even the Equerry had so much to do that he had no time to speak to me. I learnt later that while there was no friction of any sort between the new and the old Household, Marlborough House was firmly convinced that Buckingham Palace was hopelessly out of date and that none of Queen Victoria’s Household were any good. There was a great deal of truth in this, but while the other Departments were certainly out of date, the Private Secretary’s office under Sir Arthur Bigge was not only up to date, but far better managed than the equivalent office in the Prince of Wales’ Household. Bigge had organized the office on business lines, and since I had joined the Household typewriting and shorthand had been introduced, and the filing of papers brought up to date. The new appointment of myself as an Assistant Private Secretary was not altogether popular. In the first place Francis Knollys considered it quite unnecessary, and in the second place a member of Queen Victoria’s old-fashioned Household had been nominated, which probably meant incompetence. There were clerks at Buckingham Palace, but they were probably hopeless, therefore quite obviously the Marlborough House staff had better carry on as before, and the Assistant Private Secretary, who was a fifth wheel to a coach, could join the ceremonial Household.<ref name=":1" />{{rp|101–102}}</blockquote> More on the households: <blockquote> The Lord Chamberlain’s office under Sir Arthur Ellis, which was concerned with ceremonial issues, was also reorganized and all sorts of innovations and improvements were introduced: the most effective of these was the ‘Drawing-rooms’, as they were then called, being held in the evening instead of the afternoon. They were in future called ‘Courts’ and were held in the ballroom instead of the throne-room. Ellis, who had visited every Court in Europe with King Edward when he was Prince of Wales, had an unrivalled knowledge of the way things were done abroad and was able to adapt the best features of the Continental receptions. The King went into every detail and between them they stage-managed perfectly a new piece of pageantry. When I attended a ‘Court’ I was always struck by the incongruous music the band played, and determined to do what I could to have this remedied. The majority of the Household, being quite unmusical, clamoured for popular airs, and Sir Walter Parratt, the Master of the Music, who cared only for classical music and looked down on any other sort of music, complied with the demand. I argued that these popular airs robbed the ceremony of all dignity. A presentation at Court was often a great event in a lady’s life, but if she went past the King and Queen to the tune of ‘His nose was redder than it was’, the whole impression was spoilt. I maintained that minuets and old-fashioned airs, operatic music with a ‘mysterious’ touch, were what was wanted. I wrote to Sir Walter Parratt, who welcomed my opposition, to counter the pressure for popular airs; not that he carried out my proposals, but he played music that he liked.<ref name=":1" />{{rp|125}} </blockquote> On the Drawing-Rooms: <blockquote>King Edward liked French and Viennese light operas, whereas Queen Alexandra preferred grand opera, particularly Wagner. One morning the bandmaster received a message from the King to play Offenbach, and one from the Queen to play Wagner. Finding himself unable to comply with both, he thought he would hit upon the happy medium and selected Gilbert and Sullivan operas, and as always with people who compromise, he got into trouble with both the King and the Queen.<ref name=":1" />{{rp|125}}</blockquote> == Bibliography == * Ponsonby, Arthur [Arthur Augustus William Harry]. ''Henry Ponsonby, Queen Victoria's Private Secretary: His Life from His Letters''. London: Macmillan, 1943. * Ponsonby, Frederick, Sir, first Lord Sysonby. ''Recollections of Three Reigns''. Introd. and notes, Colin Welch. London, Eyre & Spottiswoode, 1951. ==Footnotes== {{reflist}} qdkr3rsvyc1tz5npf4i916zr8vgli8e 0 265303 2812098 2179260 2026-05-30T02:46:25Z CommonsDelinker 9184 Replacing Civilian_Conservation_Corps_Third_Corps_Area_typing_class_with_Works_Progress_Administration_instructor_USA_1933.gif with [[File:Civilian_Conservation_Corps,_Third_Corps_Area-_typing_class_with_W.P.A._instructor_-_DPLA_-_b67f53f0a13ec576f149e37e 2812098 wikitext text/x-wiki {{Infobox person|birth_date=July 30, 1889|death_date=March 14, 1971|occupation=Farmer, Car mechanic, Factory Worker|residence=Leighton, Alabama, South Bend, Indiana (briefly), later Colbert, Alabama||birth_place=Barton County, Kansas|death_place=Colbert, Alabama|spouse=Mae Jones}} == Biography == === Overview === Charles Frederick Gerber was interviewed for the [[wikipedia: Federal_Writers'_Project|Federal Writers' Project]] by Charles M. Donigan. Having lived in Colbert, Alabama for most of his life, he mainly worked as a tenant farmer and lived through the industrial and economic transformation of the South throughout the 20th century. All through his life, Charles claimed that "his chief ambition is to own a house".<ref>Interviewer, Donigan, Charles M. on Charles Frederick Gerber, Folder 21, in the Federal Writers' Project papers #3709, Southern Historical Collection, The Wilson Library, the University of North Carolina at Chapel Hill.</ref> He believed that owning a house would help an individual and a family have stable lifestyles. === Early Life === Charles Frederick Gerber was born on July 30, 1889, in Barton County, Kansas to German immigrant parents.<ref>“Charles Frederick Gerber (1889-1971) • FamilySearch.” Accessed July 8, 2020. https://ancestors.familysearch.org/en/LBLR-ZMP/charles-frederick-gerber-1889-1971.</ref> His father, George Frederick Gerber, worked as a homestead farmer whereas his mother, Mary Hester Gerber, worked as a housewife. George immigrated to America on account of the California Gold discovery in 1849. Using the earnings he earned from the [https://www.history.com/topics/westward-expansion/gold-rush-of-1849 Gold Rush], George settled in Ohio and continued farming while living in a farmhouse. There, he met the Hester family and got married to Mary. They then moved to Kansas and continued farming while George started a German-language newspaper. However, during those years, the family experienced several crop failures. In 1898, they, therefore, decided to move to Leighton, Alabama, after hearing about the state’s rich farmland and warm climate. They bought 310 acres of farmland a mile south of the Tennessee River and established roots. In his childhood, Charles decided to follow his father’s footsteps and become a farmer.<ref>Interviewer, Donigan, Charles M. on Charles Frederick Gerber, Folder 21, in the Federal Writers' Project papers #3709, Southern Historical Collection, The Wilson Library, the University of North Carolina at Chapel Hill.</ref> === Later Life === As a farmer, Charles grew cotton, vegetables, and raised livestock. He got married to Mae Jones in 1911 and rented an 80-acre property in Leighton, Alabama. In exchange for receiving loans from landowners and the government for the property, Charles sold shares of his cotton crops so that he could afford to live in a large house.<ref>RAJAN, RAGHURAM G., and RODNEY RAMCHARAN. "Land and Credit: A Study of the Political Economy of Banking in the United States in the Early 20th Century." The Journal of Finance66, no. 6 (2011): 1895-931. Accessed July 8, 2020. www.jstor.org/stable/41305180.</ref> He also began to work in the garage and automobile repair business in Leighton, Alabama, working as a car mechanic. He earned around $2,500.00 a year, which he believed was a stable income. During this time, he had two daughters, named Jessie Lee Gerber and May Lane Gerber.<ref>Interviewer, Donigan, Charles M. on Charles Frederick Gerber, Folder 21, in the Federal Writers' Project papers #3709, Southern Historical Collection, The Wilson Library, the University of North Carolina at Chapel Hill.</ref> However, due to the rapid industrialization of the 20th century and the consequent decline of agricultural production, many farmers like Charles moved to urban areas to take up jobs that would better support themselves financially. In addition, the Great Depression hit the United States, causing thousands of workers to lose their jobs. As a result, Charles had to sell his car to avoid losses. Fortunately, Charles and his family moved to South Bend, Indiana just before the depression. There, he worked as a factory worker in a manufacturing plant that helped assemble automobiles and other industrial goods. After two years, when the Depression ended, the family returned to Alabama, settling in Colbert County and continuing to farm. In Colbert, his daughters went to Colbert County High School. However, because Charles wanted his daughters to finish high school, he contributed a part of his earnings to their education. His younger daughter, May Helen, decided to leave the eleventh grade and get a job in order to help Jessie Lee graduate. Both his daughters eventually went on to get jobs in the business industry.<ref>Interviewer, Donigan, Charles M. on Charles Frederick Gerber, Folder 21, in the Federal Writers' Project papers #3709, Southern Historical Collection, The Wilson Library, the University of North Carolina at Chapel Hill.</ref> Through hard work and dedication, eventually, Charles was able to successfully pay off debts and return to farming full-time. Furthermore, many relief programs, including the New Deal, were passed by the Democratic Party, which helped workers like Charles remain financially stable.<ref>Ganzel, Bill. “Farming in the 1930's.” The FSA, Farm Security Administration Helps Tenant Farmers, 2003. https://livinghistoryfarm.org/farminginthe30s/water_13.html.</ref> These reforms motivated Charles to become an avid supporter of the Democratic Party. He earned enough money to live in a large house with a large farm. He used portions of his savings to donate to the local Church in Colbert. He continued farming in Colbert until his death on March 14, 1971.<ref>“Charles Frederick Gerber (1889-1971) • FamilySearch.” Accessed July 8, 2020. https://ancestors.familysearch.org/en/LBLR-ZMP/charles-frederick-gerber-1889-1971.</ref> == Social Issues == [[File:1850 Woman and Men in California Gold Rush.jpg|thumb|left|1850 Woman and Men in California Gold Rush]] === California Gold Rush === During the westward expansion of the United States, gold nuggets were found in the Sacramento Valley in 1848. It became a significant event in American history as thousands of immigrants from around the globe came to California to mine for gold. Gold was considered to be a popular commodity during the 1800s and as described by "The World Impact of the California Gold Rush 1849-1857", “The increased supply of gold meant prosperity, and their optimism encouraged widespread speculation".<ref>Roske, Ralph J. "The World Impact of the California Gold Rush 1849-1857." Arizona and the West5, no. 3 (1963): 187-232. Accessed July 8, 2020. www.jstor.org/stable/40167071.</ref> This caused thousands of immigrants to come to America because they believed working there would help them achieve their dreams and their destiny. As a result, California’s foreign population increased from less than 1,000 to 100,000.<ref>History.com Editors. “California Gold Rush.” History.com. A&E Television Networks, April 6, 2010. https://www.history.com/topics/westward-expansion/gold-rush-of-1849.</ref> In 1849, when the Gold Rush reached its peak, many individuals mortgaged their savings, took loans, or spent their savings to make the journey to California for mining. The miners, mainly men, immigrated to California to extract gold whereas women stayed back home and took care of household responsibilities. The miners mainly came from Mexico, South America, and Europe and they became known as the 49ers.<ref>Roske, Ralph J. "The World Impact of the California Gold Rush 1849-1857." Arizona and the West5, no. 3 (1963): 187-232. Accessed July 8, 2020. www.jstor.org/stable/40167071.</ref> After 1850, the gold mines in California disappeared as gold became more difficult to extract. Although the technique of [https://www.history.com/topics/westward-expansion/gold-rush-of-1849. hydraulic mining] was developed in 1853, the process destroyed much of California’s landscape. Dams that were designed to provide water to mining areas had to divert water away from the landscape and sediment clogged areas of water. In addition, fossil fuels were used to construct canals, depleting natural resources. Moreover, the conditions of mineworkers were harsh as they worked for extensive hours without many breaks, got harsh punishments from their bosses, and faced dangerous working conditions. Even though the mineworkers suffered immensely, many of them managed to provide better lifestyles for themselves and their families.<ref>History.com Editors. “California Gold Rush.” History.com. A&E Television Networks, April 6, 2010. https://www.history.com/topics/westward-expansion/gold-rush-of-1849.</ref> [[File:Civilian Conservation Corps, Third Corps Area- typing class with W.P.A. instructor - DPLA - b67f53f0a13ec576f149e37ec6d1e6a8.gif|thumb|right|Civilian Conservation Corps Third Corps Area typing class with Works Progress Administration instructor USA 1933]] === Great Depression and Democratic Party Reforms === During the [https://iowaculture.gov/history/education/educator-resources/primary-source-sets/great-depression. Great Depression], crop prices fell at extreme rates and land prices grew exponentially, preventing farmers from paying their debts or expenses. As a result, many farmers lost their land. Furthermore, excessive planting and plowing weakened the soil, leading to the [http://www.jstor.org/stable/3874910. Dust Bowl], a disaster that depleted water for vegetation production and destroyed thousands of crops. As the Depression hit its peak, the Democratic Party came to power under the Presidency of Franklin Delano Roosevelt in 1933.<ref>Federico, Giovanni. "Not Guilty? Agriculture in the 1920s and the Great Depression." The Journal of Economic History 65, no. 4 (2005): 949-76. Accessed July 14, 2020. www.jstor.org/stable/3874910.</ref> During Roosevelt's Presidency, the federal government's main priorities were to help civilians, especially farmers, get their jobs back. He noticed that farmers were forced to sell their crops at low prices even though they produced crops at high rates. This was because "The supply of crops and livestock was much higher than the demand for those products, and so the prices dropped". <ref>Ganzel, Bill. “Farming in the 1930s,” 2003. https://livinghistoryfarm.org/farminginthe30s/water_10.html.</ref> To create a balance between crop supply and crop demand, the government decided to increase crop prices and decrease crop production. As a result, many farmers could manage their farms effectively without wasting farmland. However, for the farmers who lost their farmland or jobs, Roosevelt provided jobs to these workers through the help of government organizations. Through their party platforms, the Democratic Party took larger control of the government by creating federal agencies to protect farmers and other employees by insuring their losses or expenses. For instance, the Civilian Conservation Corps was created to give employment to workers for infrastructure projects and constructing government buildings. Furthermore, the Farmer Security Administration gave loans to tenant farmers at low-interest rates, enabling them to buy land and advanced machinery for farming. The relief programs helped farmers lead financially stable lifestyles.<ref>“Great Depression and the Dust Bowl,” June 30, 2020. https://iowaculture.gov/history/education/educator-resources/primary-source-sets/great-depression.</ref> ==References== mc2vyrjknn5v1g6nosrv697p7otm7gg 0 267737 2812115 2811692 2026-05-30T10:49:08Z Saltrabook 1417466 2812115 wikitext text/x-wiki [[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|Prediabetes-remission research program]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]] = '''' The John Snow Research Institute -2026'''' = Billions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(not blood sugar).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity.[[File:Lifestyle Medicine Pillars.png|300px|right|The focus of Lifestyle Medicine is on these 6 pillars.]] [[File:John Snow.jpg|thumb|left| John Snow in the early nineteenth century]] [[File:Cholera in London 1866.gif|thumb|250px|Map of a later cholera outbreak in London, in 1866]] [[File:Choleramaplondon1866.png|thumb|right|250px|Legend for the map above]] 1. '<nowiki/>'''Prevalence studies'''' 1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref> 1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref> 1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref> 1.4. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> - General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> - Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> - John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref> 3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref> 4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(no FPG). A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained: '''5. Minutes from meetings''' <ref> https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 </ref> '''Prediabetes-Remission Research Network:''' <small>Prof. Ing. MSc. Nailet Delgado; Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> = Online Meeting , May 11, 2026 = Prediabetes – Remission in Small - and medium economy countries is the target. Keeping eyes open for applications for lifestyle medicine, sporadic supplement metformin Prepare documentation to apply for funding. Clearly define the project title, objectives, scope (countries, communities, ages), strategy (how to collect data, with what equipment, what variables), required materials, and required personnel. Meeting with Lene Daugaard director SIMAC Svendborg to do a similar to Panama study. Periodically search for organizations that could fund our project. Apply for funding when the opportunity arises. Obtain those funds. In parallel, without interruption, continue data collection according to the current methodology with prevalence data, and a comparative study between countries can be conducted using this collected data. Intervention study 16 weeks in one or two of the target populations. Proposed budget for 5 years:  5 mill Dkr. The first year can be to collect data from two other countries, Denmark and Turkey (Istanbul) compare with the data from Panama, UMIP including a short review study on similar data and an 16 week intervention study with the goal to produce a strategic model for prevalence and effectful intervention to be reported in 1-2 international  articles. PLEASE ADD THE MISSING POINTS. Possible funding entities: Innovation Fund Denmark EIFO DANIDA CROWDFOUNDING European Commission programs. SKOV website Lundbeckfonden 3vxtvaftfibxpprttmqgww0qtf1f9gs 2812122 2812115 2026-05-30T11:27:39Z Saltrabook 1417466 2812122 wikitext text/x-wiki [[ widescreen File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|Prediabetes-remission research program]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]] = '''' The John Snow Research Institute -2026'''' = Billions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(not blood sugar).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity.[[File:Lifestyle Medicine Pillars.png|300px|right|The focus of Lifestyle Medicine is on these 6 pillars.]] [[File:John Snow.jpg|thumb|left| John Snow in the early nineteenth century]] [[File:Cholera in London 1866.gif|thumb|250px|Map of a later cholera outbreak in London, in 1866]] [[File:Choleramaplondon1866.png|thumb|right|250px|Legend for the map above]] 1. '<nowiki/>'''Prevalence studies'''' 1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref> 1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref> 1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref> 1.4. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> - General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> - Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> - John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref> 3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref> 4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(no FPG). A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained: '''5. Minutes from meetings''' <ref> https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 </ref> '''Prediabetes-Remission Research Network:''' <small>Prof. Ing. MSc. Nailet Delgado; Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> = Online Meeting , May 11, 2026 = Prediabetes – Remission in Small - and medium economy countries is the target. Keeping eyes open for applications for lifestyle medicine, sporadic supplement metformin Prepare documentation to apply for funding. Clearly define the project title, objectives, scope (countries, communities, ages), strategy (how to collect data, with what equipment, what variables), required materials, and required personnel. Meeting with Lene Daugaard director SIMAC Svendborg to do a similar to Panama study. Periodically search for organizations that could fund our project. Apply for funding when the opportunity arises. Obtain those funds. In parallel, without interruption, continue data collection according to the current methodology with prevalence data, and a comparative study between countries can be conducted using this collected data. Intervention study 16 weeks in one or two of the target populations. Proposed budget for 5 years:  5 mill Dkr. The first year can be to collect data from two other countries, Denmark and Turkey (Istanbul) compare with the data from Panama, UMIP including a short review study on similar data and an 16 week intervention study with the goal to produce a strategic model for prevalence and effectful intervention to be reported in 1-2 international  articles. PLEASE ADD THE MISSING POINTS. Possible funding entities: Innovation Fund Denmark EIFO DANIDA CROWDFOUNDING European Commission programs. SKOV website Lundbeckfonden 95lpv0mvhztozmmki51h37hgifzwk1x 2812124 2812122 2026-05-30T11:28:47Z Saltrabook 1417466 2812124 wikitext text/x-wiki [[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png| wide |Prediabetes-remission research program]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]] = '''' The John Snow Research Institute -2026'''' = Billions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(not blood sugar).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity.[[File:Lifestyle Medicine Pillars.png|300px|right|The focus of Lifestyle Medicine is on these 6 pillars.]] [[File:John Snow.jpg|thumb|left| John Snow in the early nineteenth century]] [[File:Cholera in London 1866.gif|thumb|250px|Map of a later cholera outbreak in London, in 1866]] [[File:Choleramaplondon1866.png|thumb|right|250px|Legend for the map above]] 1. '<nowiki/>'''Prevalence studies'''' 1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref> 1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref> 1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref> 1.4. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> - General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> - Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> - John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref> 3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref> 4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(no FPG). A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained: '''5. Minutes from meetings''' <ref> https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 </ref> '''Prediabetes-Remission Research Network:''' <small>Prof. Ing. MSc. Nailet Delgado; Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> = Online Meeting , May 11, 2026 = Prediabetes – Remission in Small - and medium economy countries is the target. Keeping eyes open for applications for lifestyle medicine, sporadic supplement metformin Prepare documentation to apply for funding. Clearly define the project title, objectives, scope (countries, communities, ages), strategy (how to collect data, with what equipment, what variables), required materials, and required personnel. Meeting with Lene Daugaard director SIMAC Svendborg to do a similar to Panama study. Periodically search for organizations that could fund our project. Apply for funding when the opportunity arises. Obtain those funds. In parallel, without interruption, continue data collection according to the current methodology with prevalence data, and a comparative study between countries can be conducted using this collected data. Intervention study 16 weeks in one or two of the target populations. Proposed budget for 5 years:  5 mill Dkr. The first year can be to collect data from two other countries, Denmark and Turkey (Istanbul) compare with the data from Panama, UMIP including a short review study on similar data and an 16 week intervention study with the goal to produce a strategic model for prevalence and effectful intervention to be reported in 1-2 international  articles. PLEASE ADD THE MISSING POINTS. Possible funding entities: Innovation Fund Denmark EIFO DANIDA CROWDFOUNDING European Commission programs. SKOV website Lundbeckfonden cb6klbzrh9oyk7i8krptcblwoyqcmnr 2812125 2812124 2026-05-30T11:29:56Z Saltrabook 1417466 2812125 wikitext text/x-wiki [[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png| wide|200|Prediabetes-remission research program]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]] = '''' The John Snow Research Institute -2026'''' = Billions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(not blood sugar).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity.[[File:Lifestyle Medicine Pillars.png|300px|right|The focus of Lifestyle Medicine is on these 6 pillars.]] [[File:John Snow.jpg|thumb|left| John Snow in the early nineteenth century]] [[File:Cholera in London 1866.gif|thumb|250px|Map of a later cholera outbreak in London, in 1866]] [[File:Choleramaplondon1866.png|thumb|right|250px|Legend for the map above]] 1. '<nowiki/>'''Prevalence studies'''' 1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref> 1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref> 1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref> 1.4. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> - General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> - Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> - John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref> 3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref> 4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(no FPG). A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained: '''5. Minutes from meetings''' <ref> https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 </ref> '''Prediabetes-Remission Research Network:''' <small>Prof. Ing. MSc. Nailet Delgado; Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> = Online Meeting , May 11, 2026 = Prediabetes – Remission in Small - and medium economy countries is the target. Keeping eyes open for applications for lifestyle medicine, sporadic supplement metformin Prepare documentation to apply for funding. Clearly define the project title, objectives, scope (countries, communities, ages), strategy (how to collect data, with what equipment, what variables), required materials, and required personnel. Meeting with Lene Daugaard director SIMAC Svendborg to do a similar to Panama study. Periodically search for organizations that could fund our project. Apply for funding when the opportunity arises. Obtain those funds. In parallel, without interruption, continue data collection according to the current methodology with prevalence data, and a comparative study between countries can be conducted using this collected data. Intervention study 16 weeks in one or two of the target populations. Proposed budget for 5 years:  5 mill Dkr. The first year can be to collect data from two other countries, Denmark and Turkey (Istanbul) compare with the data from Panama, UMIP including a short review study on similar data and an 16 week intervention study with the goal to produce a strategic model for prevalence and effectful intervention to be reported in 1-2 international  articles. PLEASE ADD THE MISSING POINTS. Possible funding entities: Innovation Fund Denmark EIFO DANIDA CROWDFOUNDING European Commission programs. SKOV website Lundbeckfonden 5e2gtragcq856l1skkds2x9m0bhxesm 2812126 2812125 2026-05-30T11:30:36Z Saltrabook 1417466 2812126 wikitext text/x-wiki [[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|200|Prediabetes-remission research program]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]] = '''' The John Snow Research Institute -2026'''' = Billions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(not blood sugar).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity.[[File:Lifestyle Medicine Pillars.png|300px|right|The focus of Lifestyle Medicine is on these 6 pillars.]] [[File:John Snow.jpg|thumb|left| John Snow in the early nineteenth century]] [[File:Cholera in London 1866.gif|thumb|250px|Map of a later cholera outbreak in London, in 1866]] [[File:Choleramaplondon1866.png|thumb|right|250px|Legend for the map above]] 1. '<nowiki/>'''Prevalence studies'''' 1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref> 1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref> 1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref> 1.4. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> - General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> - Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> - John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref> 3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref> 4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(no FPG). A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained: '''5. Minutes from meetings''' <ref> https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 </ref> '''Prediabetes-Remission Research Network:''' <small>Prof. Ing. MSc. Nailet Delgado; Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> = Online Meeting , May 11, 2026 = Prediabetes – Remission in Small - and medium economy countries is the target. Keeping eyes open for applications for lifestyle medicine, sporadic supplement metformin Prepare documentation to apply for funding. Clearly define the project title, objectives, scope (countries, communities, ages), strategy (how to collect data, with what equipment, what variables), required materials, and required personnel. Meeting with Lene Daugaard director SIMAC Svendborg to do a similar to Panama study. Periodically search for organizations that could fund our project. Apply for funding when the opportunity arises. Obtain those funds. In parallel, without interruption, continue data collection according to the current methodology with prevalence data, and a comparative study between countries can be conducted using this collected data. Intervention study 16 weeks in one or two of the target populations. Proposed budget for 5 years:  5 mill Dkr. The first year can be to collect data from two other countries, Denmark and Turkey (Istanbul) compare with the data from Panama, UMIP including a short review study on similar data and an 16 week intervention study with the goal to produce a strategic model for prevalence and effectful intervention to be reported in 1-2 international  articles. PLEASE ADD THE MISSING POINTS. Possible funding entities: Innovation Fund Denmark EIFO DANIDA CROWDFOUNDING European Commission programs. SKOV website Lundbeckfonden j0y8ze2dwt9qtvo8q8e7y1nhbqru84f 2812127 2812126 2026-05-30T11:31:38Z Saltrabook 1417466 2812127 wikitext text/x-wiki [[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thump|]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]] = '''' The John Snow Research Institute -2026'''' = Billions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(not blood sugar).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity.[[File:Lifestyle Medicine Pillars.png|300px|right|The focus of Lifestyle Medicine is on these 6 pillars.]] [[File:John Snow.jpg|thumb|left| John Snow in the early nineteenth century]] [[File:Cholera in London 1866.gif|thumb|250px|Map of a later cholera outbreak in London, in 1866]] [[File:Choleramaplondon1866.png|thumb|right|250px|Legend for the map above]] 1. '<nowiki/>'''Prevalence studies'''' 1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref> 1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref> 1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref> 1.4. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> - General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> - Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> - John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref> 3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref> 4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(no FPG). A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained: '''5. Minutes from meetings''' <ref> https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 </ref> '''Prediabetes-Remission Research Network:''' <small>Prof. Ing. MSc. Nailet Delgado; Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> = Online Meeting , May 11, 2026 = Prediabetes – Remission in Small - and medium economy countries is the target. Keeping eyes open for applications for lifestyle medicine, sporadic supplement metformin Prepare documentation to apply for funding. Clearly define the project title, objectives, scope (countries, communities, ages), strategy (how to collect data, with what equipment, what variables), required materials, and required personnel. Meeting with Lene Daugaard director SIMAC Svendborg to do a similar to Panama study. Periodically search for organizations that could fund our project. Apply for funding when the opportunity arises. Obtain those funds. In parallel, without interruption, continue data collection according to the current methodology with prevalence data, and a comparative study between countries can be conducted using this collected data. Intervention study 16 weeks in one or two of the target populations. Proposed budget for 5 years:  5 mill Dkr. The first year can be to collect data from two other countries, Denmark and Turkey (Istanbul) compare with the data from Panama, UMIP including a short review study on similar data and an 16 week intervention study with the goal to produce a strategic model for prevalence and effectful intervention to be reported in 1-2 international  articles. PLEASE ADD THE MISSING POINTS. Possible funding entities: Innovation Fund Denmark EIFO DANIDA CROWDFOUNDING European Commission programs. SKOV website Lundbeckfonden drk73wwdbwqre5mgsiuu4vrr82m07yy 2812128 2812127 2026-05-30T11:32:26Z Saltrabook 1417466 2812128 wikitext text/x-wiki [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]] = '''' The John Snow Research Institute -2026'''' = Billions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(not blood sugar).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity.[[File:Lifestyle Medicine Pillars.png|300px|right|The focus of Lifestyle Medicine is on these 6 pillars.]] [[File:John Snow.jpg|thumb|left| John Snow in the early nineteenth century]] [[File:Cholera in London 1866.gif|thumb|250px|Map of a later cholera outbreak in London, in 1866]] [[File:Choleramaplondon1866.png|thumb|right|250px|Legend for the map above]] 1. '<nowiki/>'''Prevalence studies'''' 1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref> 1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref> 1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref> 1.4. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> - General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> - Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> - John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref> 3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref> 4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(no FPG). A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained: '''5. Minutes from meetings''' <ref> https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 </ref> '''Prediabetes-Remission Research Network:''' <small>Prof. Ing. MSc. Nailet Delgado; Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> = Online Meeting , May 11, 2026 = Prediabetes – Remission in Small - and medium economy countries is the target. Keeping eyes open for applications for lifestyle medicine, sporadic supplement metformin Prepare documentation to apply for funding. Clearly define the project title, objectives, scope (countries, communities, ages), strategy (how to collect data, with what equipment, what variables), required materials, and required personnel. Meeting with Lene Daugaard director SIMAC Svendborg to do a similar to Panama study. Periodically search for organizations that could fund our project. Apply for funding when the opportunity arises. Obtain those funds. In parallel, without interruption, continue data collection according to the current methodology with prevalence data, and a comparative study between countries can be conducted using this collected data. Intervention study 16 weeks in one or two of the target populations. Proposed budget for 5 years:  5 mill Dkr. The first year can be to collect data from two other countries, Denmark and Turkey (Istanbul) compare with the data from Panama, UMIP including a short review study on similar data and an 16 week intervention study with the goal to produce a strategic model for prevalence and effectful intervention to be reported in 1-2 international  articles. PLEASE ADD THE MISSING POINTS. Possible funding entities: Innovation Fund Denmark EIFO DANIDA CROWDFOUNDING European Commission programs. SKOV website Lundbeckfonden daa3gq0kzd1jzmy77ouif9l7ypokywz 2812129 2812128 2026-05-30T11:40:38Z Saltrabook 1417466 2812129 wikitext text/x-wiki [[ widescreen File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|Prediabetes-remission research program]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]] = '''' The John Snow Research Institute -2026'''' = Billions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(not blood sugar).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity.[[File:Lifestyle Medicine Pillars.png|300px|right|The focus of Lifestyle Medicine is on these 6 pillars.]] [[File:John Snow.jpg|thumb|left| John Snow in the early nineteenth century]] [[File:Cholera in London 1866.gif|thumb|250px|Map of a later cholera outbreak in London, in 1866]] [[File:Choleramaplondon1866.png|thumb|right|250px|Legend for the map above]] 1. '<nowiki/>'''Prevalence studies'''' 1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref> 1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref> 1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref> 1.4. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> - General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> - Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> - John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref> 3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref> 4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(no FPG). A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained: '''5. Minutes from meetings''' <ref> https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 </ref> '''Prediabetes-Remission Research Network:''' <small>Prof. Ing. MSc. Nailet Delgado; Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> = Online Meeting , May 11, 2026 = Prediabetes – Remission in Small - and medium economy countries is the target. Keeping eyes open for applications for lifestyle medicine, sporadic supplement metformin Prepare documentation to apply for funding. Clearly define the project title, objectives, scope (countries, communities, ages), strategy (how to collect data, with what equipment, what variables), required materials, and required personnel. Meeting with Lene Daugaard dir. SIMAC Svendborg. Periodically search for organizations that could fund our project. Apply for funding when the opportunity arises. Obtain those funds. In parallel, without interruption, continue data collection according to the current methodology with prevalence data, and a comparative study between countries can be conducted using this collected data. Intervention study 16 weeks in one or two of the target populations. Proposed budget for 5 years:  5 mill Dkr. The first year can be to collect data from two other countries, Denmark and Turkey (Istanbul) compare with the data from Panama, UMIP including a short review study on similar data and an 16 week intervention study with the goal to produce a strategic model for prevalence and effectful intervention to be reported in 1-2 international  articles. Possible funding entities: Innovation Fund Denmark EIFO DANIDA CROWDFOUNDING European Commission programs. SKOV website Lundbeckfonden ort0g4czetkecn13heor0t0xvh40yz7 2812130 2812129 2026-05-30T11:43:20Z Saltrabook 1417466 2812130 wikitext text/x-wiki [[File:ChatGPT Image 30 may 2026, 12 29 12 p.m.png|ChatGPT_Image_30_may_2026,_12_29_12_p.m]] [[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|Prediabetes-remission research program]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]] = '''' The John Snow Research Institute -2026'''' = Billions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(not blood sugar).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity.[[File:Lifestyle Medicine Pillars.png|300px|right|The focus of Lifestyle Medicine is on these 6 pillars.]] [[File:John Snow.jpg|thumb|left| John Snow in the early nineteenth century]] [[File:Cholera in London 1866.gif|thumb|250px|Map of a later cholera outbreak in London, in 1866]] [[File:Choleramaplondon1866.png|thumb|right|250px|Legend for the map above]] 1. '<nowiki/>'''Prevalence studies'''' 1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref> 1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref> 1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref> 1.4. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> - General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> - Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> - John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref> 3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref> 4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(no FPG). A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained: '''5. Minutes from meetings''' <ref> https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 </ref> '''Prediabetes-Remission Research Network:''' <small>Prof. Ing. MSc. Nailet Delgado; Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> = Online Meeting , May 11, 2026 = Prediabetes – Remission in Small - and medium economy countries is the target. Keeping eyes open for applications for lifestyle medicine, sporadic supplement metformin Prepare documentation to apply for funding. Clearly define the project title, objectives, scope (countries, communities, ages), strategy (how to collect data, with what equipment, what variables), required materials, and required personnel. Meeting with Lene Daugaard dir. SIMAC Svendborg. Periodically search for organizations that could fund our project. Apply for funding when the opportunity arises. Obtain those funds. In parallel, without interruption, continue data collection according to the current methodology with prevalence data, and a comparative study between countries can be conducted using this collected data. Intervention study 16 weeks in one or two of the target populations. Proposed budget for 5 years:  5 mill Dkr. The first year can be to collect data from two other countries, Denmark and Turkey (Istanbul) compare with the data from Panama, UMIP including a short review study on similar data and an 16 week intervention study with the goal to produce a strategic model for prevalence and effectful intervention to be reported in 1-2 international  articles. Possible funding entities: Innovation Fund Denmark EIFO DANIDA CROWDFOUNDING European Commission programs. SKOV website Lundbeckfonden eakwfkw3lonulb9c518388l9mj00nkt 2812131 2812130 2026-05-30T11:44:29Z Saltrabook 1417466 2812131 wikitext text/x-wiki [[File:ChatGPT Image 30 may 2026, 12 29 12 p.m.png|thumb|ChatGPT_Image_30_may_2026,_12_29_12_p.m]] [[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|Prediabetes-remission research program]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]] = '''' The John Snow Research Institute -2026'''' = Billions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(not blood sugar).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity.[[File:Lifestyle Medicine Pillars.png|300px|right|The focus of Lifestyle Medicine is on these 6 pillars.]] [[File:John Snow.jpg|thumb|left| John Snow in the early nineteenth century]] [[File:Cholera in London 1866.gif|thumb|250px|Map of a later cholera outbreak in London, in 1866]] [[File:Choleramaplondon1866.png|thumb|right|250px|Legend for the map above]] 1. '<nowiki/>'''Prevalence studies'''' 1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref> 1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref> 1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref> 1.4. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> - General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> - Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> - John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref> 3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref> 4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(no FPG). A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained: '''5. Minutes from meetings''' <ref> https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 </ref> '''Prediabetes-Remission Research Network:''' <small>Prof. Ing. MSc. Nailet Delgado; Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> = Online Meeting , May 11, 2026 = Prediabetes – Remission in Small - and medium economy countries is the target. Keeping eyes open for applications for lifestyle medicine, sporadic supplement metformin Prepare documentation to apply for funding. Clearly define the project title, objectives, scope (countries, communities, ages), strategy (how to collect data, with what equipment, what variables), required materials, and required personnel. Meeting with Lene Daugaard dir. SIMAC Svendborg. Periodically search for organizations that could fund our project. Apply for funding when the opportunity arises. Obtain those funds. In parallel, without interruption, continue data collection according to the current methodology with prevalence data, and a comparative study between countries can be conducted using this collected data. Intervention study 16 weeks in one or two of the target populations. Proposed budget for 5 years:  5 mill Dkr. The first year can be to collect data from two other countries, Denmark and Turkey (Istanbul) compare with the data from Panama, UMIP including a short review study on similar data and an 16 week intervention study with the goal to produce a strategic model for prevalence and effectful intervention to be reported in 1-2 international  articles. Possible funding entities: Innovation Fund Denmark EIFO DANIDA CROWDFOUNDING European Commission programs. SKOV website Lundbeckfonden 3pem4yrhc6gk2vrepkct4hdg8uppuas 2812132 2812131 2026-05-30T11:45:34Z Saltrabook 1417466 2812132 wikitext text/x-wiki [[File:ChatGPT Image 30 may 2026, 12 29 12 p.m.png|120|ChatGPT_Image_30_may_2026,_12_29_12_p.m]] [[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|Prediabetes-remission research program]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]] = '''' The John Snow Research Institute -2026'''' = Billions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(not blood sugar).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity.[[File:Lifestyle Medicine Pillars.png|300px|right|The focus of Lifestyle Medicine is on these 6 pillars.]] [[File:John Snow.jpg|thumb|left| John Snow in the early nineteenth century]] [[File:Cholera in London 1866.gif|thumb|250px|Map of a later cholera outbreak in London, in 1866]] [[File:Choleramaplondon1866.png|thumb|right|250px|Legend for the map above]] 1. '<nowiki/>'''Prevalence studies'''' 1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref> 1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref> 1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref> 1.4. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> - General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> - Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> - John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref> 3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref> 4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(no FPG). A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained: '''5. Minutes from meetings''' <ref> https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 </ref> '''Prediabetes-Remission Research Network:''' <small>Prof. Ing. MSc. Nailet Delgado; Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> = Online Meeting , May 11, 2026 = Prediabetes – Remission in Small - and medium economy countries is the target. Keeping eyes open for applications for lifestyle medicine, sporadic supplement metformin Prepare documentation to apply for funding. Clearly define the project title, objectives, scope (countries, communities, ages), strategy (how to collect data, with what equipment, what variables), required materials, and required personnel. Meeting with Lene Daugaard dir. SIMAC Svendborg. Periodically search for organizations that could fund our project. Apply for funding when the opportunity arises. Obtain those funds. In parallel, without interruption, continue data collection according to the current methodology with prevalence data, and a comparative study between countries can be conducted using this collected data. Intervention study 16 weeks in one or two of the target populations. Proposed budget for 5 years:  5 mill Dkr. The first year can be to collect data from two other countries, Denmark and Turkey (Istanbul) compare with the data from Panama, UMIP including a short review study on similar data and an 16 week intervention study with the goal to produce a strategic model for prevalence and effectful intervention to be reported in 1-2 international  articles. Possible funding entities: Innovation Fund Denmark EIFO DANIDA CROWDFOUNDING European Commission programs. SKOV website Lundbeckfonden lfwlf9ku6hygfbby29cjjpmnmqlpkuf 2812133 2812132 2026-05-30T11:46:33Z Saltrabook 1417466 2812133 wikitext text/x-wiki [[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|Prediabetes-remission research program]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]] = '''' The John Snow Research Institute -2026'''' = Billions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(not blood sugar).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity.[[File:Lifestyle Medicine Pillars.png|300px|right|The focus of Lifestyle Medicine is on these 6 pillars.]] [[File:John Snow.jpg|thumb|left| John Snow in the early nineteenth century]] [[File:Cholera in London 1866.gif|thumb|250px|Map of a later cholera outbreak in London, in 1866]] [[File:Choleramaplondon1866.png|thumb|right|250px|Legend for the map above]] 1. '<nowiki/>'''Prevalence studies'''' 1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref> 1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref> 1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref> 1.4. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> - General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> - Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> - John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref> 3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref> 4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(no FPG). A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained: '''5. Minutes from meetings''' <ref> https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 </ref> '''Prediabetes-Remission Research Network:''' <small>Prof. Ing. MSc. Nailet Delgado; Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> = Online Meeting , May 11, 2026 = Prediabetes – Remission in Small - and medium economy countries is the target. Keeping eyes open for applications for lifestyle medicine, sporadic supplement metformin Prepare documentation to apply for funding. Clearly define the project title, objectives, scope (countries, communities, ages), strategy (how to collect data, with what equipment, what variables), required materials, and required personnel. Meeting with Lene Daugaard dir. SIMAC Svendborg. Periodically search for organizations that could fund our project. Apply for funding when the opportunity arises. Obtain those funds. In parallel, without interruption, continue data collection according to the current methodology with prevalence data, and a comparative study between countries can be conducted using this collected data. Intervention study 16 weeks in one or two of the target populations. Proposed budget for 5 years:  5 mill Dkr. The first year can be to collect data from two other countries, Denmark and Turkey (Istanbul) compare with the data from Panama, UMIP including a short review study on similar data and an 16 week intervention study with the goal to produce a strategic model for prevalence and effectful intervention to be reported in 1-2 international  articles. Possible funding entities: Innovation Fund Denmark EIFO DANIDA CROWDFOUNDING European Commission programs. SKOV website Lundbeckfonden td7jdzpp03x8mu70cq2juk6wsejqgtd 2812134 2812133 2026-05-30T11:52:14Z Saltrabook 1417466 /* ' The John Snow Research Institute -2026' */ 2812134 wikitext text/x-wiki == '''The John Snow Prediabetes Research Institute''' == [[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|Prediabetes-remission research program]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]] '''The Prediabetes-remission research program''' Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (BMI) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(not blood sugar).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity.[[File:Lifestyle Medicine Pillars.png|300px|right|The focus of Lifestyle Medicine is on these 6 pillars.]] [[File:John Snow.jpg|thumb|left| John Snow in the early nineteenth century]] [[File:Cholera in London 1866.gif|thumb|250px|Map of a later cholera outbreak in London, in 1866]] [[File:Choleramaplondon1866.png|thumb|right|250px|Legend for the map above]] 1. '<nowiki/>'''Prevalence studies'''' 1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref> 1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref> 1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref> 1.4. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> - General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> - Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> - John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref> 3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref> 4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(no FPG). A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained: '''5. Minutes from meetings''' <ref> https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 </ref> '''Prediabetes-Remission Research Network:''' <small>Prof. Ing. MSc. Nailet Delgado; Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> = Online Meeting , May 11, 2026 = Prediabetes – Remission in Small - and medium economy countries is the target. Keeping eyes open for applications for lifestyle medicine, sporadic supplement metformin Prepare documentation to apply for funding. Clearly define the project title, objectives, scope (countries, communities, ages), strategy (how to collect data, with what equipment, what variables), required materials, and required personnel. Meeting with Lene Daugaard dir. SIMAC Svendborg. Periodically search for organizations that could fund our project. Apply for funding when the opportunity arises. Obtain those funds. In parallel, without interruption, continue data collection according to the current methodology with prevalence data, and a comparative study between countries can be conducted using this collected data. Intervention study 16 weeks in one or two of the target populations. Proposed budget for 5 years:  5 mill Dkr. The first year can be to collect data from two other countries, Denmark and Turkey (Istanbul) compare with the data from Panama, UMIP including a short review study on similar data and an 16 week intervention study with the goal to produce a strategic model for prevalence and effectful intervention to be reported in 1-2 international  articles. Possible funding entities: Innovation Fund Denmark EIFO DANIDA CROWDFOUNDING European Commission programs. SKOV website Lundbeckfonden ei5t2pvqd4nfjydsmlujredtodt1aah 2812135 2812134 2026-05-30T11:56:49Z Saltrabook 1417466 /* The John Snow Prediabetes Research Institute */ 2812135 wikitext text/x-wiki == '''The John Snow Prediabetes Research Institute''' == [[File:ChatGPT Image 30 may 2026, 11 58 20 a.m.png|thumb|Prediabetes-remission research program]] [[File:ChatGPT Image 24 abr 2026, 08 16 04 a.m.png|thumb|]] Millions are at increased risk of developing metabolic syndromes with prediabetes, diabetes type 2, high blood pressure and overweight. All can lower their risks by staying physical active and eating well. For early identification of the risks we propose to register weight and height (BMI) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(not blood sugar).The 16-weeks '''intervention studies''' include learnings by short video sequences and self-monitoring of blood sugar with glucometer, and self-evaluation of diet and physical activity.[[File:Lifestyle Medicine Pillars.png|300px|right|The focus of Lifestyle Medicine is on these 6 pillars.]] [[File:John Snow.jpg|thumb|left| John Snow in the early nineteenth century]] [[File:Cholera in London 1866.gif|thumb|250px|Map of a later cholera outbreak in London, in 1866]] [[File:Choleramaplondon1866.png|thumb|right|250px|Legend for the map above]] 1. '<nowiki/>'''Prevalence studies'''' 1.1 The-International-Maritime-Health-Database <ref>https://www.dropbox.com/scl/fi/z3cq5ciiev06y8v9duw7u/A-International-Maritime-Health-Database.docx?cloud_editor=word&dl=0&rlkey=pt0kdesvmagcxaa2wez3tmza3 </ref> 1.2 Nursing Students Health Database <ref> https://www.dropbox.com/scl/fi/tcznmmd2y3nona5e3h1ro/The-Nursing-students-health-database.docx?cloud_editor=word&dl=0&rlkey=onbjh4o8ko1lzdvgyi8nlrotk </ref> 1.3. Medical student's Health Database <ref>https://www.dropbox.com/scl/fi/f16h9b60u4gxgt56un2jf/The-Medical-students-Health-database.docx?cloud_editor=word&dl=0&rlkey=xyfqen5trdc5lniaovipl548n </ref> 1.4. School childrens Health database <ref> https://www.dropbox.com/scl/fi/u6u50c8bxwhte9t2t6ck8/The-School-children-s-Health-database.docx?cloud_editor=word&dl=0&rlkey=zlyz5wn673wf7owettq3nx3h5 </ref> 2. '''Intervention studies''' Englsh <ref>https://www.dropbox.com/scl/fi/oi6cx6tlwwvoko3ed37tn/Invitation-to-the-course-English.docx?cloud_editor=word&dl=0&rlkey=7kzg91tqfgjskxf5aji8khicx </ref> Danish <ref>https://www.dropbox.com/scl/fi/2qahc3q9hmf4skbvk77ab/Invitation-to-the-course-in-Danish.docx?cloud_editor=word&dl=0&rlkey=x63w8oqvarz284zg2btq2johv </ref> Spanish <ref> https://www.dropbox.com/scl/fi/bn71inqeeth4o4mc1fjth/Invitation-to-the-course-Spanish.docx?cloud_editor=word&dl=0&rlkey=popmr1fnodh1v951v9l7k9ezv </ref> - General research protocol draft <ref> https://www.dropbox.com/scl/fi/gau25oy5y1s57046icjt2/Research-protocol-draft.docx?cloud_editor=word&dl=0&rlkey=wat63e25ritmujwcpss8s4v0s </ref> - Health Promoting Schools <ref> https://www.dropbox.com/scl/fi/0rm7honrezbjwrcy3h3yk/Health-promoting-schools.docx?cloud_editor=word&dl=0&rlkey=673jyzcmwbfw7k9ui9nmtp0zh </ref> - John Snow Institute bylaws <ref> https://www.dropbox.com/scl/fi/lccr7jtnga1u0x75117zn/John-Snow-revision-2-March-11.doc?cloud_editor=word&dl=0&rlkey=lz2gi7mslcoay5dzygg8h6n6r </ref> 3. '''Publications and pptx''' 2016-2026 <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/The_International_Type_2_Diabetes_Mellitus_and_Hypertension_Research_Group#The_John_Snow_Institute </ref><ref name=":0"> https://www.dropbox.com/scl/fi/mw7ft423lkkpjoxywd2bf </ref> 4. '''Strategies for research and implementation''' For early identification of the risks we propose to register weight and height (Body Mass Index) and the fasting blood sugar in the '''Prevalence studies''' at the schools for seafarers, nurses, medical students and the kids schools.(no FPG). A practical strategy for prediabetes remission in low- and middle-income countries (LMICs) must assume that laboratory capacity, workforce, and financing are constrained: '''5. Minutes from meetings''' <ref> https://www.dropbox.com/t/3ZfLGngkS3pSlAQ3 </ref> '''Prediabetes-Remission Research Network:''' <small>Prof. Ing. MSc. Nailet Delgado; Dr. Olaf Jensen, MD, MPH, PhD, o147248@gmail.com; MSc.Ph.D. Bishal Gyawali (SDU); MSc.PhD Vivi Just-Nørregaard; Dr. Johan Hviid Andersen MD, PhD. Prof Århus University; Prof. MSc. Agnes Flores, UMECIT, Panama; Dr. Maite, Vacamonte, Panama; Bruno Nørdam, Randers; Dr. Maite Duque, Venezuela; Dr. Indira Santos Panama; Med.Stud. Ashley Lezcano, Panama; Dr. Antonio Roberto Abaya MD Filippines; Dr. Jen Mendoza, MD, Filippines; Dr. Andra Ergle MD, Latvia; Prof. MSc. Ingrid Morató, Tarragona/Cadiz, Spain; MBA Christian Acheampong, Turkey; Dr. Alejandro Martinez, MPH, Costa Rica; Dr. Med. Sci Finn Gyntelberg; NFA.and Bispebj. Hosp. Denmark,</small> ==References== [[Category:Prediabetes ]] <references />Education 1: Research Methodology <ref>https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/EDUCATION/Education_module_links</ref> <references /> = Online Meeting , May 11, 2026 = Prediabetes – Remission in Small - and medium economy countries is the target. Keeping eyes open for applications for lifestyle medicine, sporadic supplement metformin Prepare documentation to apply for funding. Clearly define the project title, objectives, scope (countries, communities, ages), strategy (how to collect data, with what equipment, what variables), required materials, and required personnel. Meeting with Lene Daugaard dir. SIMAC Svendborg. Periodically search for organizations that could fund our project. Apply for funding when the opportunity arises. Obtain those funds. In parallel, without interruption, continue data collection according to the current methodology with prevalence data, and a comparative study between countries can be conducted using this collected data. Intervention study 16 weeks in one or two of the target populations. Proposed budget for 5 years:  5 mill Dkr. The first year can be to collect data from two other countries, Denmark and Turkey (Istanbul) compare with the data from Panama, UMIP including a short review study on similar data and an 16 week intervention study with the goal to produce a strategic model for prevalence and effectful intervention to be reported in 1-2 international  articles. Possible funding entities: Innovation Fund Denmark EIFO DANIDA CROWDFOUNDING European Commission programs. SKOV website Lundbeckfonden 67exiz5fjk1fntxxjsw6zf61dc7j0h2 2 275853 2812099 2298595 2026-05-30T02:47:35Z CommonsDelinker 9184 Replacing Woman_with_Quilt_GReat_Depression_27-0934a.gif with [[File:Works_Progress_Administration-_Crafts_-_Quilting_Project_-_DPLA_-_cd0f62d819dc880b8e67944f76b39d7c.gif]] (by [[:c:User:CommonsDelinker|CommonsDelinker]] because: [[:c:COM:FR|File renamed 2812099 wikitext text/x-wiki = Engl 105 Unit 2 Project, Lolly Bleu = {| class="wikitable" ! colspan="2" |Lolly Bleu |- !Birth Date |Unknown, Late 1800s |- !Birth Place |Unknown, On the Gulf Coast of Texas |- !Died |Unknown |- !Nationality |American |- !Race |Caucasian |- !Residence |Venus, Florida |- !Education |None |} == Overview == [[File:Map of Florida highlighting Highlands County.svg|thumb|149x149px|Highlands County, Florida where the Bleu family lived]] Lolly Bleu was born on the [[wikipedia:Gulf_Coast_of_the_United_States|Gulf Coast of Texas]] in the late 19th century. Her exact date of birth is unknown. Bleu was interviewed by Barbara Darsey of the [[Federal Writers' Project – Life Histories|Federal Writer's Project]] on November 29th, 1938 on a small farm in rural [[wikipedia:Venus,_Florida|Venus, Florida]]. == Biography == === Personal Life === Bleu's exact location and date of birth is unknown but both her and her husband grew up in poor, farming families on the Texas Gulf Coast. The Bleu family eventually moved as squatters to an abandoned plot of land in Venus, Florida after hearing rumors of the rich farmlands. Bleu and her family found an old farmhouse on this land and converted it into their new home. Bleu was a mother of 13 children in total, one of whom, Edie, suffered from a [[wikipedia:Developmental_disorder|developmental disorder]]. Edie required special attention from Bleu including assisted feedings of a special diet she created after receiving little access to good doctors. Bleu always valued education, however, she never pursued degrees, enjoying country life more. Bleu had high hopes for her children’s education as she pushed them to make a long 2 mile walk from their rural farm to the bus stop every day. Furthermore, Bleu encouraged her daughter's to take any courses they could find, including business and [[wikipedia:Shorthand|stenography]]<ref name=":0">{{Cite web|url=https://dc.lib.unc.edu/cdm/ref/collection/03709/id/930|title=Folder 101: Darsey, Barbara (interviewer): Lolly Bleu, Florida Squatter :: Federal Writers Project Papers|website=dc.lib.unc.edu|access-date=2021-07-18}}</ref>. [[File:Federal Emergency Relief Administration, FERA camps for unemployed women. Negro camp in Atlanta, GA - NARA - 196584.tif|thumb|220x220px|FERA Camp for the unemployed during the Great Depression]] Besides being a farmer, Bleu found enjoyment in sewing quilts and canning some of the produce grown on their farm. When the family needed money, she sold her handmade goods. In addition to this source of income and money brought home by the kids, Bleu's husband, who's name is unknown, relied on work assistance from [[wikipedia:Federal_Emergency_Relief_Administration|FERA]] after the [[wikipedia:Great_Depression|Great Depression]] hit. Bleu was fond of the government's help. But, she firmly believed that women should stay out of politics, leaving all voting up to her husband's decisions. Despite both being roughly in their 50's, Bleu and her husband were quite healthy. Bleu attributed this to farm work and long walks into town. Furthermore, she noted that she studied food values and made sure to feed her family a balanced diet. Bleu's cause, location, and date of death are all unknown<ref name=":0" />. == Social Issues == === Gender Roles === [[File:Works Progress Administration- Crafts - Quilting Project - DPLA - cd0f62d819dc880b8e67944f76b39d7c.gif|thumb|Woman with a quilt she crafted during the Great Depression]] After wide spread devastation hit the United States during the Great Depression, household members had to adjust. Many families moved into cheaper shanty towns, called [[wikipedia:Hooverville|Hoovervilles]]. Others relied on finding more sources of income. Many times, the responsibility for this task fell on women. In fact, between 1930 and 1940, the number of married women in the [[wikipedia:Workforce|labor force]] increased by nearly 50 percent<ref name=":1">{{Cite journal|last=Bolin|first=Winifred D. Wandersee|date=1978|title=The Economics of Middle-Income Family Life: Working Women During the Great Depression|url=https://www.jstor.org/stable/1888142|journal=The Journal of American History|volume=65|issue=1|pages=60–74|doi=10.2307/1888142|issn=0021-8723}}</ref>. The stereotype that men made the money and women focused solely on caring for the home began to vanish as women's work outside of the home reflected dire economic need<ref name=":1" />. For example, impoverished mothers had to prioritize their family's finances at times by selling whatever they could make at home. In short, "flexibility and creativity<ref>{{Cite journal|last=Helmbold|first=Lois Rita|date=1987|title=Beyond the Family Economy: Black and White Working-Class Women during the Great Depression|url=https://www.jstor.org/stable/3177885|journal=Feminist Studies|volume=13|issue=3|pages=629–655|doi=10.2307/3177885|issn=0046-3663}}</ref>" defined women's actions during this time. === Access to Education === [[File:Civilian Conservation Corps Camp S-52.jpg|thumb|294x294px|A camp of the Civilian Conservation Corps, which provided education and work assistance to young men throughout the Great Depression]] “People are moving to farm areas not because there are explicit market-based employment opportunities available there, but because the farmland offers some other means of [[wikipedia:Subsistence_agriculture|subsistence]]<ref>{{Cite journal|last=Boone|first=Christopher|last2=Wilse-Samson|first2=Laurence|date=2019-01-14|title=Farm Mechanization and Rural Migration in the Great Depression|url=https://ecommons.cornell.edu/handle/1813/71383|language=en-US}}</ref>." This observation sums up human migration throughout the Great Depression. Families were desperate to find land in areas that were not only cheap to buy but cheap to live at in the long run. Despite people spreading out into rural farm areas, schools did not follow with them. In fact, by 1934 almost 20,000 schools nationwide had closed down due to exorbitant administrative costs<ref name=":2">{{Cite web|url=https://www.theedadvocate.org/comprehending-great-depression-influenced-american-education/|title=Comprehending How The Great Depression Influenced American Education|date=2016-09-02|website=The Edvocate|language=en-US|access-date=2021-07-18}}</ref>. This crisis affected poorer southern areas, such as Venus, Florida where Bleu lived, more intensely with many schools forcing children to pay tuition<ref name=":2" />. Mothers were unable to provide their children with much education as they were already juggling responsibilities like family doctor, providing 24/7 healthcare to their families, free of charge, to save money<ref>{{Cite web|url=https://www.richmondregister.com/opinion/columns/health-care-during-the-great-depression/article_664cefd5-c948-5917-a4c5-bab9fa37938c.html|title=Health care during The Great Depression|last=Columnist|first=Glenmore Jones|website=Richmond Register|language=en|access-date=2021-07-18}}</ref>. In order to combat this, the [[wikipedia:New_Deal|New Deal policy]], created by [[wikipedia:Franklin_D._Roosevelt|President Franklin D. Roosevelt]] included organizations to assist in making work education more accessible<ref name=":2" />. For example, the [[wikipedia:Civilian_Conservation_Corps|Civilian Conservation Core]] was chartered to provide young men with environmental conservation work. == Notes == <references /> == References == # Darsey, Barbara. 1938. “Folder 101: Darsey, Barbara (Interviewer): Lolly Bleu, Florida Squatter.” Federal Writer’s Project Papers Series 1. Life Histories, 1936-1940 and undated (Subseries 1.2. Florida). <nowiki>https://dc.lib.unc.edu/cdm/singleitem/collection/03709/id/930/rec/1</nowiki>. 15 Jul. 2021. # Bolin, Winifred D. Wandersee. 2013. "The Economics of Middle-Income Family Life: Working Women During the Great Depression" In ''Volume 5/2 The Intersection of Work and Family Life'' edited by Nancy F. Cott, 566-580. Berlin, Boston: K. G. Saur. <nowiki>https://www.jstor.org/stable/1888142</nowiki>. 15 Jul. 2021. # Helmbold, Lois Rita. 1987. “Beyond the Family Economy: Black and White Working-Class Women during the Great Depression.” Feminist Studies 13 (3): 629. <nowiki>https://www.jstor.org/stable/3177885.15</nowiki> Jul. 2021. # Boone, Christopher and Wilse-Samson, Lawrence. 2019. ''Farm Mechanization and Rural Migration in the Great Depression''. Cornell University Library: SC Johnson School of Business. <nowiki>https://ecommons.cornell.edu/handle/1813/71383</nowiki>. 15 Jul 2021. # Lynch, Matthew. 2016. “Comprehending How The Great Depression Influenced American Education.” The Advocate. 2016. <nowiki>https://www.theedadvocate.org/comprehending-great-depression-influenced-american-education/</nowiki>. 15 Jul. 2021. # Jones, Glenmore. 2020. “Health Care during The Great Depression.” Richmond Register. 2020. <nowiki>https://www.richmondregister.com/opinion/columns/health-care-during-the-great-depression/article_664cefd5-c948-5917-a4c5-bab9fa37938c.html</nowiki>. 15 Jul. 2021. edbm0v1r45belenfbgle2dy2qptb8ht 0 275859 2812097 2299344 2026-05-30T02:45:45Z CommonsDelinker 9184 Replacing Farm_Security_Administration_school_in_Alabama_USA_1935.gif with [[File:Farm_Security_Administration-_School_in_Alabama_-_DPLA_-_3c404c0ee16ff88f0ecf4a81dc65463d.gif]] (by [[:c:User:CommonsDelinker|CommonsDelinker]] because: [[:c:COM:FR|File ren 2812097 wikitext text/x-wiki == Overview == Dick Striker was a Caucasian American male born around 1875 in Goldsboro, North Carolina. He was interviewed by William Forster on September 12, 1938, at the reported age of 63. == Biography == {{Infobox person|ethnicity=Caucasian American|occupation=Farmer|spouse=Mary (Smith-Jackson) Striker|birthdate=c. 1875-76|birthplace=Goldsboro, NC|death date=Unkown}} === Personal Life === Dick Striker was born and raised on a farm near Goldsboro and Newton Grove, North Carolina. At the age of twenty, Striker married and had four children. Unfortunately, three of those children, and his first wife died at a young age. Six months after his fist wife’s death, Striker married a widow named Mary (Smith) Jackson. Mary Smith had her first marriage at the age of fifteen and had three children, two of whom died at a young age. Dick and Mary Striker had one daughter together, who was born a year after they married. In 1928, Dick’s brother, John Striker, died, leaving two small boys, as John’s wife had also passed away shortly before. As a result, Dick and Mary Striker adopted the children.<ref>Interview, Forster, William O. on Dick Striker, September 12, 1938, Folder 397, Federal Writing Project Papers, Southern Historical Collection, UNC Chapel Hill.</ref> === Livelihood === As a farmer in the early-twentieth century, Striker struggled immensely to live and support his family because of the extreme poverty farmers faced. Every year Striker maintained at least 15 acres of farmland tilling cotton, corn, and tobacco, among other products. For this work, Striker’s annual wage was a meager fifty dollars (about eight hundred dollars today), plus ownership of a small, rustic house and garden. Thus, Striker had little to no discretionary income to raise and support, which had a major impact as the family struggled to get food, clothing, or shelter. In fact, Striker’s adopted sons had so little clothing that Reverend Anthony Wilkins, a pastor at a nearby church and Striker’s nearest neighbor, considered requesting welfare workers to “get the boys some clothes”. Striker’s house was very poorly maintained. Striker himself stated, “‘I don’t suppose the owner could get anyone to live here except me.’”<ref>Forster, 4</ref> This issue of poverty did not only affect Dick Striker, but all farmers in the Great Depression. == Social Issues == [[File:Poor mother and children, Oklahoma, 1936 by Dorothea Lange.jpg|thumb|A displaced mother and her children struggling from poverty during the Great Depression]] === Employment/Poverty === The problem began in World War One. During the war farmers–especially those from Western countries, “substantially increased their productive capacity, augmenting inputs and adopting several innovations, most notably mechanical ones.”<ref>Giovanni Federico, “Not guilty? Agriculture in the 1920s and the Great Depression” ''Journal of Economic History'' 65, no. 4 (2005) 949</ref> However, demand stagnated as the consumption of agricultural products remained constant (price and income [[wikipedia:Price_elasticity_of_demand|inelasticity]]) despite greater supply. In addition, the population was increasing much more slowly than before the war. The excess production either converted to stocks, was financed with short-term loans, or, most significantly, caused prices of agricultural products to decline steeply.<ref>Jakob Madsen, “Agricultural crises and the international transmission of the Great Depression” ''Journal of Economic History'' 61, no. 2 (2001) 329</ref> Agriculture price declines had a significant influence on the rest of the economy, as they lowered overall demand. Prices fell further, lowering investments and diminishing exporters' purchasing power, compelling them to quit the [[wikipedia:Gold_standard|gold standard]].<ref>Barry Eichengreen, “Golden Fetters. The Gold Standard and the Great Depression, 1919-1939.” ''The Economic History Review'' 47, no. 1 (1994) 407</ref> As a result, the [[wikipedia:Great_Depression|Great Depression]] ensued, and farmers were highly vulnerable to any additional decrease in prices, even a relatively mild one.<ref>Federico, 950</ref> [[File:Farm Security Administration- School in Alabama - DPLA - 3c404c0ee16ff88f0ecf4a81dc65463d.gif|thumb|Typical school in rural Alabama during the Great Depression]] === Education === One of the major issues that stemmed from these economic disparities in rural areas was education. Education in rural America during the 1930s was virtually non-existent, for a few reasons. Villages and rural areas were difficult to access, school facilities were insufficient, and student performance was low when compared to other schools.<ref>Michael Gardiner, “Education in Rural Areas” ''Issues in Education Policy'' no. 4 (2008) </ref> These conditions were exacerbated during the Great Depression. During the Great Depression, the value of farm land and agricultural products fell drastically. This directly impacted the education system as taxes that were drawn on those sectors and used to support schools fell.<ref>Claudia Reinhardt, Bill Ganzel, “Going to School in Rural America during the 1930s”, Ganzel Group, 2003. <nowiki>https://livinghistoryfarm.org/farminginthe30s/life_21.html</nowiki>.</ref> However, it also had a large indirect effect on education. In the 1930s, children whose families farmed for a living–as did most North Carolinians–had a harder difficulty attending school than children who lived in towns. These rural youngsters may be required to miss school for planting, hoeing, and harvest. In order to live, most families need all members to work.<ref>Anita Davis, “Public Schools in North Carolina in the Great Depression | NCpedia”, Tar Heel Junior Historian, 2010. <nowiki>https://www.ncpedia.org/public-schools-great-depression</nowiki>.</ref> In the case of Dick Striker, this was picking cotton. For more than three weeks, Striker kept his kids from the consolidated school at Newton Grove, and forced them to pick cotton.<ref>Forster, 7</ref> == References == <references /> == Bibliography == Aldcroft, Derek H., and Barry Eichengreen. 1994. “Golden Fetters. The Gold Standard and the Great Depression, 1919-1939.” The Economic History Review 47 (1). <nowiki>https://doi.org/10.2307/2598244</nowiki>. Davis, Anita. “Public Schools in North Carolina in the Great Depression | NCpedia”. Tar Heel Junior Historian, 2010. <nowiki>https://www.ncpedia.org/public-schools-great-depression</nowiki>. Federico, Giovanni. 2005. “Not Guilty? Agriculture in the 1920s and the Great Depression.” Journal of Economic History 65 (4). <nowiki>https://doi.org/10.1017/S0022050705000367</nowiki>. Gardiner, Michael. “Education in Rural Areas”, ''Issues in Education Policy'' no. 4 (2008): 1-33 Interview, Forster, William O. on Dick Striker, September 12, 1938, Folder 397, Federal Writing Project Papers, Southern Historical Collection, UNC Chapel Hill. Madsen, J. B. 2001. “Agricultural Crises and the International Transmission of the Great Depression.” Journal of Economic History 61 (2). <nowiki>https://doi.org/10.1017/S0022050701028030</nowiki>. Reinhardt Claudia, Ganzel Bill, “Going to School in Rural America during the 1930s”. LivingHistoryFarm. Ganzel Group, 2003. https://livinghistoryfarm.org/farminginthe30s/life_21.html.= nm1z6yacabaevib3u94or5p20ywvgt9 0 275885 2812096 2535570 2026-05-30T02:44:25Z CommonsDelinker 9184 Replacing Works_Progress_Administration_arts_and_crafts_display_Kansas_USA_1930s.gif with [[File:WPA_Display_of_Arts_and_Crafts_Projects_in_Kansas_-_DPLA_-_2594a9566c411d79ab5b9af4c3c056ff.gif]] (by [[:c:User:CommonsDelinker|CommonsDelinker]] because: [[: 2812096 wikitext text/x-wiki {| class="wikitable" ! colspan="2" |Lolly Bleu |- !Born |Unknown, Late 1800s along Gulf Coast of Texas |- !Death |Unknown |- !Nationality |American |- !Ethnicity |Caucasian |- !Spouse |Unknown |- !Residence |Venus, Florida |- !Education |None |} == Overview == Lolly Bleu was born on the [[wikipedia:Gulf_Coast_of_the_United_States|Gulf Coast of Texas]] in the late 19th century. Her exact date of birth is unknown. Bleu was interviewed by Barbara Darsey of the [[Federal Writers' Project – Life Histories|Federal Writer's Project]] on November 29th, 1938 on a small farm in rural [[wikipedia:Venus,_Florida|Venus, Florida]]. [[File:Map of Florida highlighting Highlands County.svg|thumb|166x166px|Highlands County, Florida where the Bleu family lived]] == Biography == === Personal Life === [[File:Federal Emergency Relief Administration, FERA camp for unemployed women in Arcola, PA - NARA - 196583.tif|thumb|187x187px|FERA camp for the unemployed during the Great Depression]] Bleu's exact location and date of birth is unknown but both her and her husband grew up in poor, farming families on the Texas Gulf Coast. The Bleu family eventually moved as squatters to an abandoned plot of land in Venus, Florida after hearing rumors of the rich farmlands. They found an old farmhouse on this land and converted it into their new home. Bleu was a mother of 13 children in total, one of whom, Edie, suffered from a [[wikipedia:Developmental_disorder|developmental disorder.]] Edie required special attention from Bleu including assisted feedings of a special diet she created after receiving little access to doctors. Bleu always valued education, however, she never pursued degrees, enjoying country life more. Bleu had high hopes for her children’s education as she pushed them to make a long 2 mile walk from their rural farm to the bus stop every day. Furthermore, Bleu encouraged her daughter's to take any courses they could find, including business and [[wikipedia:Shorthand|stenography]]<ref name=":0">{{Cite web|url=https://dc.lib.unc.edu/cdm/ref/collection/03709/id/930|title=Folder 101: Darsey, Barbara (interviewer): Lolly Bleu, Florida Squatter :: Federal Writers Project Papers|website=dc.lib.unc.edu|access-date=2021-07-20}}</ref>. Besides being a farmer, Bleu found enjoyment in sewing quilts and canning some of the produce grown on their farm. When the family needed money, she sold her handmade goods. In addition to this source of income and money brought home by the kids, Bleu's husband, who's name is unknown, relied on work assistance from [[wikipedia:Federal_Emergency_Relief_Administration|FERA]] after the [[wikipedia:Great_Depression|Great Depression]] hit. Bleu was fond of the government's help. But she firmly believed that women should stay out of politics, leaving all voting up to her husband's decisions. Despite both being roughly in their 50's, Bleu and her husband were quite healthy. Bleu attributed this to farm work and long walks into town. Furthermore, she noted that she studied food values and made sure to feed her family a balanced diet. Bleu's cause, location, and date of death are all unknown<ref name=":0" />. == Social Issues == [[File:WPA Display of Arts and Crafts Projects in Kansas - DPLA - 2594a9566c411d79ab5b9af4c3c056ff.gif|thumb|262x262px|Working woman during the Great Depression]] === Gender Roles === After widespread devastation hit the United States during the Great Depression, household members had to adjust. Many families moved into cheaper shanty towns, called [[wikipedia:Hooverville|Hoovervilles]]. Others relied on finding more sources of income. Many times, the responsibility for this task fell on women. In fact, between 1930 and 1940, the number of married women in the [[wikipedia:Workforce|labor force]] increased by nearly 50 percent<ref name=":1">{{Cite journal|last=Bolin|first=Winifred D. Wandersee|date=1978|title=The Economics of Middle-Income Family Life: Working Women During the Great Depression|url=https://www.jstor.org/stable/1888142|journal=The Journal of American History|volume=65|issue=1|pages=60–74|doi=10.2307/1888142|issn=0021-8723}}</ref>. The stereotype that men made the money and women focused solely on caring for the home began to vanish as women's work outside of the home reflected dire economic need<ref name=":1" />. For example, impoverished mothers had to prioritize their family's finances at times by selling whatever they could make at home. In short, "flexibility and creativity" defined women's actions during this time<ref>{{Cite journal|last=Helmbold|first=Lois Rita|date=1987|title=Beyond the Family Economy: Black and White Working-Class Women during the Great Depression|url=https://www.jstor.org/stable/3177885|journal=Feminist Studies|volume=13|issue=3|pages=629–655|doi=10.2307/3177885|issn=0046-3663}}</ref>. === Access to Education === [[File:Schoolinthe1930s.jpg|thumb|261x261px|School during the Great Depression]] “People are moving to farm areas not because there are explicit market-based employment opportunities available there, but because the farmland offers some other means of [[wikipedia:Subsistence_agriculture|subsistence]]<ref>{{Cite journal|last=Boone|first=Christopher|last2=Wilse-Samson|first2=Laurence|date=2019-01-14|title=Farm Mechanization and Rural Migration in the Great Depression|url=https://ecommons.cornell.edu/handle/1813/71383|language=en-US}}</ref>." This observation perfectly sums up human migration throughout the Great Depression. Families were desperate to find land in areas that were not only cheap to buy but cheap to live at in the long run. Despite people spreading out into rural farm areas, schools did not follow with them. In fact, by 1934 almost 20,000 schools nationwide had closed down due to exorbitant administrative costs<ref name=":2">{{Cite web|url=https://www.theedadvocate.org/comprehending-great-depression-influenced-american-education/|title=Comprehending How The Great Depression Influenced American Education|date=2016-09-02|website=The Edvocate|language=en-US|access-date=2021-07-20}}</ref>. This crisis affected poorer southern areas, such as Venus, Florida where Bleu lived, more intensely with many schools forcing children to pay tuition<ref name=":2" />. Mothers were unable to provide their children with much education as they were already juggling responsibilities like family doctor, providing 24/7 healthcare to their families, free of charge, to save money<ref>{{Cite web|url=https://www.richmondregister.com/opinion/columns/health-care-during-the-great-depression/article_664cefd5-c948-5917-a4c5-bab9fa37938c.html|title=Health care during The Great Depression|last=Columnist|first=Glenmore Jones|website=Richmond Register|language=en|access-date=2021-07-20}}</ref>. In order to combat this, the [[wikipedia:New_Deal|New Deal policy]], created by [[wikipedia:Franklin_D._Roosevelt|President Franklin D. Roosevelt]], included organizations to assist in making work education more accessible<ref name=":2" />. For example, the [[wikipedia:Civilian_Conservation_Corps|Civilian Conservation Corps]] was chartered to provide young men with environmental conservation work. == Notes == <references /> == References == # Darsey, Barbara. 1938. “Folder 101: Darsey, Barbara (Interviewer): Lolly Bleu, Florida Squatter.” Federal Writer’s Project Papers Series 1. Life Histories, 1936-1940 and undated (Subseries 1.2. Florida). <nowiki>https://dc.lib.unc.edu/cdm/singleitem/collection/03709/id/930/rec/1</nowiki>. 15 Jul. 2021. # Bolin, Winifred D. Wandersee. 2013. "The Economics of Middle-Income Family Life: Working Women During the Great Depression" In ''Volume 5/2 The Intersection of Work and Family Life''edited by Nancy F. Cott, 566-580. Berlin, Boston: K. G. Saur. <nowiki>https://www.jstor.org/stable/1888142</nowiki>. 15 Jul. 2021. # Helmbold, Lois Rita. 1987. “Beyond the Family Economy: Black and White Working-Class Women during the Great Depression.” Feminist Studies 13 (3): 629. <nowiki>https://www.jstor.org/stable/3177885.15</nowiki> Jul. 2021. # Boone, Christopher and Wilse-Samson, Lawrence. 2019. ''Farm Mechanization and Rural Migration in the Great Depression''. Cornell University Library: SC Johnson School of Business. <nowiki>https://ecommons.cornell.edu/handle/1813/71383</nowiki>. 15 Jul 2021. # Lynch, Matthew. 2016. “Comprehending How The Great Depression Influenced American Education.” The Advocate. 2016. <nowiki>https://www.theedadvocate.org/comprehending-great-depression-influenced-american-education/</nowiki>. 15 Jul. 2021. # Jones, Glenmore. 2020. “Health Care during The Great Depression.” Richmond Register. 2020. <nowiki>https://www.richmondregister.com/opinion/columns/health-care-during-the-great-depression/article_664cefd5-c948-5917-a4c5-bab9fa37938c.html</nowiki>. 15 Jul. 2021. {{CourseCat}} ray6ce65npvggpy2l9v4huh4eck8bop 0 275946 2812100 2535566 2026-05-30T02:47:56Z CommonsDelinker 9184 Replacing Franklin_D._Roosevelt_Delivers_1943_State_of_the_Union_Address.jpg with [[File:Franklin_D._Roosevelt_Delivers_1943_State_of_the_Union_Address_-_DPLA_-_494838cf5e98520cc4f09199543eb3c0.jpg]] (by [[:c:User:CommonsDelinker|CommonsDelinker]] because 2812100 wikitext text/x-wiki == Arthur Graham Harris == == Overview == Arthur Graham Harris was a general practitioner from [[wikipedia:Fairfield,_Hyde_County,_North_Carolina|Fairfield, North Carolina]]. He was interviewed by W.O. Saunders for the Federal Writers' Project. == Biography == === Early Life === Harris was one of eight children to his father who was a corn and pig farmer. He graduated from the [[wikipedia:University_of_Maryland,_College_Park|University of Maryland]] in 1905 in medicine and pharmacy. He then interned at a hospital in [[wikipedia:Maryland|Maryland]] for one year before returning home to the rural town of Fairfield, NC, to form a [[wikipedia:Medicine|medical practice]]. === Career === After Harris’s father died, he took over ownership of his farm. During the period of the [[wikipedia:World_War_I|First World War]], Harris would make as much as $5,000 a year by leasing the farm out to tenants. He also supplemented his income by working part-time for the government. He maintained a profitable practice and through his various sources of income, he was able to lead an extravagant lifestyle. He would buy expensive cars and would wear them out in a year. He was also known to be an affluent and generous person in the community, regularly handing out expensive cigars to people. As the [[wikipedia:Great_Depression|Great Depression]] hit Fairfield, he was unable to make money off the farm and was no longer employed by the government. As such, his income was restricted to less than $700 a year he now made from his private practice. As a rural doctor, he also had the cost of procuring and stocking the medication he sold. Many of the families in Fairfield now lived in poverty and were working for the [[wikipedia:Works_Progress_Administration|Works Progress Administration]]. During the Great Depression, 80% of the residents could not afford medicine or healthcare, which greatly affected the profitability of his practice. [[wikipedia:Typhoid_fever|Typhoid]] and [[wikipedia:Dysentery|Dysentery]] were endemic in the town, and some families were so poor, they lived practically on [[wikipedia:Cornbread|cornbread]]. === Later Life === Harris and his wife had two sons who both went to college, one who went on to work in [[wikipedia:New_York_(state)|New York]], and the other, John, who returned home to take over the farm from Harris. Harris’s wife was keen for John to be in a [[wikipedia:White-collar_worker|white-collar profession]], but Harris thought that kind of work was too insecure and insisted he returned home. Harris found a hobby of raising and selling turkeys. He continued to run his medical practice afterwards, but his income was never as much as before the Great Depression. === Views on Universal Healthcare === Harris was a strong supporter of [[wikipedia:Socialized_medicine|socialized healthcare]] and the [[wikipedia:Wagner-Murray-Dingell_Bill|Wagner National Health Bill]]. He claimed that both doctors and patients would benefit. Doctors would be happier on a fixed income paid by the State and free to give services to anyone who needs them, and patients would no longer bear the cost of healthcare and wait until they are gravely ill until they visit a doctor. == Social Issues == === Public Health and the Affordability of Healthcare in the 1930s === The rise in unemployment and decrease in wages caused by the Great Depression greatly affected the ability of many people to afford medical care.<ref name=":0">1.     Gorman, Linda. "The history of health care costs and health insurance." ''Wisconsin Policy Research Institute Report'' 19, no. 10 (2006): 1-31.</ref>This was in addition to a rise in the cost of health care throughout the 1920s. Hospital costs rose from 7.6% of total family medical bills in 1918, to 13% in 1929. Costs continued to rise throughout the depression, eventually reaching 40% in 1934.<ref name=":0" /> Counterintuitively, the negative effects the depression had on the affordability of public health contrasted with a decline in [[wikipedia:Mortality_rate|mortality]] and rise in [[wikipedia:Life_expectancy|life expectancy]] in people throughout the US from 1929-1933. During the Great Depression, the life expectancy of white males and females increased by 6 years, and non-white males and females saw an increase of 8 years.<ref name=":1">1.     Tapia Granados, J. A., and A. V. Diez Roux. “Life and Death during the Great Depression.” Proceedings of the National Academy of Sciences 106, no. 41 (2009): 17290–95. <nowiki>https://doi.org/10.1073/pnas.0904491106</nowiki>.</ref> Of the six causes of death that caused two-thirds of total mortality, only [[wikipedia:Suicide|suicides]] increased during the Great Depression, accounting for around 2% of all deaths.<ref name=":1" /> [[File:Franklin D. Roosevelt Delivers 1943 State of the Union Address - DPLA - 494838cf5e98520cc4f09199543eb3c0.jpg|thumb|200px|Franklin D. Roosevelt delivering the State of the Union Address which mentions the Second Bill of Rights]]The rise in suicides was resolved in part due to Franklin D. Roosevelt’s [[wikipedia:New_Deal|New Deal]] relief program starting from 1933, which resulted in a drastic decrease in suicide rates throughout the 1930s.<ref>1.     Price V. Fishback, Michael R. Haines, Shawn Kantor; Births, Deaths, and New Deal Relief during the Great Depression. ''The Review of Economics and Statistics'' 2007; 89 (1): 1–14. doi: <nowiki>https://doi.org/10.1162/rest.89.1.1</nowiki></ref> === Push for Socialized Healthcare After The Great Depression === During his first term of office, [[wikipedia:Franklin_D._Roosevelt|Franklin D. Roosevelt]] sought to include publicly funded health care programs in his Social Security legislation. The reforms were strongly opposed by the [[wikipedia:American_Medical_Association|American Medical Association]].<ref name=":2">1.     A Brief History: Universal Health Care Efforts in the US.” PNHP, May 3, 2021. <nowiki>https://pnhp.org/a-brief-history-universal-health-care-efforts-in-the-us</nowiki></ref> Roosevelt's Committee on Economic Security feared that the inclusion of health insurance on the bill would threaten the passage of the entire [[wikipedia:Welfare|Social Security]] legislation, and so it was therefore excluded. Any further changes in social policy were extremely difficult after a [[wikipedia:Conservatism|conservative]] resurgence in the 1938 election.<ref name=":2" /> Roosevelt publicly endorsed socialized healthcare in his [[wikipedia:1944_United_States_presidential_election|1944 election campaign]]. He proposed the [[wikipedia:Second_Bill_of_Rights|Second Bill of Rights]] which would have guaranteed universal medical care and “the right to adequate medical care and the opportunity to achieve and enjoy good health”. He also supported the [[wikipedia:Wagner-Murray-Dingell_Bill|Wagner-Murray-Dingell Bill]] of 1945 which proposed to create a national medical and hospitalization program, but the bill was ultimately rejected.<ref>1.     “Social Security.” Social Security History. Accessed July 19, 2021. <nowiki>https://www.ssa.gov/history/corningchap3.html</nowiki></ref> == References == {{reflist}} {{CourseCat}} sbzim60ed9ybqh96dmgvxxxhjsfcz2c 0 285723 2812058 2811727 2026-05-29T20:25:04Z Scogdill 1331941 2812058 wikitext text/x-wiki Especially with respect to fashion, the newspapers at the end of the 19th century in the UK often used specialized terminology. The definitions on this page are to provide a sense of what someone in the late 19th century might have meant by the term rather than a definition of what we might mean by it today. In the absence of a specialized glossary from the end of the 19th century in the U.K., we use the ''Oxford English Dictionary'' because the senses of a word are illustrated with examples that have dates so we can be sure that the senses we pick are appropriate for when they are used in the quotations we have. We also sometimes use the French ''Wikipédia'' to define a word because many technical terms of fashion were borrowings from the French. Also, often the French ''Wikipédia'' provides historical context for the uses of a word similar to the way the ''OED'' does. == Articles or Parts of Clothing: Men's == [[Social Victorians/Terminology#Military|Men's military uniforms]] are discussed below. === À la Romaine === [[File:Johann Baptist Straub - Mars um 1772-1.jpg|thumb|left|alt=Old and damaged marble statue of a Roman god of war with flowing cloak, big helmet with a plume on top, and armor|Johann Baptist Straub's 1772 ''à la romaine'' ''Mars'']] A few people who attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball in 1897]] personated Roman gods or people. They were dressed not as Romans, however, but ''à la romaine'', which was a standardized style of depicting Roman figures that was used in paintings, sculpture and the theatre for historical dress from the 17th until the 20th century. The codification of the style was developed in France in the 17th century for theatre and ballet, when it became popular for masked balls. Women as well as men could be dressed ''à la romaine'', but much sculpture, portraiture and theatre offered opportunities for men to dress in Roman style — with armor and helmets — and so it was most common for men. In large part because of the codification of the style as well as the painting and sculpture, the style persisted and remained influential into the 20th century and can be found in museums and galleries and on monuments. For example, Johann Baptist Straub's 1772 statue of Mars (left), now in the Bayerisches Nationalmuseum, Munich, missing part of an arm, shows Mars ''à la romaine''. In London, an early 17th-century example of a figure of Mars ''à la romaine'', with a helmet, is "at the foot of the Buckingham tomb in Henry VII's Chapel at Westminster Abbey."<ref>Webb, Geoffrey. “Notes on Hubert Le Sueur-II.” ''The Burlington Magazine for Connoisseurs'' 52, no. 299 (1928): 81–89. http://www.jstor.org/stable/863535.</ref>{{rp|81, Col. 2c}} [[File:Sir-Anthony-van-Dyck-Lord-John-Stuart-and-His-Brother-Lord-Bernard-Stuart.jpg|thumb|alt=Old painting of 2 men flamboyantly and stylishly dressed in colorful silk, with white lace, high-heeled boots and long hair|Van Dyck's c. 1638 painting of cavaliers Lord John Stuart and his brother Lord Bernard Stuart]] [[File:Frans_Hals_-_The_Meagre_Company_(detail)_-_WGA11119.jpg|thumb|Frans Hals - The Meagre Company (detail) - WGA11119.jpg]] === Cavalier === As a signifier in the form of clothing of a royalist political and social ideology begun in France in the early 17th century, the cavalier style established France as the leader in fashion and taste. Adopted by [[Social Victorians/Terminology#Military|wealthy royalist British military officers]] during the time of the Restoration, the style signified a political and social position, both because of the loyalty to Charles I and II as well the wealth required to achieve the cavalier look. The style spread beyond the political, however, to become associated generally with dress as well as a style of poetry.<ref>{{Cite journal|date=2023-04-25|title=Cavalier poet|url=https://en.wikipedia.org/w/index.php?title=Cavalier_poet&oldid=1151690299|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Cavalier_poet.</ref> Van Dyck's 1638 painting of two brothers (right) emphasizes the cavalier style of dress. The cavalier style included gloves with large gauntlets, lace on boots, more loosely fitted breeches, coats or doublets, which were slashed so the shirt beneath was visible. Men who dressed in cavalier style also wore large and, later, powdered wigs, like those of Louis XIV, having taken the French style back to Britain. Neck treatments in the cavalier style were falling bands, wide lace collars and jabots. These were all looser, unsupported with wires, the way the earlier ruffs were, and unstarched. === Coats === ==== Doublet ==== * In the 19th-century newspaper accounts we have seen that use this word, doublet seems always to refer to a garment worn by a man, but historically women may have worn doublets. In fact, a doublet worn by Queen Elizabeth I — the golden doublet — exists and is in the Elizabeth Day McCormick collection at the Boston Museum of Fine Arts (but no image of it is in the public domain). * Technically doublets were long sleeved, although we cannot be certain what this or that Victorian tailor would have done for a costume. For example, the [[Social Victorians/People/Spencer Compton Cavendish#Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball|Duke of Devonshire's costume as Charles V]] shows long sleeves that may be part of the surcoat but should be the long sleeves of the doublet. ==== Pourpoint ==== A padded doublet worn under armor to protect the warrior from the metal chafing. A pourpoint could also be worn without the armor. ==== Surcoat ==== Sometimes just called ''coat''. [[File:Oscar Wilde by Sarony 1882 18.jpg|thumb|alt=Old photograph of a young man wearing a velvet jacket, knee breeches, silk hose and shiny pointed shoes with bows, seated on a sofa and leaning on his left hand and holding a book in his right| Oscar Wilde, 1882, by Napoleon Sarony]] === Hose, Stockings and Tights === Newspaper accounts from the late 19th century of men's clothing use the term ''hose'' for what we might call stockings or tights. In fact, the terminology is specific. ''Stockings'' is the more general term and could refer to hose or tights. With knee breeches men wore hose, which ended above the knee, and women wore hose under their dresses. The ''Oxford English Dictionary'' defines tights as "Tight-fitting breeches, worn by men in the 18th and early 19th centuries, and still forming part of court-dress."<ref>“Tights, N.” ''Oxford English Dictionary'', Oxford UP, July 2023, https://doi.org/10.1093/OED/2693287467.</ref> By 1897, the term was in use for women's stockings, which may have come up only to the knee. Tights were also worn by dancers and acrobats. This general sense of ''tights'' does not assume that they were knitted. ''Clocking'' is decorative embroidery on hose, usually, at the ankles on either the inside or the outside of the leg. It started at the ankle and went up the leg, sometimes as far as the knee. On women's hose, the clocking could be quite colorful and elaborate, while the clocking on men's hose was more inconspicuous. In many photographs men's hose are wrinkled, especially at the ankles and the knees, because they were shaped from woven fabric. Silk hose were knitted instead of woven, which gave them elasticity and reduced the wrinkling. The famous Sarony carte de visite photograph of Oscar Wilde (right) shows him in 1882 wearing knee breeches and silk hose, which are shiny and quite smoothly fitted although they show a few wrinkles at the ankles and knees. In the portraits of people in costume at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]], the men's hose are sometimes quite smooth, which means they were made of knitted silk and may have been smoothed for the portrait. In painted portraits the hose are almost always depicted as smooth, part of the artist's improvement of the appearance of the subject. === Shoes and Boots === == Articles or Parts of Clothing: Women's == === '''Chérusque''' === According to the French ''Wikipedia'', ''chérusque'' is a 19th-century term for the kind of standing collar like the ones worn by ladies in the Renaissance.<ref>{{Cite journal|date=2021-06-26|title=Collerette (costume)|url=https://fr.wikipedia.org/w/index.php?title=Collerette_(costume)&oldid=184136746|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Collerette_(costume)#Au+xixe+siècle+:+la+Chérusque.</ref> === Corsage === According to the ''Oxford English Dictionary'', the corsage is the "'body' of a woman's dress; a bodice."<ref>"corsage, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/42056. Accessed 7 February 2023.</ref> This sense is well documented in the ''OED'' for the mid and late 19th-century, used this way in fiction as well as in a publication like ''Godey's Lady's Book'', which would be expected to use appropriate terminology associated with fashion and dress making. The sense of "a bouquet worn on the bodice" is, according to the ''OED'', American. === [[Social Victorians/Terminology/Foundation Garments#Corset|Corset]] === === Décolletage === === Girdle === === Mancheron === According to the ''Oxford English Dictionary'', a ''mancheron'' is a "historical" word for "A piece of trimming on the upper part of a sleeve on a woman's dress."<ref>"mancheron, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/113251. Accessed 17 April 2023.</ref> At the present, in French, a ''mancheron'' is a cap sleeve "cut directly on the bodice."<ref>{{Cite journal|date=2022-11-28|title=Manche (vêtement)|url=https://fr.wikipedia.org/w/index.php?title=Manche_(v%C3%AAtement)&oldid=199054843|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Manche_(v%C3%AAtement).</ref> === Paletot === A cloak or jacket worn by both women and men in different periods. In the late 19th century, we see Queen Victoria wearing them frequently, sometimes dressed for outdoors but not always. Paletot-redingote:<blockquote>United Kingdom. Introduced in 1867, ladies' fitted long coat cut without a waist seam. It had revers and buttoned down the front. They sometimes had capes.<ref name=":7" />{{rp|217}}</blockquote> According to the French ''Wikipédia'', a paletot is longer than hip length, has long sleeves, opens in the front.<ref>{{Cite journal|date=2026-02-20|title=Manteau (vêtement)|url=https://fr.wikipedia.org/w/index.php?title=Manteau_(v%C3%AAtement)&oldid=233467144|journal=Wikipédia|language=fr}}</ref> === Petticoat === According to the ''O.E.D.'', a petticoat is a <blockquote>skirt, as distinguished from a bodice, worn either externally or showing beneath a dress as part of the costume (often trimmed or ornamented); an outer skirt; a decorative underskirt. Frequently in ''plural'': a woman's or girl's upper skirts and underskirts collectively. Now ''archaic'' or ''historical''.<ref>“petticoat, n., sense 2.b”.  ''Oxford English Dictionary'', Oxford University Press,  September 2023, <https://doi.org/10.1093/OED/1021034245></ref> </blockquote>This sense is, according to the ''O.E.D.'', "The usual sense between the 17th and 19th centuries." However, while petticoats belong in both outer- and undergarments — that is, meant to be seen or hidden, like underwear — they were always under another garment, for example, underneath an open overskirt. The primary sense seems to have shifted through the 19th century so that, by the end, petticoats were underwear and the term ''underskirt'' was used to describe what showed under an open overskirt. In the 19th century, women wore their chemises, bloomers and [[Social Victorians/Terminology/Foundation Garments#Hoops|hoops]] under their petticoats. === Stomacher === According to the ''O.E.D.'', a stomacher is "An ornamental covering for the chest (often covered with jewels) worn by women under the lacing of the bodice,"<ref>“stomacher, n.¹, sense 3.a”. ''Oxford English Dictionary'', Oxford University Press, September 2023, <https://doi.org/10.1093/OED/1169498955></ref> although by the end of the 19th century, the bodice did not often have visible laces. Some stomachers were so decorated that they were thought of as part of the jewelry. === Train === According to Debrett's,<blockquote>A peeress's coronation robe is a long-trained crimson velvet mantle, edged with miniver pure, with a miniver pure cape. The length of the train varies with the rank of the wearer: * Duchess: for rows of ermine; train to be six feet * Marchioness: three and a half rows of ermine; train to be three and three-quarters feet * Countess: three rows of ermine; train to be three and a half feet * Viscountess: two and a half rows of ermine; train to be three and a quarter feet * Baroness: two rows of ermine; train to be three feet<ref name=":2">{{Cite web|url=https://debretts.com/royal-family/dress-codes/|title=Dress Codes|website=debretts.com|language=en-US|access-date=2023-07-27}} https://debretts.com/royal-family/dress-codes/.</ref> </blockquote>The pattern on the coronet worn was also quite specific, similar but not exactly the same for peers and peeresses. Debrett's also distinguishes between coronets and tiaras, which were classified more like jewelry, which was regulated only in very general terms. Peeresses put on their coronets after the Queen or Queen Consort has been crowned. ['''peers?'''] A train is The Length of the Train '''For the monarch [or a royal?]''' == Hats, Bonnets and Headwear == === Women's === The dresses in the 1892 production of Reyer's Salammbo, based on the Flaubert novel, were influential and occasioned a lot of newspaper coverage:<blockquote>Among the concessions to women made recently in Paris, and over which old-fashioned folk shake their heads as being a terrible innovation, is the permission given to sit in the orchestra stalls at the theatre. Though only in the two last rows of the spectators, women of the first class had place, they are still obliged to appear in demi-toilette, which includes the wearing of a bonnet. It was on the occasion of the first performance of “Salammbo” that the change was allowed, and there are not wanting people who think that after such a departure a deluge, or some such visitation, may be looked for.<ref>"Ladies Column." ''Kilburn Times'' 8 July 1892, Friday: 7 [of 8], Col. 2b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001813/18920708/175/0007. Print title: ''The Kilburn Times, Hampstead and North-Western Post'', p. 7</ref></blockquote>[[Social Victorians/People/Bourke|Gwendolen Bourke]] was dressed as Salammbo at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]]. ==== Fontanges ==== [[File:Recueil de modes - Tome 4 - cent-quatre-vingt-cinq planches - estampes - btv1b105296325 (083 of 195).jpg|thumb|Recueil de modes - Tome 4 - cent-quatre-vingt-cinq planches - estampes - btv1b105296325 (083 of 195).jpg]][[File:Madame de Ludre en Stenkerke et falbala - (estampe) (2e état) - N. arnoult fec - btv1b53265886c.jpg|none|thumb|Madame de Ludre en Stenkerke et falbala - (estampe) (2e état) - N. arnoult fec - btv1b53265886c.jpg]] ==== Widow's Cap ==== or mourning bonnet According to Kate Strasdin, widow's caps were "white crinkled crape [sic] objects with long streamers flowing down the back, ... customarily worn by single old women who had never remarried."<ref>Strasdin, Kate. ''The Dress Diary: Secrets from a Victorian Woman's Wardrobe''. Pegasus, 2023.</ref>{{rp|734 of 1124}} [[Social Victorians/People/Queen Victoria#Widow's Cap|Queen Victoria's widow's caps]] and [[Social Victorians/People/Queen Victoria#Headdresses|other headdresses]] are discussed on her page. === Men's === == Cinque Cento == According to the ''Oxford English Dictionary'', ''Cinque Cento'' is a shortening of ''mil cinque cento'', or 1500.<ref>"cinquecento, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/33143. Accessed 7 February 2023.</ref> The term, then would refer, perhaps informally, to the sixteenth century. == Court Dress == Also Levee and drawing-room == Crevé == ''Creve'', without the accent, is an old word in English (c. 1450) for burst or split.<ref>"creve, v." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/44339. Accessed 8 February 2023.</ref> ['''With the acute accent, it looks like a past participle in French.'''] == Elaborations == In her 1973 ''The Best Circles: Society, Etiquette and the Season'', Leonore Davidoff notes that women’s status was indicated by dress and especially ornament: “Every cap, bow, streamer, ruffle, fringe, bustle, glove and other elaboration,” she says, “symbolised some status category for the female wearer.”<ref name=":1">Davidoff, Leonore. ''The Best Circles: Society Etiquette and the Season''. Intro., Victoria Glendinning. The Cressett Library (Century Hutchinson), 1986 (orig 1973).</ref>{{rp|93}} Looking at these elaborations as meaningful rather than dismissing them as failed attempts at "historical accuracy" reveals a great deal about the individual women who wore or carried them — and about the society women and political hostesses in their roles as managers of the social world. In her review of ''The House of Worth: Portrait of an Archive'', Mary Frances Gormally says,<blockquote>In a socially regulated year, garments custom made with a Worth label provided women with total reassurance, whatever the season, time of day or occasion, setting them apart as members of the “Best Circles” dressed in luxurious, fashionable and always appropriate attire (Davidoff 1973). The woman with a Worth wardrobe was a woman of elegance, lineage, status, extreme wealth and faultless taste.<ref>Gormally, Mary Frances. Review essay of ''The House of Worth: Portrait of an Archive'', by Amy de la Haye and Valerie D. Mendes (V&A Publishing, 2014). ''Fashion Theory'' 2017 (21, 1): 109–126. DOI: 10.1080/1362704X.2016.1179400.</ref>{{rp|117}}</blockquote> [[File:Aglets from Spanish portraits - collage by shakko.jpg|thumb|alt=A collage of 12 different ornaments typically worn by elite people from Spain in the 1500s and later|Aglets — Detail from Spanish Portraits]] === Aglet, Aiglet === Historically, an aglet is a "point or metal piece that capped a string [or ribbon] used to attach two pieces of the garment together, i.e., sleeve and bodice."<ref name=":7">Lewandowski, Elizabeth J. ''The Complete Costume Dictionary''. Scarecrow Press, 2011.</ref>{{rp|4}} Although they were decorative, they were not always visible on the outside of the clothing. They were often stuffed inside the layers at the waist (for example, attaching the bodice to a skirt or breeches). Alonso Sánchez Coello's c. 1584<ref name=":11" />{{rp|316}} portrait (above right, in the [[Social Victorians/Terminology#16th Century|Hoops section]]) shows infanta Isabel Clara Eugenia wearing a vertugado, with its "typically Spanish smooth cone-shaped contour," with "handsome aiglets cascad[ing] down center front."<ref name=":11">Payne, Blanche. ''History of Costume from the Ancient Egyptians to the Twentieth Century''. Harper & Row, 1965.</ref>{{rp|315}} === Berthe === Can be spelled ''bertha''. A wide collar made of lace and gathered at the neckline, sometimes covering the arms. Lewandowski says,<blockquote>Wide collar popular on women's gowns. Accented dropped shoulder line. Often made of lace.<ref name=":7" />{{rp|29}}</blockquote> === Dags === Popular in European dress 1450–1550, dagging was a "hanging end or shred" decoration on the edges of outer clothing, with a similar term used for "a row of decorative strips of cloth that may ornament a tent, booth or fairground."<ref>{{Cite journal|date=2026-05-14|title=dag|url=https://en.wiktionary.org/w/index.php?title=dag&oldid=90785397|journal=Wiktionary, the free dictionary|language=en}}</ref> Often dagging would be used to hem the bottom edges of hoods, doublets, tabards and chain mail. === Flounce === A ruffle that is gathered on one edge, the bottom edge is free. Flounces are typically part of the decoration on a skirt.[[File:SarahBernhardt alsKameliendame1881.jpg|thumb|Bernhardt, 1881]] === Frou-frou === In French, ''frou-frou'' or, spelled as ''froufrou'', is the sound of the rustling of silk or sometimes of fabrics in general.<ref>{{Cite journal|date=2023-07-25|title=frou-frou|url=https://fr.wiktionary.org/w/index.php?title=frou-frou&oldid=32508509|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/frou-frou.</ref> The first use the French ''Wiktionnaire'' lists is Honoré Balzac, ''La Cousine Bette'', 1846.<ref>{{Cite journal|date=2023-06-03|title=froufrou|url=https://fr.wiktionary.org/w/index.php?title=froufrou&oldid=32330124|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/froufrou.</ref> ''Frou-frou'' is also a 1869 French drawing-room comedy by Henri Meilhac and Ludovic Halévy<ref>{{Cite journal|date=2025-04-19|title=Henri Meilhac|url=https://en.wikipedia.org/w/index.php?title=Henri_Meilhac&oldid=1286340698|journal=Wikipedia|language=en}}</ref> and performed by Sarah Bernhardt in London in 1881 (Bernhardt, left, in a costume elaborate enough to be described with the term frou-frou ['''conflicting info, is a photo of Bernhardt in ''La Dame aux Camélias'' instead'''?]). ''Frou-frou'' is a term clothing historians use to describe decorative additions to an article of clothing; often the term has a slight negative connotation, suggesting that the additions are superficial and, perhaps, excessive. === Plastics === Small poufs of fabric connected in a strip in the 18th century, Rococo styles. === Pouf, Puff, Poof === According to the French ''Wikipédia'', a pouf was, beginning in 1744, a "kind of women's hairstyle":<blockquote>The hairstyle in question, known as the “pouf”, had launched the reputation of the enterprising Rose Bertin, owner of the Grand Mogol, a very prominent fashion accessories boutique on Rue Saint-Honoré in Paris in 1774. Created in collaboration with the famous hairdresser, Monsieur Léonard, the pouf was built on a scaffolding of wire, fabric, gauze, horsehair, fake hair, and the client's own hair held up in an almost vertical position. — (Marie-Antoinette, ''Queen of Fashion'', translated from the American by Sylvie Lévy, in ''The Rules of the Game'', n° 40, 2009)</blockquote>''Puff'' and ''poof'' are used to describe clothing. === Shirring === ''Shirring'' is the gathering of fabric to make poufs or puffs. The 19th century is known for its use of this decorative technique. Even men's clothing had shirring: at the shoulder seam. === Sequins === Sequins, paillettes, spangles Sequins — or paillettes — are "small, scalelike glittering disks."<ref name=":7" />{{rp|216}} The French ''Wiktionnaire'' defines ''paillette'' as "Lamelle de métal, brillante, mince, percée au milieu, ordinairement ronde, et qu’on applique sur une étoffe pour l’orner [A strip of metal, shiny, thin, pierced in the middle, usually round, and which is applied to a fabric in order to decorate it.]"<ref name=":8">{{Cite journal|date=2024-03-18|title=paillette|url=https://fr.wiktionary.org/w/index.php?title=paillette&oldid=33809572|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/paillette.</ref> According to the ''OED'', the use of ''sequin'' as a decorative device for clothing (as opposed to gold coins minted and used for international trade) goes back to the 1850s.<ref>“Sequin, N.” ''Oxford English Dictionary'', Oxford UP, September 2023, https://doi.org/10.1093/OED/4074851670.</ref> The first instance of ''spangle'' as "A small round thin piece of glittering metal (usually brass) with a hole in the centre to pass a thread through, used for the decoration of textile fabrics and other materials of various sorts" is from c. 1420.<ref>“Spangle, N. (1).” ''Oxford English Dictionary'', Oxford UP, July 2023, https://doi.org/10.1093/OED/4727197141.</ref> The first use of ''paillette'' listed in the French ''Wiktionnaire'' is in Jules Verne in 1873 to describe colored spots on icy walls.<ref name=":8" /> Currently many distinguish between sequins (which are smaller) and paillettes (which are larger). Before the 20th century, sequins were metal discs or foil leaves, and so of course if they were silver or copper, they tarnished. It is not until well into the 20th century that plastics were invented and used for sequins. === Trim and Lace === ''A History of Feminine Fashion'', published sometime before 1927 and probably commissioned by [[Social Victorians/People/Dressmakers and Costumiers#Worth, of Paris|the Maison Worth]], describes Charles Frederick Worth's contributions to the development of embroidery and [[Social Victorians/Terminology#Passementerie|passementerie]] (trim) from about the middle of the 19th century:<blockquote>For it must be remembered that one of M. Worth's most important and lasting contributions to the prosperity of those who cater for women's needs, as well as to the variety and elegance of his clients' garments, was his insistence on new fabrics, new trimmings, new materials of every description. In his endeavours to restore in Paris the splendours of the days of La Pompadour, and of Marie Antoinette, he found himself confronted at the outset with a grave difficulty, which would have proved unsurmountable to a man of less energy, resource and initiative. The magnificent materials of those days were no longer to be had! The Revolution had destroyed the market for beautiful materials of this, type, and the Restoration and regime of Louis Philippe had left a dour aspect in the City of Light. ... On parallel lines [to his development of better [[Social Victorians/Terminology#Satin|satin]]], [Worth] stimulated also the manufacture of embroidery and ''passementerie''. It was he who first started the manufacture of laces copied from the designs of the real old laces. He was the / first dressmaker to use fur in the trimming of light materials — but he employed only the richer furs, such as sable and ermine, and had no use whatever for the inferior varieties of skins.<ref name=":9">[Worth, House of.] {{Cite book|url=http://archive.org/details/AHistoryOfFeminineFashion|title=A History Of Feminine Fashion (1800s to 1920s)}} Before 1927. [Likely commissioned by Worth. Link is to Archive.org; info from Wikimedia Commons: https://commons.wikimedia.org/wiki/File:Worth_Biarritz_salon.jpg.]</ref>{{rp|6–7}}</blockquote> ==== Gold and Silver Fabric and Lace ==== The ''Encyclopaedia Britannica'' (9th edition) has an article on gold and silver fabric, threads and lace attached to the article on gold. (This article is based on knowledge that would have been available toward the end of the 19th century and does not, obviously, reflect current knowledge or ways of talking.)<blockquote>GOLD AND SILVER LACE. Under this heading a general account may be given of the use of the precious metals in textiles of all descriptions into which they enter. That these metals were used largely in the sumptuous textiles of the earliest periods of civilization there is abundant testimony; and to this day, in the Oriental centres whence a knowledge and the use of fabrics inwoven, ornamented, and embroidered with gold and silver first spread, the passion for such brilliant and costly textiles is still most strongly and generally prevalent. The earliest mention of the use of gold in a woven fabric occurs in the description of the ephod made for Aaron (Exod. xxxix. 2, 3) — "And he made the ephod of gold, blue, and purple, and scarlet, and fine twined linen. And they did beat the gold into thin plates, and cut it into wires (strips), to work it in the blue, and in the purple, and in the scarlet, and in the fine linen, with cunning work." In both the ''Iliad'' and the ''Odyssey'' distinct allusion is frequently made to inwoven and embroidered golden textiles. Many circumstances point to the conclusion that the art of weaving and embroidering with gold and silver originated in India, where it is still principally prosecuted, and that from one great city to another the practice travelled westward, — Babylon, Tarsus, Baghdad, Damascus, the islands of Cyprus and Sicily, Con- / stantinople and Venice, all in the process of time becoming famous centres of these much prized manufactures. Alexander the Great found Indian kings and princes arrayed in robes of gold and purple; and the Persian monarch Darius, we are told, wore a war mantle of cloth of gold, on which were figured two golden hawks as if pecking at each other. There is reason, according to Josephus, to believe that the “royal apparel" worn by Herod on the day of his death (Acts xii. 21) was a tissue of silver. Agrippina, the wife of the emperor Claudius, had a robe woven entirely of gold, and from that period downwards royal personages and high ecclesiastical dignitaries used cloth and tissues of gold and silver for their state and ceremonial robes, as well as for costly hangings and decorations. In England, at different periods, various names were applied to cloths of gold, as ciclatoun, tartarium, naques or nac, baudekiu or baldachin, Cyprus damask, and twssewys or tissue. The thin flimsy paper known as tissue paper, is so called because it originally was placed between the folds of gold "tissue" to prevent the contiguous surfaces from fraying each other. At what time the drawing of gold wire for the preparation of these textiles was first practised is not accurately known. The art was probably introduced and applied in different localities at widely different dates, but down till mediaeval times the method graphically described in the Pentateuch continued to be practised with both gold and silver. Fabrics woven with gold and silver continue to be used on the largest scale to this day in India; and there the preparation of the varieties of wire, and the working of the various forms of lace, brocade, and embroidery, is at once an important and peculiar art. The basis of all modern fabrics of this kind is wire, the "gold wire" of the manufacturer being in all cases silver gilt wire, and silver wire being, of course, composed of pure silver. In India the wire is drawn by means of simple draw-plates, with rude and simple appliances, from rounded bars of silver, or gold-plated silver, as the case may be. The wire is flattened into the strip or ribbon-like form it generally assumes by passing it, fourteen or fifteen strands simultaneously, over a fine, smooth, round-topped anvil, and beating it as it passes with a heavy hammer having a slightly convex surface. From wire so flattened there is made in India soniri, a tissue or cloth of gold, the web or warp being composed entirely of golden strips, and ruperi, a similar tissue of silver. Gold lace is also made on a warp of thick yellow silk with a weft of flat wire, and in the case of ribbons the warp or web is composed of the metal. The flattened wires are twisted around orange (in the case of silver, white) coloured silk thread, so as completely to cover the thread and present the appearance of a continuous wire; and in this form it is chiefly employed for weaving into the rich brocades known as kincobs or kinkhábs. Wires flattened, or partially flattened, are also twisted into exceedingly fine spirals, and in this form they are the basis of numerous ornamental applications. Such spirals drawn out till they present a waved appearance, and in that state flattened, are much used for rich heavy embroideries termed karchobs. Spangles for embroideries, &c., are made from spirals of comparatively stout wire, by cutting them down ring by ring, laying each C-like ring on an anvil, and by a smart blow with a hammer flattening it out into a thin round disk with a slit extending from the centre to one edge. Fine spirals are also used for general embroidery purposes. The demand for various kinds of loom-woven and embroidered gold and silver work in India is immense; and the variety of textiles so ornamented is also very great. "Gold and silver," says Dr Birdwood in his ''Handbook to the British-Indian Section, Paris Exhibition'', 1878, "are worked into the decoration of all the more costly loom-made garments and Indian piece goods, either on the borders only, or in stripes throughout, or in diapered figures. The gold-bordered loom embroideries are made chiefly at Sattara, and the gold or silver striped at Tanjore; the gold figured ''mashrus'' at Tanjore, Trichinopoly, and Hyderabad in the Deccau; and the highly ornamented gold-figured silks and gold and silver tissues principally at Ahmedabad, Benares, Murshedabad, and Trichinopoly." Among the Western communities the demand for gold and silver lace and embroideries arises chiefly in connexion with naval and military uniforms, court costumes, public and private liveries, ecclesiastical robes and draperies, theatrical dresses, and the badges and insignia of various orders. To a limited extent there is a trade in gold wire and lace to India and China. The metallic basis of the various fabrics is wire round and flattened, the wire being of three kinds — 1st, gold wire, which is invariably silver gilt wire; 2d, copper gilt wire, used for common liveries and theatrical purposes; and 3d, silver wire. These wires are drawn by the ordinary processes, and the flattening, when done, is accomplished by passing the wire between a pair of revolving rollers of fine polished steel. The various qualities of wire are prepared and used in precisely the same way as in India, — round wire, flat wire, thread made of flat gold wire twisted round orange-coloured silk or cotton, known in the trade as "orris," fine spirals and spangles, all being in use in the West as in the East. The lace is woven in the same manner as ribbons, and there are very numerous varieties in richness, pattern, and quality. Cloth of gold, and brocades rich in gold and silver, are woven for ecclesiastical vestments and draperies. The proportions of gold and silver in the gold thread for the lace trade varies, but in all cases the proportion of gold is exceedingly small. An ordinary gold lace wire is drawn from a bar containing 90 parts of silver and 7 of copper, coated with 3 parts of gold. On an average each ounce troy of a bar so plated is drawn into 1500 yards of wire; and therefore about 16 grains of gold cover a mile of wire. It is estimated that about 250,000 ounces of gold wire are made annually in Great Britain, of which about 20 per cent, is used for the headings of calico, muslin, &c., and the remainder is worked up in the gold lace trade.<ref>William Chandler Roberts-Austen and H. Bauerman [W.C.R. — H.B.]. "Gold and Silver Lace." In "Gold." ''Encyclopaedia Britannica'', 9th Edition (1875–1889). Vol. 10 (X). Adam and Charles Black (Publisher). https://archive.org/details/encyclopaedia-britannica-9ed-1875/Vol%2010%20%28G-GOT%29%20193592738.23/page/753/mode/1up (accessed January 2023): 753, Col. 2c – 754, Cols. 1a–b – 2a–b.</ref></blockquote> [[File:Royal Lace detail.jpg|alt=Modern photograph of a piece of old lace|thumb|Detail of Royal Lace]] ==== Guipure ==== According to the French ''Wikipédia'',<blockquote>Guipure is a type of lace in which the background is formed using bars or connecting threads, rather than a mesh or net structure. The production of crocheted Irish Guipure developed in Ireland as a means of generating supplementary income — essential for survival — during the Great Famine caused by the potato blight, which began in 1845. This technique subsequently spread to Brittany — specifically to the fishing ports of southern Finistère (the *Pays Bigouden* region) — during the sardine fishing crisis of the early 20th century[4]. In this region, it is known as "Picot Bigouden," as the *picot* stitch is extensively used to create the background fill.<ref>{{Cite journal|date=2026-04-05|title=Dentelle|url=https://fr.wikipedia.org/w/index.php?title=Dentelle&oldid=234839309|journal=Wikipédia|language=fr}}</ref></blockquote>The detail of royal lace, right, shows the absence of a woven structure and the complexity of this particular lace pattern. ==== Honiton Lace ==== Kate Stradsin says,<blockquote>Honiton lace was the finest English equivalent of Brussels bobbin lace and was constructed in small ‘sprigs, in the cottages of lacemakers[.'] These sprigs were then joined together and bleached to form the large white flounces that were so sought after in the mid-nineteenth century.<ref>Strasdin, Kate. "Rediscovering Queen Alexandra’s Wardrobe: The Challenges and Rewards of Object-Based Research." ''The Court Historian'' 24.2 (2019): 181-196. Rpt http://repository.falmouth.ac.uk/3762/15/Rediscovering%20Queen%20Alexandra%27s%20Wardrobe.pdf: 13, and (for the little quotation) n. 37, which reads "Margaret Tomlinson, ''Three Generations in the Honiton Lace Trade: A Family History'', self-published, 1983."</ref></blockquote> [[File:Strook in Alençon naaldkant, 1750-1775.jpg|thumb|alt=A long piece of complex white lace with garlands, flowers and bows|Point d'Alençon lace, 1750-1775]] ==== Passementerie ==== ''Passementerie'' is the French term for trim on clothing or furniture. The 19th century (especially during the First and Second Empire) was a time of great "''exubérance''" in passementerie in French design, including the development and widespread use of the Jacquard loom.<ref>{{Cite journal|date=2023-06-10|title=Passementerie|url=https://fr.wikipedia.org/w/index.php?title=Passementerie&oldid=205068926|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Passementerie.</ref> ==== Point d'Alençon Lace ==== A lace made by hand using a number of complex steps and layers. The lacemakers build the point d'Alençon design on some kind of mesh and sometimes leave some of the mesh in as part of the lace and perhaps to provide structure. Elizabeth Lewandowski defines point d'Alençon lace and Alençon lace separately. Point lace is needlepoint lace,<ref name=":7" />{{rp|233}} so Alençon point is "a two thread [needlepoint] lace."<ref name=":7" />{{rp|7}} Alençon lace has a "floral design on [a] fine net ground [and is] referred to as [the] queen of French handmade needlepoint laces. The original handmade Alençon was a fine needlepoint lace made of linen thread."<ref name=":7" />{{rp|7}} The sample of point d'Alençon lace (right), from 1750–1775, shows the linen mesh that the lace was constructed on.<ref>{{Cite web|url=http://openfashion.momu.be/#9ce5f00e-8a06-4dab-a833-05c3371f3689|title=MoMu - Open Fashion|website=openfashion.momu.be|access-date=2024-02-26}} ModeMuseum Antwerpen. http://openfashion.momu.be/#9ce5f00e-8a06-4dab-a833-05c3371f3689.</ref> The consistency in this sample suggests it may have been made by machine. == Elastic == Elastic had been invented and was in use by the end of the 19th century. For the sense of "Elastic cord or string, usually woven with india-rubber,"<ref name=":6">“elastic, adj. & n.”.  ''Oxford English Dictionary'', Oxford University Press,  September 2023, <https://doi.org/10.1093/OED/1199670313>.</ref> the ''Oxford English Dictionary'' has usage examples beginning in 1847. The example for 1886 is vivid: "The thorough-going prim man will always place a circle of elastic round his hair previous to putting on his college cap."<ref name=":6" /> == Fabric == === Brocatelle === Brocatelle is a kind of brocade, more simple than most brocades because it uses fewer warp and weft threads and fewer colors to form the design. The article in the French ''Wikipédia'' defines it like this:<blockquote>La '''brocatelle''' est un type de tissu datant du <abbr>xvi^e</abbr> siècle qui comporte deux chaînes et deux trames, au minimum. Il est composé pour que le dessin ressorte avec un relief prononcé, grâce à la chaîne sur un fond en sergé. Les brocatelles les plus anciennes sont toujours fabriquées avec une des trames en lin.<ref>{{Cite journal|date=2023-06-01|title=Brocatelle|url=https://fr.wikipedia.org/w/index.php?title=Brocatelle&oldid=204796410|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Brocatelle.</ref></blockquote>Which translates to this:<blockquote>Brocatelle is a type of fabric dating from the 16th century that has two warps and two wefts, at a minimum. It is composed so that the design stands out with a pronounced relief, thanks to the weft threads on a twill background. The oldest brocades were always made with one of the wefts being linen.</blockquote>The ''Oxford English Dictionary'' says, brocatelle is an "imitation of brocade, usually made of silk or wool, used for tapestry, upholstery, etc., now also for dresses. Both the nature and the use of the stuff have changed" between the late 17th century and 1888, the last time this definition was revised.<ref>"brocatelle, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/23550. Accessed 4 July 2023.</ref> === Broché === Lewandowski says, "to be woven with a raised figure or to be embossed."<ref name=":7" />{{rp|39}} In English, the word might be spelled with or without the acute accent on the final ''e''. Generally, the term was used loosely to describe fabric with a pattern woven into it, either in the same color or a color different from that of the background. That is, the weave that produces the pattern is different from the weave that produces the background. S. F. A. Caulfeild and B. C. Saward published this definition of ''broché'' in their 1887 ''Dictionary of Needlework'', according to the ''Oxford English Dictionary'' (the ''face'' being the side of the fabric facing the viewer):<blockquote>Broché. A French term denoting a velvet or silk textile, with a satin figure thrown up on the face.<ref>“Broché, Adj.” ''Oxford English Dictionary'', Oxford UP, December 2024, https://doi.org/10.1093/OED/1054215522.</ref></blockquote> === Chiffon === A lightweight, somewhat sheer silk fabric, chiffon would have been worn only by the social elite at the end of the 19th century.<ref name=":25">{{Cite journal|date=2025-10-12|title=Chiffon (fabric)|url=https://en.wikipedia.org/w/index.php?title=Chiffon_(fabric)&oldid=1316464288|journal=Wikipedia|language=en}}</ref> Synthetic fibers were not invented until the 20th century — nylon chiffon in 1938 and polyester chiffon not until 1958.<ref name=":25" /> === Ciselé === In the late 19th century, ciselé is a silk velvet whose loops have been treated (some cut, some uncut) to give the fabric a brocade-like pattern by "varying the pile height" [quoting Google Translate].<ref>{{Cite journal|date=2021-10-18|title=velours ciselé|url=https://fr.wiktionary.org/w/index.php?title=velours_cisel%C3%A9&oldid=29903717|journal=Wiktionnaire, le dictionnaire libre|language=fr}}</ref> === Crape === The ''Oxford English Dictionary'' distinguishes the use of ''crêpe'' (using a circumflex rather than an acute accent over the first ''e'') from ''crape'' in textiles, saying ''crêpe'' is "often borrowed [from the French] as a term for all crapy fabrics other than ordinary [[Social Victorians/Mourning|black mourning crape]],"<ref name=":24">"crêpe, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/44242. Accessed 10 February 2023.</ref> with usage examples ranging from 1797 to the mid 20th century. This distinction seems more prescriptive than descriptive since texts from the 19th century to now do not make it reliably. Sometimes 19th-century newspapers put an acute accent on the ''e'' and spelled it crépe. The fabric used for full mourning was black crape, a fabric with a dull texture, but writers continue to vary in how to spell it. Julia Baird uses ''crêpe'', defining it as "a thick black rustling material made of silk, crimped to make it look dull."<ref>Baird, Julia. ''Victoria the Queen, an Intimate Biography of the Woman Who Ruled an Empire''. Random House, 2016. https://books.apple.com/us/book/victoria-the-queen/id953835024.</ref>{{rp|584 of 1203}} However it is spelled, crêpe is<blockquote>Any number of fabrics with characteristic crinkled or puckered surface.<ref name=":7" /> (77)</blockquote> ==== Crepe de Chine ==== Crêpe de chine, the ''OED'' says, is "a white or other coloured crape made of raw silk."<ref name=":24" /> Lewandowski defines it as "a very lightweight, fine, plain weave silk fabric. ... Introduced in 1866, China crepe with soft, silky surface."<ref name=":7" /> (77) ==== Crepon de Chine ==== Crepon is a fabric heavier than the usual crape but treated like crape to be crinkly. Lewandowski says,<blockquote>Introduced in 1882, wool, silk, or blend fabric like very heavy crepe. ... Gay Nineties (1890–1900 C.E.). Popular in 1890s, woolen fabric creped to appear puffed between stripes [or] squares.<ref name=":7" /> (77)</blockquote>According to Lewandowski, ''crepon'' can also be another word for bustle (1865–1890 C.E. to present).<ref name=":7" /> (77) === Crinoline === Technically, crinoline was a fabric made mostly of horsehair and sometimes linen, stiffened with starch or glue, similar to buckram today, used in men's military collars and [[Social Victorians/Terminology#Crinolines|women's foundation garments]]. Lewandowski defines crinoline as <blockquote>(1840–1865 C.E.). France. Originally horsehair cloth used for officers' collars. Later used for women's underskirts to support skirts. Around 1850, replaced by many petticoats, starched and boned. Around 1856, [[Social Victorians/Terminology#Crinoline Hoops|light metal cage]] was developed.<ref name=":7" />{{rp|78}}</blockquote>The term has been used so consistently for the cage first introduced in the 1850s that held the skirt out from the body, however, that it is important to say ''crinoline cage'' or ''crinoline fabric'' or ''crinoline petticoat'' to be clear. === Épinglé Velvet === Often spelled ''épingle'' rather than ''épinglé'', this term appears to have been used for a fabric made of wool, or at least wool along with linen or cotton, that was heavier and stiffer than silk velvet. It was associated with outer garments and men's clothing. Nowadays, épinglé velvet is an upholstery fabric in which the pile is cut into designs and patterns, and the portrait of [[Social Victorians/People/Douglas-Hamilton Duke of Hamilton|Mary, Duchess of Hamilton]] shows a mantle described as épinglé velvet that does seem to be a velvet with a woven pattern perhaps cut into the pile. === [[Social Victorians/Terminology#Trim and Lace|Lace]] === While lace also functioned sometimes as fabric — at the décolletage, for example, on the stomacher or as a veil — here we organize it as a [[Social Victorians/Terminology#Trim and Lace|part of the elaboration of clothing]]. === Liberty Fabrics === === Lisse === According to the ''Oxford English Dictionary'', the term ''lisse'' as a "kind of silk gauze" was used in the 19th-century UK and US.<ref>"lisse, n.1." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/108978. Accessed 4 July 2023.</ref> === Muslin === === Satin === The pre-1927 ''History of Feminine Fashion'', probably commissioned by Charles Frederick Worth's sons, describes Worth's "insistence on new fabrics, new trimmings, new materials of every description" at the beginning of his career in the mid 19th century:<blockquote>When Worth first entered the business of dressmaking, the only materials of the richer sort used for woman's dress were velvet, faille, and watered silk. Satin, for example, was never used. M. Worth desired to use satin very extensively in the gowns he designed, but he was not satisfied with what could be had at the time; he wanted something very much richer than was produced by the mills at Lyons. That his requirements entailed the reconstruction of mills mattered little — the mills were reconstructed under his directions, and the Lyons looms turned out a richer satin than ever, and the manufacturers prospered accordingly.<ref name=":9" />{{rp|6 in printed, 26 in digital book}}</blockquote> === Selesia === According to the ''Oxford English Dictionary'', ''silesia'' is "A fine linen or cotton fabric originally manufactured in Silesia in what is now Germany (''Schlesien'').<ref>"Silesia, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/179664. Accessed 9 February 2023.</ref> It may have been used as a lining — for pockets, for example — in garments made of more luxurious or more expensive cloth. The word ''sleazy'' — "Of textile fabrics or materials: Thin or flimsy in texture; having little substance or body."<ref>"sleazy, adj." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/181563. Accessed 9 February 2023.</ref> — may be related. === Shot Fabric === According to the ''Oxford English Dictionary'', "Of a textile fabric: Woven with warp-threads of one colour and weft-threads of another, so that the fabric (usually silk) changes in tint when viewed from different points."<ref>“Shot, ''Adj.''”  ''Oxford English Dictionary'', Oxford UP,  July 2023, https://doi.org/10.1093/OED/2977164390.</ref> A shot fabric might also be made of silk and cotton fibers. === Tissue === A lightly woven fabric like gauze or chiffon. The light weave can make the fabric translucent and make pleating and gathering flatter and less bulky. Tissue can be woven to be shot, sheer, stiff or soft. Historically, the term in English was used for a "rich kind of cloth, often interwoven with gold or silver" or "various rich or fine fabrics of delicate or gauzy texture."<ref>“Tissue, N.” ''Oxford English Dictionary'', Oxford UP, March 2024, https://doi.org/10.1093/OED/5896731814.</ref> === Tulle === In the 19th century, tulle — a very fine net — was a sheer woven tissue made of linen or silk. Tulle looms were invented in the late 18th century,<ref name=":23">{{Cite journal|date=2025-09-04|title=Tulle (tissu)|url=https://fr.wikipedia.org/w/index.php?title=Tulle_(tissu)&oldid=228712045|journal=Wikipédia|language=fr}}</ref> and the fabric "first made by machine in 1768 in Nottingham."<ref name=":7" />{{rp|299}} By 1802 English tulle was recognized as higher quality than French tulle, even though the fabric is named for the French city.<ref name=":23" /> Tulle is still used today, but it is usually made of synthetic fabric.<blockquote>It is a finer textile than the textile referred to as "net". ... It can be made of various fibres, including silk, nylon, polyester and rayon. Polyester is the most common fibre used for tulle.<ref>{{Cite journal|date=2025-08-05|title=Tulle (netting)|url=https://en.wikipedia.org/w/index.php?title=Tulle_(netting)&oldid=1304416320|journal=Wikipedia|language=en}}</ref></blockquote>Victorian silk tulle would not have been stiff unless it was treated with sizing. == Fan == The ''Encyclopaedia Britannica'' (9th edition) has an article on the fan. (This article is based on knowledge that would have been available toward the end of the 19th century and does not, obviously, reflect current knowledge or ways of talking.)<blockquote>FAN (Latin, ''vannus''; French, ''éventail''), a light implement used for giving motion to the air. ''Ventilabrum'' and ''flabellum'' are names under which ecclesiastical fans are mentioned in old inventories. Fans for cooling the face have been in use in hot climates from remote ages. A bas-relief in the British Museum represents Sennacherib with female figures carrying feather fans. They were attributes of royalty along with horse-hair fly-flappers and umbrellas. Examples may be seen in plates of the Egyptian sculptures at Thebes and other places, and also in the ruins of Persepolis. In the museum of Boulak, near Cairo, a wooden fan handle showing holes for feathers is still preserved. It is from the tomb of Amen-hotep, of the 18th dynasty, 17th century <small>B</small>.<small>C</small>. In India fans were also attributes of men in authority, and sometimes sacred emblems. A heartshaped fan, with an ivory handle, of unknown age, and held in great veneration by the Hindus, was given to the prince of Wales. Large punkahs or screens, moved by a servant who does nothing else, are in common use by Europeans in India at this day. Fans were used in the early Middle Ages to keep flies from the sacred elements during the celebrations of the Christian mysteries. Sometimes they were round, with bells attached — of silver, or silver gilt. Notices of such fans in the ancient records of St Paul’s, London, Salisbury cathedral, and many other churches, exist still. For these purposes they are no longer used in the Western church, though they are retained in some Oriental rites. The large feather fans, however, are still carried in the state processions of the supreme pontiff in Rome, though not used during the celebration of the mass. The fan of Queen Theodolinda (7th century) is still preserved in the treasury of the cathedral of Monza. Fans made part of the bridal outfit, or ''mundus muliebris'', of ancient Roman ladies. Folding fans had their origin in Japan, and were imported thence to China. They were in the shape still used—a segment of a circle of paper pasted on a light radiating frame-work of bamboo, and variously decorated, some in colours, others of white paper on which verses or sentences are written. It is a compliment in China to invite a friend or distinguished guest to write some sentiment on your fan as a memento of any special occasion, and this practice has continued. A fan that has some celebrity in France was presented by the Chinese ambassador to the Comtesse de Clauzel at the coronation of Napoleon I. in 1804. When a site was given in 1635, on an artificial island, for the settlement of Portuguese merchants in Nippo in Japan, the space was laid out in the form of a fan as emblematic of an object agreeable for general use. Men and women of every rank both in China and Japan carry fans, even artisans using them with one hand while working with the other. In China they are often made of carved ivory, the sticks being plates very thin and sometimes carved on both sides, the intervals between the carved parts pierced with astonishing delicacy, and the plates held together by a ribbon. The Japanese make the two outer guards of the stick, which cover the others, occasionally of beaten iron, extremely thin and light, damascened with gold and other metals. Fans were used by Portuguese ladies in the 14th century, and were well known in England before the close of the reign of Richard II. In France the inventory of Charles V. at the end of the 14th century mentions a folding ivory fan. They were brought into general use in that country by Catherine de’ Medici, probably from Italy, then in advance of other countries in all matters of personal luxury. The court ladies of Henry VIII.’s reign in England were used to handling fans, A lady in the Dance of Death by Holbein holds a fan. Queen Elizabeth is painted with a round leather fan in her portrait at Gorhambury; and as many as twenty-seven are enumerated in her inventory (1606). Coryat, an English traveller, in 1608 describes them as common in Italy. They also became of general use from that time in Spain. In Italy, France, and Spain fans had special conventional uses, and various actions in handling them grew into a code of signals, by which ladies were supposed to convey hints or signals to admirers or to rivals in society. A paper in the ''Spectator'' humorously proposes to establish a regular drill for these purposes. The chief seat of the European manufacture of fans during the 17th century was Paris, where the sticks or frames, whether of wood or ivory, were made, and the decorations painted on mounts of very carefully prepared vellum (called latterly ''chicken skin'', but not correctly), — a material stronger and tougher than paper, which breaks at the folds. Paris makers exported fans unpainted to Madrid and other Spanish cities, where they were decorated by native artists. Many were exported complete; of old fans called Spanish a great number were in fact made in France. Louis XIV. issued edicts at various times to regulate the manufacture. Besides fans mounted with parchment, Dutch fans of ivory were imported into Paris, and decorated by the heraldic painters in the process called “Vernis Martin,” after a famous carriage painter and inventor of colourless lac varnish. Fans of this kind belonging to the Queen and to the late baroness de Rothschild were exhibited in 1870 at Kensington. A fan of the date of 1660, representing sacred subjects, is attributed to Philippe de Champagne, another to Peter Oliver in England in the / 17th century. Cano de Arevalo, a Spanish painter of the 17th century devoted himself to fan painting. Some harsh expressions of Queen Christina to the young ladies of the French court are said to have caused an increased ostentation in the splendour of their fans, which were set with jewels and mounted in gold. Rosalba Carriera was the name of a fan painter of celebrity in the 17th century. Lebrun and Romanelli were much employed during the same period. Klingstet, a Dutch artist, enjoyed a considerable reputation for his fans from the latter part of the 17th and the first thirty years of the 18th century. The revocation of the edict of Nantes drove many fan-makers out of France to Holland and England. The trade in England was well established under the Stuart sovereigns. Petitions were addressed by the fan-makers to Charles II. against the importation of fans from India, and a duty was levied upon such fans in consequence. This importation of Indian fans, according to Savary, extended also to France. During the reign of Louis XV. carved Indian and China fans displaced to some extent those formerly imported from Italy, which had been painted on swanskin parchment prepared with various perfumes. During the 18th century all the luxurious ornamentation of the day was bestowed on fans as far as they could display it. The sticks were made of mother-of-pearl or ivory, carved with extraordinary skill in France, Italy, England, and other countries. They were painted from designs of Boucher, Watteau, Lancret, and other "genre" painters, Hébert, Rau, Chevalier, Jean Boquet, Mad. Verité, are known as fan painters. These fashions were followed in most countries of Europe, with certain national differences. Taffeta and silk, as well as fine parchment, were used for the mounts. Little circles of glass were let into the stick to be looked through, and small telescopic glasses were sometimes contrived at the pivot of the stick. They were occasionally mounted with the finest point lace. An interesting fan (belonging to Madame de Thiac in France), the work of Le Flamand, was presented by the municipality of Dieppe to Marie Antoinette on the birth of her son the dauphin. From the time of the Revolution the old luxury expended on fans died out. Fine examples ceased to be exported to England and other countries. The painting on them represented scenes or personages connected with political events. At a later period fan mounts were often prints coloured by hand. The events of the day mark the date of many examples found in modern collections. Amongst the fanmakers of the present time the names of Alexandre, Duvelleroy, Fayet, Vanier, may be mentioned as well known in Paris. The sticks are chiefly made in the department of Oise, at Le Déluge, Crèvecœur, Méry, Ste Geneviève, and other villages, where whole families are engaged in preparing them; ivory sticks are carved at Dieppe. Water-colour painters of distinction often design and paint the mounts, the best designs being figure subjects. A great impulse has been given to the manufacture and painting of fans in England since the exhibition which took place at South Kensington in 1870. Other exhibitions have since been held, and competitive prizes offered, one of which was gained by the Princess Louise. Modern collections of fans take their date from the emigration of many noble families from France at the time of the Revolution. Such objects were given as souvenirs and occasionally sold by families in straitened circumstances. A large number of fans of all sorts, principally those of the 18th century, French, English, German, Italian Spanish, &c., have been lately bequeathed to the South Kensington Museum. Regarding the different parts of folding fans it may be well to state that the sticks are called in French ''brins'', the two outer guards ''panaches'', and the mount ''feuille''.<ref>J. H. Pollen [J.H.P.]. "Fan." ''Encyclopaedia Britannica'', 9th Edition (1875–1889). Vol. '''10''' ('''X'''). Adam and Charles Black (Publisher). https://archive.org/details/encyclopaedia-britannica-9ed-1875/Vol%209%20%28FAL-FYZ%29%20193323016.23/page/26/mode/2up (accessed January 2023): 27, Col. 1b – 28, Col. 1c.</ref></blockquote>Folding fans were available and popular early and are common accessories in portraits of fashionable women through the centuries. == Costumes for Theatre and Fancy Dress == Fancy-dress (or costume) balls were popular and frequent in the U.K. and France as well as the rest of Europe and North America during the 19th century. The themes and styles of the fancy-dress balls influenced those that followed. At the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]], the guests came dressed in costume from times before 1820, as instructed on '''the invitation''', but their clothing was much more about late-Victorian standards of beauty and fashion than the standards of whatever time period the portraits they were copying or basing their costumes on. === Fancy Dress === In her ''Magnificent Entertainments: Fancy Dress Balls of Canada's Governors General, 1876-1898'', Cynthia Cooper describes the resources available to those needing help making a costume for a fancy-dress ball:<blockquote>There were a number of places eager ballgoers could turn for assistance and inspiration. Those with a scholarly bent might pore over history books or study pictures of paintings or other works of art. For more direct advice, one could turn to the barrage of published information specifically on fancy dress. Women’s magazines such as ''Godey’s Lady’s Book'' and ''The Englishwoman’s Domestic Magazine'' sometimes featured fancy dress designs and articles, and enticing specialized books were available with extensive recommendations for choosing fancy dress. By far the most complete sources were the books by [[Social Victorians/People/Ardern Holt|Ardern Holt]], a prolific British authority on the subject. Holt’s book for women, ''Fancy Dresses Described, or What to Wear at Fancy Balls'' (published in six editions between 1879 and 1896), began with the query, ‘‘But what are we to wear?” Holt’s companion book, ''Gentlemen’s Fancy Dress:'' ''How to Choose It'', was also published in six editions from 1882 to 1905. Other prominent authorities included Mrs. Aria’s ''Costume: Fanciful, Historical, and Theatrical'' and, in the US, the Butterick Company’s ''Masquerade and Carnival: Their Customs and Costumes''. The Butterick publication relied heavily on Holt, copying large sections of the introduction outright and paraphrasing other sections.<ref name=":16">Cooper, Cynthia. ''Magnificent entertainments: fancy dress balls of Canada's Governors General, 1876-1898''.Fredericton, N.B.; Hull, Quebec: Goose Lane Editions and Canadian Museum of Civilization, 1997. Internet Archive https://archive.org/details/magnificententer0000coop/.</ref>{{rp|28–29}}</blockquote> Cynthia Cooper discusses how "historical accuracy" works in historical fiction and historical dress: <blockquote>A seemingly accurate costume and coiffure bespoke a cultured individual whose most gratifying compliment would be “historically correct.” Those who were fortunate enough to own actual clothing from an earlier period might wear it with pride as a historical relic, though they would generally adapt or remake it in keeping with the aesthetics of their own period. Historical accuracy was always in the eye of beholders inclined to overlook elements of current fashion in a historical costume. Theatre had long taught the public that if a costume appeared tasteful and attractive, it could be assumed to be accurate. Even at Queen Victoria’s fancy dress balls, costume silhouette was always far more like the fashionable dress of the period than of the time portrayed. For this reason, many extant eighteenth-century dresses show evidence of extensive alterations done in the nineteenth century, no doubt for fancy dress purposes.<ref name=":16" />{{rp|25}}</blockquote> The newspaper ''The Queen'' published dress and fashion information and advice under the byline of [[Social Victorians/People/Ardern Holt|Ardern Holt]], who regularly answered questions from readers about fashion as well as about fancy dress. Holt also wrote entire articles with suggestions for what might make an appealing fancy-dress costume as well as pointing readers away from costumes that had been worn too frequently. The suggestions for costumes are based on familiar types or portraits available to readers, similar to Holt's books on fancy dress, which ran through a number of editions in the 1880s and 1890s. Fancy-dress questions sometimes asked for details about costumes worn in theatrical or operatic productions, which Holt provides. In November 1897, Holt refers to the Duchess of Devonshire's 2 July ball: "Since the famous fancy ball, given at Devonshire House during this year, historical fancy dresses have assumed a prominence that they had not hitherto known."<ref>Holt, Ardern. "Fancy Dress a la Mode." The ''Queen'' 27 November 1897, Saturday: 94 [of 145 in BNA; print p. 1026], Col. 1a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18971127/459/0094.</ref> Holt goes on to provide a number of ideas for costumes for historical fancy dress, as always with a strong leaning toward Victorian standards of beauty and style and away from any concern for historical accuracy. As Leonore Davidoff says, "Every cap, bow, streamer, ruffle, fringe, bustle, glove and other elaboration symbolised some status category for the female wearer."<ref name=":1" />{{rp|93}} [handled under [[Social Victorians/Terminology#Elaborations|Elaborations]]] === Historical Accuracy === Many of the costumes at the ball were based on portraits, especially when the guest was dressed as a historical figure. If possible, we have found the portraits likely to have been the originals, or we have found, if possible, portraits that show the subjects from the two time periods at similar ages. The way clothing was cut changed quite a bit between the 18th and 19th centuries. We think of Victorian clothing — particularly women's clothing, and particularly at the end of the century — as inflexible and restrictive, especially compared to 20th- and 21st-century customs permitting freedom of movement. The difference is generally evolutionary rather than absolute — that is, as time has passed since the 18th century, clothing has allowed an increasingly greater range of movement, especially for people who did not do manual labor. By the end of the 19th century, garments like women's bodices and men's coats were made fitted and smooth by attention to the grain of the fabric and by the use of darts (rather than techniques that assembled many small, individual pieces of fabric). * clothing construction and flat-pattern techniques * Generally, the further back in time we go, the more 2-dimensional the clothing itself was. ==== Women's Versions of Historical Accuracy at the Ball ==== As always with this ball, whatever historical accuracy might be present in a woman's costume is altered so that the wearer is still a fashionable Victorian lady. What makes the costumes look "Victorian" to our eyes is the line of the silhouette caused by the foundation undergarments as well as the many "elaborations"<ref name=":1" />{{rp|93}}, mostly in the decorations, trim and accessories. Also, the clothing hangs and drapes differently because the fabric was cut on grain and the shoulders were freed by the way the sleeves were set in. ==== Men's Versions of Historical Accuracy at the Ball ==== Because men were not wearing a Victorian foundation garment at the end of the century, the men's costumes at the ball are more historically accurate in some ways. * Trim * Mixing neck treatments * Hair * Breeches * Shoes and boots * Military uniforms, arms, gloves, boots == Feathers and Plumes == === Aigrette === Elizabeth Lewandowski defines ''aigrette'' as "France. Feather or plume from an egret or heron."<ref name=":7" />{{rp|5}} Sometimes the newspapers use the term to refer to an accessory (like a fan or ornament on a hat) that includes such a feather or plume. The straight and tapered feathers in an aigrette are in a bundle. === Prince of Wales's Feathers or White Plumes === The feathers in an aigrette came from egrets and herons; Prince of Wales's feathers came from ostriches. A fuller discussion of Prince of Wales's feathers and the white ostrich plumes worn at court appears on [[Social Victorians/Victorian Things#Ostrich Feathers and Prince of Wales's Feathers|Victorian Things]]. For much of the late 18th and 19th centuries, white ostrich plumes were central to fashion at court, and at a certain point in the late 18th century they became required for women being presented to the monarch and for their sponsors. Our purpose here is to understand why women were wearing plumes at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] as part of their costumes. First published in 1893, [[Social Victorians/People/Lady Colin Campbell|Lady Colin Campbell]]'s ''Manners and Rules of Good Society'' (1911 edition) says that<blockquote>It was compulsory for both Married and Unmarried Ladies to Wear Plumes. The married lady’s Court plume consisted of three white feathers. An unmarried lady’s of two white feathers. The three white feathers should be mounted as a Prince of Wales plume and worn towards the left hand side of the head. Colored feathers may not be worn. In deep mourning, white feathers must be worn, black feathers are inadmissible. White veils or lace lappets must be worn with the feathers. The veils should not be longer than 45 inches.<ref>{{Cite web|url=https://www.edwardianpromenade.com/etiquette/the-court-presentation/|title=The Court Presentation|last=Holl|first=Evangeline|date=2007-12-07|website=Edwardian Promenade|language=en-US|access-date=2022-12-18}} https://www.edwardianpromenade.com/etiquette/the-court-presentation/.</ref></blockquote>[[Social Victorians/Victorian Things#Ostrich Feathers and Prince of Wales's Feathers|This fashion was imported from France]] in the mid 1770s.<ref>"Abstract" for Blackwell, Caitlin. "'<nowiki/>''The Feather'd Fair in a Fright''': The Emblem of the Feather in Graphic Satire of 1776." ''Journal for Eighteenth-Century Studies'' 20 January 2013 (Vol. 36, Issue 3): 353-376. ''Wiley Online'' DOI: https://doi.org/10.1111/j.1754-0208.2012.00550.x (accessed November 2022).</ref> Separately, a secondary heraldic emblem of the Prince of Wales has been a specific arrangement of 3 ostrich feathers in a gold coronet<ref>{{Cite journal|date=2022-11-07|title=Prince of Wales's feathers|url=https://en.wikipedia.org/w/index.php?title=Prince_of_Wales%27s_feathers&oldid=1120556015|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Prince_of_Wales's_feathers.</ref> since King Edward III (1312–1377<ref>{{Cite journal|date=2022-12-14|title=Edward III of England|url=https://en.wikipedia.org/w/index.php?title=Edward_III_of_England&oldid=1127343221|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Edward_III_of_England.</ref>). Some women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] wore white ostrich feathers in their hair, but most of them are not Prince of Wales's feathers. Most of the plumes in these portraits are arrangements of some kind of headdress to accompany the costume. A few, wearing what looks like the Princes of Wales's feathers, might be signaling that their character is royal or has royal ancestry. '''One of the women [which one?] was presented to the royals at this ball?''' Here is the list of women who are wearing white ostrich plumes in their portraits in the [[Social Victorians/1897 Fancy Dress Ball/Photographs|''Diamond Jubilee Fancy Dress Ball'' album of 286 photogravure portraits]]: # Kathleen Pelham-Clinton, the [[Social Victorians/People/Newcastle|Duchess of Newcastle]] # [[Social Victorians/People/Louisa Montagu Cavendish|Luise Cavendish]], the Duchess of Devonshire # Jesusa Murrieta del Campo Mello y Urritio (née Bellido), [[Social Victorians/People/Santurce|Marquisa de Santurce]] # Lady [[Social Victorians/People/Farquhar|Emilie Farquhar]] # Princess (Laura Williamina Seymour) Victor of  [[Social Victorians/People/Gleichen#Laura%20Williamina%20Seymour%20of%20Hohenlohe-Langenburg|Hohenlohe Langenburg]] # Louisa Acheson, [[Social Victorians/People/Gosford|Lady Gosford]] # Alice Emily White Coke, [[Social Victorians/People/Leicester|Viscountess Coke]] # Lady Mary Stewart, Helen Mary Theresa [[Social Victorians/People/Londonderry|Vane-Tempest-Stewart]] #[[Social Victorians/People/Consuelo Vanderbilt Spencer-Churchill|Consuelo Vanderbilt Spencer-Churchill]], Duchess of [[Social Victorians/People/Marlborough|Marlborough]], dressed as the wife of the French Ambassador at the Court of Catherine of Russia (not white, but some color that reads dark in the black-and-white photograph) #Mrs. Mary [[Social Victorians/People/Chamberlain|Chamberlain]] (at 491), wearing white plumes, as Madame d'Epinay #Lady Clementine [[Social Victorians/People/Tweeddale|Hay]] (at 629), wearing white plumes, as St. Bris (''Les Huguenots'') #[[Social Victorians/People/Meysey-Thompson|Lady Meysey-Thompson]] (at 391), wearing white plumes, as Elizabeth, Queen of Bohemia #Mrs. [[Social Victorians/People/Grosvenor|Algernon (Catherine) Grosvenor]] (at 510), wearing white plumes, as Marie Louise #Lady [[Social Victorians/People/Ancaster|Evelyn Ewart]], at 401), wearing white plumes, as the Duchess of Ancaster, Mistress of the Robes to Queen Charlotte, 1757, after a picture by Hudson #[[Social Victorians/People/Lyttelton|Edith Sophy Balfour Lyttelton]] (at 580), wearing what might be white plumes on a large-brimmed white hat, after a picture by Romney #[[Social Victorians/People/Yznaga|Emilia Yznaga]] (at 360), wearing what might be white plumes, as Cydalise of the Comedie Italienne from the time of Louis XV #Lady [[Social Victorians/People/Ilchester|Muriel Fox Strangways]] (at 403), wearing what might be two smallish white plumes, as Lady Sarah Lennox, one of the bridesmaids of Queen Charlotte A.D. 1761 #Lady [[Social Victorians/People/Lucan|Violet Bingham]] (at 586), wearing perhaps one white plume in a headdress not related to the Prince of Wales's feathers #Rosamond Fellowes, [[Social Victorians/People/de Ramsey|Lady de Ramsey]] (at 329), wearing a headdress that includes some white plumes, as Lady Burleigh #[[Social Victorians/People/Dupplin|Agnes Blanche Marie Hay-Drummond]] (at 682), in a big headdress topped with white plumes, as Mademoiselle Andrée de Taverney A.D. 1775 #Florence Canning, [[Social Victorians/People/Garvagh|Lady Garvagh]] (at 336), wearing what looks like Prince of Wales's plumes #[[Social Victorians/People/Suffolk|Marguerite Hyde "Daisy" Leiter]] (at 684), wearing what looks like Prince of Wales's plumes #Lady [[Social Victorians/People/Spicer|Margaret Spicer]] (at 281), wearing one smallish white and one black plume, as Countess Zinotriff, Lady-in-Waiting to the Empress Catherine of Russia #Mrs. [[Social Victorians/People/Cavendish Bentinck|Arthur James]] (at 318), wearing what looks like Prince of Wales's plumes, as Elizabeth Cavendish, daughter of Bess of Hardwick #Nellie, [[Social Victorians/People/Kilmorey|Countess of Kilmorey]] (at 207), wearing three tall plumes, 2 white and one dark, as Comtesse du Barri #Daisy, [[Social Victorians/People/Warwick|Countess of Warwick]] (at 53), wearing at least 1 white plume, as Marie Antoinette More men than women were wearing plumes reminiscent of the Prince of Wales's feathers: * ==== Bibliography for Plumes and Prince of Wales's Feathers ==== * Blackwell, Caitlin. "'''The Feather'd Fair in a Fright'<nowiki/>'': The Emblem of the Feather in Graphic Satire of 1776." Journal for ''Eighteenth-Century Studies'' 20 January 2013 (Vol. 36, Issue 3): 353-376. Wiley Online DOI: https://doi.org/10.1111/j.1754-0208.2012.00550.x. * "Prince of Wales's feathers." ''Wikipedia'' https://en.wikipedia.org/wiki/Prince_of_Wales%27s_feathers (accessed November 2022). ['''Add women to this page'''] * Simpson, William. "On the Origin of the Prince of Wales' Feathers." ''Fraser's magazine'' 617 (1881): 637-649. Hathi Trust https://babel.hathitrust.org/cgi/pt?id=chi.79253140&view=1up&seq=643&q1=feathers (accessed December 2022). Deals mostly with use of feathers in other cultures and in antiquity; makes brief mention of feathers and plumes in signs and pub names that may not be associated with the Prince of Wales. No mention of the use of plumes in women's headdresses or court dress. [[File:Prince Albert - Franz Xaver Winterhalter 1842.jpg|thumb|1842 Winterhalter portrait of Prince Albert wearing the insignia of the Order of the Golden Fleece, 1842|alt=1842 Portrait of Prince Albert by Winterhalter, wearing the insignia of the Golden Fleece]] == Honors == === The Bath === The Most Honourable Order of the Bath (GCB, Knight or Dame Grand Cross; KCB or DCB, Knight or Dame Commander; CB, Companion) [[File:The Golden Fleece - collar exhibited at MET, NYC.jpg|thumb|The Golden Fleece collar and pendant for the 2019 "Last Knight" exhibition at the MET, NYC.|alt=Recent photograph of a gold necklace on a wide band, with a gold skin of a sheep hanging from it as a pendant|left]] === The Golden Fleece === To wear the golden fleece is to wear the insignia of the Order of the Golden Fleece, said to be "the most prestigious and historic order of chivalry in the world" because of its long history and strict limitations on membership.<ref name=":10">{{Cite journal|date=2020-09-25|title=Order of the Golden Fleece|url=https://en.wikipedia.org/w/index.php?title=Order_of_the_Golden_Fleece&oldid=980340875|journal=Wikipedia|language=en}}</ref> The monarchs of the U.K. were members of the originally Spanish order, as were others who could afford it, like the Duke of Wellington,<ref name=":12">Thompson, R[obert]. H[ugh]. "The Golden Fleece in Britain." Publication of the ''British Numismatic Society''. 2009 https://www.britnumsoc.org/publications/Digital%20BNJ/pdfs/2009_BNJ_79_8.pdf (accessed January 2023).</ref> the first Protestant to be admitted to the order.<ref name=":10" /> Founded in 1429/30 by Philip the Good, Duke of Burgundy, the order separated into two branches in 1714, one Spanish and the other Austrian, still led by the House of Habsburg.<ref name=":10" /> The photograph (upper left) is of a Polish badge dating from the "turn of the XV and XVI centuries."<ref>{{Citation|title=Polski: Kolana orderowa orderu Złotego Runa, przełom XV i XVI wieku.|url=https://commons.wikimedia.org/wiki/File:The_Golden_Fleece_-_collar_exhibited_at_MET,_NYC.jpg|date=2019-11-10|accessdate=2023-01-10|last=Wulfstan}}. https://commons.wikimedia.org/wiki/File:The_Golden_Fleece_-_collar_exhibited_at_MET,_NYC.jpg.</ref> The collar this Golden Fleece is hanging from might be similar to the one the [[Social Victorians/People/Spencer Compton Cavendish#The Insignia of the Order of the Golden Fleece|Duke of Devonshire is wearing in the 1897 Lafayette portrait]]. The badges and collars that Knights of the Order actually wore vary quite a bit. The 1842 Franz Xaver Winterhalter portrait (upper right) of Prince Consort Albert, Victoria's husband and father of the Prince of Wales, shows him wearing the Golden Fleece on a red ribbon around his neck and the star of the Garter on the front of his coat.<ref>Winterhalter, Franz Xaver. ''Prince Albert''. {{Cite web|url=https://www.rct.uk/collection/search#/16/collection/401412/prince-albert-1819-61|title=Explore the Royal Collection Online|website=www.rct.uk|access-date=2023-01-16}} https://www.rct.uk/collection/search#/16/collection/401412/prince-albert-1819-61.</ref>[[File:Order of the Garter badge sash (United Kingdom) - Tallinn Museum of Orders.jpg|alt=Recent photograph of a gold medal on a wide blue ribbon|thumb|Order of the Garter Badge and Sash]] === The Order of the Garter === The Most Noble Order of the Knights of the Garter (KG, Knight Companion; LG, Lady Companion). Gold badge on the typical royal-blue sash (bottom right). === Royal Victorian Order === (GCVO, Knight or Dame Grand Cross; KCVO or DCVO, Knight or Dame Commander; CVO, Commander; LVO, Lieutenant; MVO, Member) === St. John === The Order of the Knights of St. John === Star of India === Most Exalted Order of the Star of India (GCSI, Knight Grand Commander; KCSI, Knight Commander; CSI, Companion) === Thistle === The Most Ancient and Most Noble Order of the Thistle == [[Social Victorians/Terminology/Foundation Garments#Hoops|Hoops]] == == Jewelry and Stones == === Cabochon === This term describes both the treatment and shape of a precious or semiprecious stone. A cabochon treatment does not facet the stone but merely polishes it, removing "the rough parts" and the parts that are not the right stone.<ref>"cabochon, n." ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/25778. Accessed 7 February 2023.</ref> A cabochon shape is often flat on one side and oval or round, forming a mound in the setting. === Cairngorm === === Ferronnière === A revival of a Renaissance fashion for controlling the hair and headdress. Usually made of a filet, often with a single pendant stone in the center of the forehead, although the Victorians' ferronnières were often elaborate and encrusted with jewels.<ref>Boyington, Amy. "Ferronnière." ''History with Amy'' 5 November 2025. Website fb.watch/FBMyC7bqde [links to fb.watch not allowed].</ref> === Half-hoop === Usually of a ring or bracelet, a precious-metal band with a setting of stones on one side, covering perhaps about 1/3 or 1/2 of the band. Half-hoop jewelry pieces were occasionally given as wedding gifts to the bride. === Jet === === ''Orfèvrerie'' === Sometimes misspelled in the newspapers as ''orvfèvrerie''. ''Orfèvrerie'' is the artistic work of a goldsmith, silversmith, or jeweler. === Ribbon Necklace === === Solitaire === A solitaire is a ring with a single stone set as the focal point. Solitaire rings were occasionally given as wedding gifts to the bride. === Turquoise === == Mantle, Cloak, Cape == In 19th-century newspaper accounts, these terms are sometimes used without precision as synonyms. These are all outer garments. Although the terms were (and are) often used generically, a short outer wrap would be a cape, a longer one would be a cloak and, after the 17th century, a full-length one possibly buttoned down the front would be a mantle. === '''Mantle''' === A mantle — often a long outer garment — might have elements like a train, sleeves, collars, revers, fur, and a cape. A late-19th-century writer making a distinction between a mantle and a cloak might use ''mantle'' if the garment is more voluminous. == Military == Several men from the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball at Devonshire House]] were dressed in military uniforms, some historical and some, possibly, not. === Armor === At the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]], much of the armor was fictional, not located in historical time and place. Helmets, ditto. ==== Chain Mail ==== chausses, mitons, hauberk, mail coif, ==== Armor ==== greaves, gauntlet * '''Cuirass''': According to the ''Oxford English Dictionary'', the primary sense of ''cuirass'' is "A piece of armour for the body (originally of leather); ''spec.'' a piece reaching down to the waist, and consisting of a breast-plate and a back-plate, buckled or otherwise fastened together ...."<ref>"cuirass, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/45604. Accessed 17 May 2023.</ref> [[File:Harmensz van Rijn Rembrandt - Ritratto di giovane - Google Art Project.jpg|alt=Old painting of a young man wearing metal collar armor around his neck|thumb|''Tronie of a Young Man in a Gorget and Cap'', attributed to Rembrandt (c. 1639)]] * '''Gorget''': By the Elizabethan age in western Europe, the gorget was the piece of plate armor that protected the neck. <blockquote>At the beginning of the 16th century, the gorget reached its full development as a component of plate armour. Unlike previous gorget plates and bevors which sat over the cuirass and also required a separate mail collar to fully protect the neck, the developed gorget was worn under the cuirass and was intended to cover a larger area of the neck, nape, shoulders and upper chest, from which the edges of the backplate and breastplate had receded.<ref>{{Cite journal|date=2026-04-02|title=Gorget|url=https://en.wikipedia.org/w/index.php?title=Gorget&oldid=1346732005|journal=Wikipedia|language=en}}</ref></blockquote>The only visible armor worn by the subject in Rembrandt's c. 1639 portrait (right) is his gorget. *. ==== Over-clothing ==== (fabric or leather): tunic, cloak, mantle === Baldric === According to the ''Oxford English Dictionary'', the primary sense of ''baldric'' is "A belt or girdle, usually of leather and richly ornamented, worn pendent from one shoulder across the breast and under the opposite arm, and used to support the wearer's sword, bugle, etc."<ref>"baldric, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/14849. Accessed 17 May 2023.</ref> This sense has been in existence since c. 1300. A baldric could be worn over armor or court dress. The ribbon worn across the chest for honors is called a sash. [[File:Knötel IV, 04.jpg|thumb|alt=An Old drawing in color of British soldiers on horses brandishing swords in 1815.|1890 illustration of the Household Cavalry (Life Guard, left; Horse Guard, right) at the Battle of Waterloo, 1815]] === Household Cavalry === The Royal Household contains the Household Cavalry, a corps of British Army units assigned to the monarch. It is made up of 2 regiments, the Life Guards and what is now called The Blues and Royals, which were formed around the time of "the Restoration of the Monarchy in 1660."<ref name=":3">Joll, Christopher. "Tales of the Household Cavalry, No. 1. Roles." The Household Cavalry Museum, https://householdcavalry.co.uk/app/uploads/sites/2/2021/06/Household-Cavalry-Museum-video-series-large-print-text-Tales-episode-01.pdf.</ref>{{rp|1}} Regimental Historian Christopher Joll says, "the original Life Guards were formed as a mounted bodyguard for the exiled King Charles II, The Blues were raised as Cromwellian cavalry and The Royals were established to defend Tangier."<ref name=":3" />{{rp|1–2}} The 1st and 2nd Life Guards were formed from "the Troops of Horse and Horse Grenadier Guards ... in 1788."<ref name=":3" />{{rp|3}} The Life Guards were and are still official bodyguards of the queen or king, but through history they have been required to do quite a bit more than serve as bodyguards for the monarch. The Household Cavalry fought in the Battle of Waterloo on Sunday, 18 June 1815 as heavy cavalry.<ref name=":3" />{{rp|3}} Besides arresting the Cato Steet conspirators in 1820 "and guarding their subsequent execution," the Household Cavalry contributed to the "the expedition to rescue General Gordon, who was trapped in Khartoum by The Mahdi and his army of insurgents" in 1884.<ref name=":3" />{{rp|3}} In 1887 they "were involved ... in the suppression of rioters in Trafalgar Square on Bloody Sunday."<ref name=":3" />{{rp|3}} ==== Grenadier Guards ==== Three men — [[Social Victorians/People/Gordon-Lennox#Lord Algernon Gordon Lennox|Lord Algernon Gordon-Lennox]], [[Social Victorians/People/Stanley#Edward George Villiers Stanley, Lord Stanley|Lord Stanley]], and [[Social Victorians/People/Stanley#Hon. Ferdinand Charles Stanley|Hon. F. C. Stanley]] — attended the ball as officers of the Grenadier Guards, wearing "scarlet tunics, ... full blue breeches, scarlet hose and shoes, lappet wigs" as well as items associated with weapons and armor.<ref name=":14">“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.</ref>{{rp|p. 34, Col. 2a}} Founded in England in 1656 as Foot Guards, this infantry regiment "was granted the 'Grenadier' designation by a Royal Proclamation" at the end of the Napoleonic Wars.<ref>{{Cite journal|date=2023-04-22|title=Grenadier Guards|url=https://en.wikipedia.org/w/index.php?title=Grenadier_Guards&oldid=1151238350|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Grenadier_Guards.</ref> They were not called Grenadier Guards, then, before about 1815. In 1660, the Stuart Restoration, they were called Lord Wentworth's Regiment, because they were under the command of Thomas Wentworth, 5th Baron Wentworth.<ref>{{Cite journal|date=2022-07-24|title=Lord Wentworth's Regiment|url=https://en.wikipedia.org/w/index.php?title=Lord_Wentworth%27s_Regiment&oldid=1100069077|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lord_Wentworth%27s_Regiment.</ref> At the time of Lord Wentworth's Regiment, the style of the French cavalier had begun to influence wealthy British royalists. In the British military, a Cavalier was a wealthy follower of Charles I and Charles II — a commander, perhaps, or a field officer, but probably not a soldier.<ref>{{Cite journal|date=2023-04-22|title=Cavalier|url=https://en.wikipedia.org/w/index.php?title=Cavalier&oldid=1151166569|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Cavalier.</ref> The Guards were busy as infantry in the 17th century, engaging in a number of armed conflicts for Great Britain, but they also served the sovereign. According to the Guards Museum,<blockquote>In 1678 the Guards were ordered to form Grenadier Companies, these men were the strongest and tallest of the regiment, they carried axes, hatches and grenades, they were the shock troops of their day. Instead of wearing tri-corn hats they wore a mitre shaped cap.<ref>{{Cite web|url=https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-2/|title=Service to the Crown|website=The Guards Museum|language=en-GB|access-date=2023-05-15}} https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-2/.</ref></blockquote>The name comes from ''grenades'', then, and we are accustomed to seeing them in front of Buckingham Palace, with their tall mitre hats. The Guard fought in the American Revolution, and in the 19th century, the Grenadier Guards fought in the Crimean War, Sudan and the Boer War. They have roles as front-line troops and as ceremonial for the sovereign, which makes them elite:<blockquote>Queen Victoria decreed that she did not want to see a single chevron soldier within her Guards. Other then [sic] the two senior Warrant Officers of the British Army, the senior Warrant Officers of the Foot Guards wear a large Sovereigns personal coat of arms badge on their upper arm. No other regiments of the British Army are allowed to do so; all the others wear a small coat of arms of their lower arms. Up until 1871 all officers in the Foot Guards had the privilege of having double rankings. An Ensign was ranked as an Ensign and Lieutenant, a Lieutenant as Lieutenant and Captain and a Captain as Captain and Lieutenant Colonel. This was because at the time officers purchased their own ranks and it cost more to purchase a commission in the Foot Guards than any other regiments in the British Army. For example if it cost an officer in the Foot Guards £1,000 for his first rank, in the rest of the Army it would be £500 so if he transferred to another regiment he would loose [sic] £500, hence the higher rank, if he was an Ensign in the Guards and he transferred to a Line Regiment he went in at the higher rank of Lieutenant.<ref>{{Cite web|url=https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-1/|title=Formation and role of the Regiments|website=The Guards Museum|language=en-GB|access-date=2023-05-15}} https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-1/.</ref></blockquote> ==== Life Guards ==== [[Social Victorians/People/Shrewsbury#Reginald Talbot's Costume|General the Hon. Reginald Talbot]], a member of the 1st Life Guards, attended the Duchess of Devonshire's ball dressed in the uniform of his regiment during the Battle of Waterloo.<ref name=":14" />{{rp|p. 36, Col. 3b}} At the Battle of Waterloo the 1st Life Guards were part of the 1st Brigade — the Household Brigade — and were commanded by Major-General Lord Edward Somerset.<ref name=":4">{{Cite journal|date=2023-09-30|title=Battle of Waterloo|url=https://en.wikipedia.org/w/index.php?title=Battle_of_Waterloo&oldid=1177893566|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Battle_of_Waterloo.</ref> The 1st Life Guards were on "the extreme right" of a French countercharge and "kept their cohesion and consequently suffered significantly fewer casualties."<ref name=":4" /> [[File:Captain, Royal Horse Guards, Blue, England, 1879, from the Military Series (N224) issued by Kinney Tobacco Company to promote Sweet Caporal Cigarettes MET DPB874122.jpg|alt=Old drawing of a soldier wearing a white cuirass, a pointed helmet, thigh-high boots, carrying a long sword|thumb|Captain, Royal Horse Guards, Blue, 1888, a Kinney Brothers Tobacco Company card]] ==== Royal Horse Guards ==== In 1650 the Regiment of Cuirassiers was "raised by Sir Arthur Haselrig on the orders of Oliver Cromwell."<ref name=":26">{{Cite journal|date=2026-05-13|title=Royal Horse Guards|url=https://en.wikipedia.org/w/index.php?title=Royal_Horse_Guards&oldid=1353961278|journal=Wikipedia|language=en}}</ref> In 1660 "it became the Earl of Oxford's Regiment .... Based on the colour of their uniform, the regiment was nicknamed 'the Oxford Blues', or simply the 'Blues.' In 1750, it became the Royal Horse Guards Blue."<ref name=":26" /> The Royal Horse Guards Blue were moved to Windsor at the end of the 18th century and "acted as royal bodyguards" to George III, who liked them.<ref name=":26" /> While pay for the men "stagnated," requirements continued to rise, so that recruits had to come from wealth.<ref name=":26" /> Riding and hunting skills were helpful to the recruits, who had to provide their own horses, pay for messes and uniforms, not to mention the position itself.<ref name=":26" /> They fought in the Battle of Waterloo, with 44 dead, 50 wounded (of which only 6 died).<ref name=":26" /> With the Duke of Wellington at their head, they became part of the Household Cavalry in 1820.<ref name=":26" /> An 1890 illustration shows a member of the Royal Horse Guard (above right) fighting at the Battle of Waterloo. The Royal Horse Guard Blue fought in the Battle of Balaclava in 1854, fighting with the heavy brigades and thus were more successful than the famous light brigade, though conditions were very difficult.<ref name=":26" /> A tobacco card published in 1888 (right) shows a captain in the Royal Horse Guards, Blue, in 1879. In 1884–85 the Blues took part in the attempt to rescue General Gordon in Khartoum. They were sent to South Africa at the end of the 19th century.<ref name=":26" /> For those men who were in the Royal Horse Guards at the end of the 19th century, the field marshals were * 1869–1885: Field Marshal Hugh Rose, 1st Baron Strathnairn, during which time — in 1877 — the name changed to the Royal Horse Guards (The Blues)."<ref name=":26" /> * 1885–1895: Field Marshal Sir Patrick Grant * 1895–1907: Field Marshal Garnet Wolseley, 1st Viscount Wolseley In 1847 Edmund Packe published his ''[[iarchive:historicalrecord00packiala/|Historical Record of the Royal Regiment of Horse Guards, or Oxford Blues]]'', which has colored images to illustrate the development of the uniform up to the middle of the 19th century (the link goes to the ''Internet Archive''). == [[Social Victorians/Mourning|Mourning]] == == Peplum == According to the French ''Wiktionnaire'', a peplum is a "Short skirt or flared flounce layered at the waist of a jacket, blouse or dress" [translation by Google Translate].<ref>{{Cite journal|date=2021-07-02|title=péplum|url=https://fr.wiktionary.org/w/index.php?title=p%C3%A9plum&oldid=29547727|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/p%C3%A9plum.</ref> The ''Oxford English Dictionary'' has a fuller definition, although, it focuses on women's clothing because the sense is written for the present day:<blockquote>''Fashion''. ... a kind of overskirt resembling the ancient peplos (''obsolete''). Hence (now usually) in modern use: a short flared, gathered, or pleated strip of fabric attached at the waist of a woman's jacket, dress, or blouse to create a hanging frill or flounce.<ref name=":5">“peplum, n.”. ''Oxford English Dictionary'', Oxford University Press, September 2023, <https://doi.org/10.1093/OED/1832614702>.</ref></blockquote>Men haven't worn peplums since the 18th century, except when wearing costumes based on historical portraits. The ''Daily News'' reported in 1896 that peplums had been revived as a fashion item for women.<ref name=":5" /> == Revers == According to the ''Oxford English Dictionary'', ''revers'' are the "edge[s] of a garment turned back to reveal the undersurface (often at the lapel or cuff) (chiefly in ''plural''); the material covering such an edge."<ref>"revers, n." ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/164777. Accessed 17 April 2023.</ref> The term is French and was used this way in the 19th century (according to the ''Wiktionnaire'').<ref>{{Cite journal|date=2023-03-07|title=revers|url=https://fr.wiktionary.org/w/index.php?title=revers&oldid=31706560|journal=Wiktionnaire|language=fr}} https://fr.wiktionary.org/wiki/revers.</ref> == Traditional vs Progressive Style == === Progressive Style === The terms ''artistic dress'' and ''aesthetic dress'' — as well as ''rational dress'' or ''dress reform'' — are not synonymous and were in use at different times to refer to different groups of people in different contexts, but we recognize them as referring to a similar kind of personal style in clothing, a style we call progressive dress or the progressive style. Used in a very precise way, ''artistic dress'' is associated with the Pre-Raphaelite artists and the women in their circle beginning in the 1860s. Similarly, ''aesthetic dress'' is associated with the 1880s and 1890s and dress reform movements, as is ''rational dress'', a movement located largely among women in the middle classes from the middle to the end of the century. In general, what we are calling the progressive style is characterized by its resistance to the highly structured fashion of its day, especially corseting, aniline dyes and an extremely close fit. This group of styles was more about individual choices and approaches than the consistent vision offered by couturiers like Maison Worth. * [[Social Victorians/People/Dressmakers and Costumiers#Alice Comyns Carr and Ada Nettleship|Ada Nettleship]]: Constance Wilde and Ellen Terry; an 1883 exhibition of dress by the Rational Dress Society featured her work, including trousers for women (with a short overskirt)<ref>{{Cite journal|date=2025-04-21|title=Ada Nettleship|url=https://en.wikipedia.org/w/index.php?title=Ada_Nettleship&oldid=1286707541|journal=Wikipedia|language=en}}</ref> * [[Social Victorians/People/Dressmakers and Costumiers#Alice Comyns Carr and Ada Nettleship|Alice Comyns Carr]]<ref>{{Cite journal|date=2025-06-06|title=Alice Comyns Carr|url=https://en.wikipedia.org/w/index.php?title=Alice_Comyns_Carr&oldid=1294283929|journal=Wikipedia|language=en}}</ref> * Grosvenor Gallery === Traditional Style === [[File:Victoria Hesse NPG 95941 crop.jpg|alt=Old photograph of a white woman wearing a very tight and fitted bodice with her skirts swept to the back|thumb|Princess Victoria, Marchioness of Milford-Haven (1863–1950), Granddaughter of Queen Victoria; wife of Prince Louis of Battenberg, 1st Marquess, c. 1878]] Images * Smooth bodice, fabric draped to the back or covering a bustle with a small cage beneath it: By the end of the century designs from the [[Social Victorians/People/Dressmakers and Costumiers#The House of Worth|House of Worth]] (or Maison Worth) define what we think of as the traditional Victorian look, which was very stylish and expensive. Queen Victoria's granddaughter Princess Victoria is shown (right) wearing a traditional but very stylish c. 1878 dress like one designed by Maison Worth. Blanche Payne describes an example of the 1895 "high style" in a gown by Worth with "the idiosyncrasies of the [1890s] full blown":<blockquote>The dress is white silk with wine-red stripes. Sleeves, collars, bows, bag, hat, and hem border match the stripes. The sleeve has reached its maximum volume; the bosom full and emphasized with added lace; the waistline is elongated, pointed, and laced to the point of distress; the skirt is smooth over the hips, gradually swinging out to sweep the floor. This is the much vaunted hourglass figure.<ref name=":11" />{{rp|530}}</blockquote> The Victorian-looking gowns at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] are stylish in a way that recalls the designs of the House of Worth. The elements that make their look so Victorian are anachronisms on the costumes representing fashion of earlier eras. The women wearing these gowns preferred the standards of beauty from their own day to a more-or-less historically accurate look. The style competing at the very end of the century with the Worth look was not the historical, however, but a progressive style called at the time ''artistic'' or ''aesthetic''. William Powell Frith's 1883 painting ''A Private View at the Royal Academy, 1881'' (discussion below) pits this kind of traditional style against the progressive or artistic style. === The Styles === [[File:Frith A Private View.jpg|thumb|William Powell Frith, ''A Private View at the Royal Academy, 1881'']] We typically think of the late-Victorian silhouette as universal but, in the periods in which corsets dominated women's dress, not all women wore corsets and not all corsets were the same, as William Powell Frith's 1883 ''A Private View at the Royal Academy, 1881'' (right) illustrates. Frith is clear in his memoir that this painting — "recording for posterity the aesthetic craze as regards dress" — deliberately contrasts what he calls the "folly" of the Aesthetic Dress movement and the look of the traditional corseted waist.<ref>Frith, William Powell. ''My Autobiography and Reminiscences''. 1887.</ref> Frith considered the Aesthetic Movement and Aesthetic Dress "ephemeral," but its rejection of corsetry looks far more consequential to us in hindsight than it did in the 19th century. As Frith sees it, his painting critiques the "craze" associated with the women in this set of identifiable portraits who are not corseted, but his commitment to realism shows us a spectrum, a range, of conservatism and if not political then at least stylistic progressivism among the women. The progressives, oddly, are the women wearing artistic (that is, somewhat historical) dress, because they’re not corseted. It is a misreading to see the presentation of the women’s fashion as a simple opposition. Constance, Countess of Lonsdale — situated at the center of this painting with Frederick Leighton, president of the Royal Academy of Art — is the most conservatively dressed of the women depicted, with her narrow sleeves, tight waist and almost perfectly smooth bodice, which tells us that her corset has eyelets so that it can be laced precisely and tightly, and it has stays (or "bones") to prevent wrinkles or natural folds in the overclothing. Lillie Langtry, in the white dress, with her stylish narrow sleeves, does not have such a tightly bound waist or smooth bodice, suggesting she may not be corseted at all, as we know she sometimes was not.['''citation'''] Jenny Trip, a painter’s model, is the woman in the green dress in the aesthetic group being inspected by Anthony Trollope, who may be taking notes. She looks like she is not wearing a corset. Both Langtry and Trip are toward the middle of this spectrum: neither is dressed in the more extreme artistic dress of, say, the two figures between Trip and Trollope. A lot has been written about the late-Victorian attraction to historical dress, especially in the context of fancy-dress balls and the Gothic revival in social events as well as art and music. Part of the appeal has to have been the way those costumes could just be beautiful clothing beautifully made. Historical dress provided an opportunity for some elite women to wear less-structured but still beautiful and influential clothing. ['''Calvert'''<ref>Calvert, Robyne Erica. ''Fashioning the Artist: Artistic Dress in Victorian Britain 1848-1900''. Ph.D. thesis, University of Glasgow, 2012. <nowiki>https://theses.gla.ac.uk/3279/</nowiki></ref>] The standards for beauty, then, with historical dress were Victorian, with the added benefit of possibly less structure. So, at the Duchess of Devonshire's ball, "while some attendees tried to hew closely to historical precedent, many rendered their historical or mythological personage in the sartorial vocabulary they knew best. The [photographs of people in their costumes at the ball offer] a glimpse into how Victorians understood history, not a glimpse into the costume of an authentic historical past."<ref>Mitchell, Rebecca N. "The Victorian Fancy Dress Ball, 1870–1900." ''Fashion Theory'' 2017 (21: 3): 291–315. DOI: 10.1080/1362704X.2016.1172817.</ref>{{rp|294}} * historical dress: beautiful clothing. * the range at the ball, from Minnie Paget to Gwladys * "In light of such efforts, the ball remains to this day one of the best documented outings of the period, and a quick glance at the album shows that ..." * The costume of the Duchess of Devonshire does not have a defined waist and may suggest that she herself is not corseted, although that would be a notable departure for her. Women had more choices about their waists than the simple opposition between no corset and tightlacing can accommodate. The range of choices is illustrated in Frith's painting, with a woman locating herself on it at a particular moment for particular reasons. Much analysis of 19th-century corsetry focuses on its sexualizing effects — corsets dominated Victorian photographic pornography ['''citations'''] and at the same time, the absence of a corset was sexual because it suggested nudity.['''citations'''] A great deal of analysis of 19th-century corsetry, on the other hand, assumes that women wore corsets for the male gaze ['''citations'''] or that they tightened their waists to compete with other women.['''citations'''] But as we can see in Frith's painting, the sexualizing effect was not universal or sweeping, and these analyses do not account for the choices women had in which corset to wear or how tightly to lace it. Especially given the way that some photographic portraits were mechanically altered to make the waist appear smaller, the size of a woman's waist had to do with how she was presenting herself to the world. That is, the fact that women made choices about the size of or emphasis on their waists suggests that they had agency that needs to be taken into account. As they navigated the complex social world, women's fashion choices had meaning. Society or political hostesses had agency not only in their clothing but generally in that complex social world. They had roles managing social events of the upper classes, especially of the upper aristocracy and oligarchy, like the Duchess of Devonshire's ball. Their class and rank, then, were essential to their agency, including to some degree their freedom to choose what kind of corset to wear and how to wear it. Also, by the end of the century lots of different kinds of corsets were available for lots of different purposes. Special corsets existed for pregnancy, sports (like tennis, bicycling, horseback riding, golf, fencing, archery, stalking and hunting), theatre and dance and, of course, for these women corsets could be made to support the special dress worn over it. Women's choices in how they presented themselves to the world included more than just their foundation garments, of course. "Every cap, bow, streamer, ruffle, fringe, bustle, glove," that is, the trim and decorations on their garments, their jewelry and accessories — which Davidoff calls "elaborations"<ref name=":1" />{{rp|93}} — pointed to a host of status categories, like class, rank, wealth, age, marital status, engagement with the empire, how sexual they wanted to seem, political alignment and purpose at the social event. For example, when women were being presented to the monarch, they were expected to wear three ostrich plumes, often called the [[Social Victorians/Terminology#Prince of Wales's Feathers or White Plumes|Prince of Wales's feathers]]. Like all fashions, the corset, which was quite long-lasting in all its various forms, eventually went out of style. Of the many factors that might have influenced its demise, perhaps most important was the women's movement, in which women's rights, freedom, employment and access to their own money and children were less slogan-worthy but at least as essential as votes for women. The activities of the animal-rights movements drew attention not only to the profligate use of the bodies and feathers of birds but also to the looming extinction of the baleen whale, which made whale bone scarce and expensive. Perhaps the century's debates over corseting and especially tightlacing were relevant to some decisions not to be corseted. And, of course, perhaps no other reason is required than that the nature of fashion is to change. == Undergarments == Unlike undergarments, Victorian women's [[Social Victorians/Terminology/Foundation Garments|foundation garments]] created the distinctive silhouette. Victorian undergarments included the chemise, the bloomers, the corset cover — articles that are not structural.<p> The corset was an important element of the understructure of foundation garments — hoops, bustles, petticoats and so on — but it has never been the only important element. === Undergarments === * Chemise * Corset cover * Bloomers * [[Social Victorians/Terminology#Petticoat|Petticoats]] (distinguish between the outer- and undergarment type of petticoat) * Combinations * [[Social Victorians/Terminology#Hose, Stockings and Tights|Hose, stockings and tights]] * Men's shirts * Men's unders ==== Bloomers ==== ==== Chemise ==== A chemise is a garment "linen, homespun, or cotton knee-length garment with [a] square neck" worn under all the other garments except the bloomers or combinations.<ref name=":7" /> (61) According to Lewandowski, combinations replaced the chemise by 1890. ==== Combinations ==== === [[Social Victorians/Terminology/Foundation Garments|Foundation Garments]] === Foundation structures changed the shape of the body by metal, cane, boning. Men wore corsets as well. * [[Social Victorians/Terminology/Foundation Garments#Corset|Corset]] * [[Social Victorians/Terminology/Foundation Garments#Hoops|Hoops]] * Padding == Footnotes == {{reflist}} moen9sdq4kzcvo0vla0w9g1ok4ef64m 0 300067 2812064 2637369 2026-05-29T21:16:32Z Scogdill 1331941 2812064 wikitext text/x-wiki [[File:Charles Beresford Vanity Fair 3 January 1895.JPG|thumb|alt=Old colored drawing of an aging man in a 19th-century Navy officer's uniform with big gold epaulettes, a long sword hanging from his belt and resting on the ground, at least 5 medals across the chest of his coat, a flat black (bicorn) officer's hat, white gloves in his right hand, his left hand on his waist, facing 3/4 to his right|"Steam Reserve" (Charles Beresford) by "Spy," ''Vanity Fair'' 3 January 1895]] [[File:Charles Beresford Vanity Fair 6 July 1899.jpg|thumb|alt=Old colored drawing of an aging white man dressed in the costume of a Chinese nobleman but with British military medals across his chest, holding a small dog in his right hand and a book in his left|"The Commercial Traveller" (Charles Beresford) ''Vanity Fair'' 6 July 1899]] ==Also Known As== * Family name: Beresford * Admiral Charles William de la Poer Beresford, 1st and last Baron Beresford * Lord Charles Beresford (1846–1916)<ref>"Admiral Charles William de la Poer Beresford, 1st and last Baron Beresford." {{Cite web|url=https://www.thepeerage.com/p2490.htm#i24892|title=Person Page|website=www.thepeerage.com|access-date=2023-10-01}} https://www.thepeerage.com/p2490.htm#i24892.</ref> * This is the family of the Marquess of Waterford in the Peerage of Ireland. ==Overview== ==Acquaintances, Friends and Enemies== ===Acquaintances=== ===Friends=== ===Enemies=== ==Organizations== ==Timeline== '''1895 January 3''', a caricature portrait of "Steam Reserve" (Charles Bereford) by Leslie Ward ("Spy") appeared in the 3 January 1895 issue of ''Vanity Fair'', Number 609 in its "Men of the Day" series.<ref name=":0">{{Cite journal|date=2024-01-14|title=List of Vanity Fair (British magazine) caricatures (1895–1899)|url=https://en.wikipedia.org/w/index.php?title=List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899)&oldid=1195518024|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/List_of_Vanity_Fair_(British_magazine)_caricatures_(1895%E2%80%931899).</ref> '''1899 July 6''', a caricature portrait of "The Commercial Traveller" (Charles Bereford) by "Cloister" appeared in the 6 July 1899 issue of ''Vanity Fair'', Number 709 in its "Statesmen" series.<ref name=":0" /> The cover of the book he is holding reads ''On Broken China and How to Mend It''. ==Demographics== * Nationality: ===Residences=== ==Family== *Admiral Charles William de la Poer Beresford, 1st and last Baron Beresford (10 February 1846 – 6 September 1919) * ===Relations=== ==Notes and Questions== ==Memoirs and Biographies== * Beresford, Charles. ''The Memoirs of Admiral Lord Charles Beresford, Written by Himself''. Vol. 1. Little, Brown, 1914. https://commons.wikimedia.org/w/index.php?title=File:The_memoirs_of_Admiral_Lord_Charles_Beresford_(IA_memoirsofadmiral01bereiala).pdf&page=13. Also at Internet Archive. ==Footnotes== {{reflist}} rd02bt9mzkzk4fg8pubo08cumgov5it 0 302815 2811985 2811977 2026-05-29T13:13:47Z Jtneill 10242 + embedded link to [[psychology]] 2811985 wikitext text/x-wiki {{underconstruction}} {{psychology}} {{75% done}} {{tertiary}} {{course}} {{op|[[User:Atcovi|Atcovi]]}} '''Child psychology''' is the branch of [[psychology]] that deals with the way children behave, think, socialize, and develop. This course aims to familiarize students with the science and history of child psychology and the biological, social, emotional, and cognitive development of a child, from conception to early childhood. Understanding your child and how to interact with them throughout their life stages is crucial to not only giving them the best learning environment to grow, but to fostering a positive, peaceful, and supportive parent-child relationship. This lesson is also intended for professionals who work with children, university students, and anyone else who is interested in learning child psychology. == Content == === History/Background === * [[Child psychology/Ch. 1]] - History, theories, methods {{stage|100}} * [[Child psychology/Ch. 2]] - Heredity, diseases, disorders, syndromes, conception, infertility {{stage|100}} === Prenatal Development === * [[Child psychology/Ch. 3]] - Prenatal development: Germinal stage, embryonic stage, fetal stage {{stage|100}} * [[Child psychology/Ch. 4]] - Three stages of childbirth, methods of childbirth, birth problems, post-partum period, neonates {{stage|25}} === Infancy === * [[Child psychology/Ch. 5]] - Infancy (physical development) {{stage|25}} * [[Child psychology/Ch. 6]] - Infancy (cognitive development) {{stage|25}} * [[Child psychology/Ch. 7 - Infancy: Social and Emotional Development|Child psychology/Ch. 7]] - Infancy (social and emotional development) {{stage|25}} === Early Childhood === * [[Child psychology/Chapter 8: Early Childhood: Physical Development|Child psychology/Ch. 8]] - Early childhood (physical development) {{stage|25}} * [[Child psychology/Chapter 9: Early Childhood: Cognitive Development|Child psychology/Ch. 9]] - Early childhood (cognitive development) {{stage|25}} * [[Child psychology/Chapter 10: Early Childhood: Social and Emotional Development|Child psychology/Ch. 10]] - Early childhood (social and emotional development) {{stage|25}} === Overview/Cheat-Sheet === * [[Child psychology/Summary of child psychology (cheat-sheet)]] - ''not created by the instructor.'' {{stage|25}} == Resources == * ''[https://www.amazon.com/Childhood-Adolescence-Voyages-Development-MindTap/dp/035737410X Rathus' Childhood and Adolescence: Voyages in Development, 7th Edition] -'' the textbook from which the instructor derived his notes. * [[w:Child_psychology|Child psychology]] - Wikipedia * [https://www.alohabdonline.com/wp-content/uploads/2020/05/The-Psychology-Of-The-Child.pdf Psychology of the Child (textbook)] - Jean Piaget & [[w:Bärbel_Inhelder|Bärbel Inhelder]] [[Category:Child psychology]] [[Category:Atcovi/Spring 2024]] bbgdpl9bm12gap0t3qdqwbobt8wue9u 2812118 2811985 2026-05-30T11:11:56Z Jtneill 10242 Simplify 2nd sentence 2812118 wikitext text/x-wiki {{underconstruction}} {{psychology}} {{75% done}} {{tertiary}} {{course}} {{op|[[User:Atcovi|Atcovi]]}} '''Child psychology''' is the branch of [[psychology]] that deals with the way children behave, think, socialize, and develop. This course aims to familiarize students with the history and science of biological, social, emotional, and cognitive development of a child, from conception to early childhood. Understanding your child and how to interact with them throughout their life stages is crucial to not only giving them the best learning environment to grow, but to fostering a positive, peaceful, and supportive parent-child relationship. This lesson is also intended for professionals who work with children, university students, and anyone else who is interested in learning child psychology. == Content == === History/Background === * [[Child psychology/Ch. 1]] - History, theories, methods {{stage|100}} * [[Child psychology/Ch. 2]] - Heredity, diseases, disorders, syndromes, conception, infertility {{stage|100}} === Prenatal Development === * [[Child psychology/Ch. 3]] - Prenatal development: Germinal stage, embryonic stage, fetal stage {{stage|100}} * [[Child psychology/Ch. 4]] - Three stages of childbirth, methods of childbirth, birth problems, post-partum period, neonates {{stage|25}} === Infancy === * [[Child psychology/Ch. 5]] - Infancy (physical development) {{stage|25}} * [[Child psychology/Ch. 6]] - Infancy (cognitive development) {{stage|25}} * [[Child psychology/Ch. 7 - Infancy: Social and Emotional Development|Child psychology/Ch. 7]] - Infancy (social and emotional development) {{stage|25}} === Early Childhood === * [[Child psychology/Chapter 8: Early Childhood: Physical Development|Child psychology/Ch. 8]] - Early childhood (physical development) {{stage|25}} * [[Child psychology/Chapter 9: Early Childhood: Cognitive Development|Child psychology/Ch. 9]] - Early childhood (cognitive development) {{stage|25}} * [[Child psychology/Chapter 10: Early Childhood: Social and Emotional Development|Child psychology/Ch. 10]] - Early childhood (social and emotional development) {{stage|25}} === Overview/Cheat-Sheet === * [[Child psychology/Summary of child psychology (cheat-sheet)]] - ''not created by the instructor.'' {{stage|25}} == Resources == * ''[https://www.amazon.com/Childhood-Adolescence-Voyages-Development-MindTap/dp/035737410X Rathus' Childhood and Adolescence: Voyages in Development, 7th Edition] -'' the textbook from which the instructor derived his notes. * [[w:Child_psychology|Child psychology]] - Wikipedia * [https://www.alohabdonline.com/wp-content/uploads/2020/05/The-Psychology-Of-The-Child.pdf Psychology of the Child (textbook)] - Jean Piaget & [[w:Bärbel_Inhelder|Bärbel Inhelder]] [[Category:Child psychology]] [[Category:Atcovi/Spring 2024]] 9n2req1admqta3po03mmrg8xgdmgbf2 0 304718 2812061 2810828 2026-05-29T21:07:14Z Dc.samizdat 2856930 /* Golden chords */ caption improvements 2812061 wikitext text/x-wiki {{Short description|Four-dimensional analog of the icosahedron}} {{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}} {{Infobox 4-polytope | Name=600-cell| Image_File=Schlegel_wireframe_600-cell_vertex-centered.png| Image_Caption=[[W:Schlegel diagram|Schlegel diagram]], vertex-centered<br>(vertices and edges)| Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]| Last=[[W:Rectified 600-cell|34]]| Index=35| Next=[[W:Truncated 120-cell|36]]| Schläfli={3,3,5}| CD={{Coxeter–Dynkin diagram|node_1|3|node|3|node|5|node}}| Cell_List=600 ([[W:Tetrahedron|{3,3}]]) [[Image:Tetrahedron.png|20px]]| Face_List=1200 [[W:triangle|{3}]]| Edge_Count=720| Vertex_Count= 120| Petrie_Polygon=[[W:Triacontagon#Petrie polygons|30-gon]]| Coxeter_Group=H₄, [3,3,5], order 14400| Vertex_Figure=[[Image:600-cell verf.svg|80px]]<br>[[W:icosahedron|icosahedron]]| Dual=[[120-cell|120-cell]]| Property_List=[[W:Convex polytope|convex]], [[W:isogonal figure|isogonal]], [[W:isotoxal figure|isotoxal]], [[W:isohedral figure|isohedral]] }} [[File:600-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]] In [[geometry]], the '''600-cell''' is the [[W:convex regular 4-polytope|convex regular 4-polytope]] (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,3,5}. It is also known as the '''C₆₀₀''', '''hexacosichoron'''<ref>[[W:Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite Symmetry Groups'', 11.5 ''Spherical Coxeter groups'', p.249</ref> and '''hexacosihedroid'''.<ref>Matila Ghyka, ''The Geometry of Art and Life'' (1977), p.68</ref> It is also called a '''tetraplex''' (abbreviated from "tetrahedral complex") and a '''[[W:polytetrahedron|polytetrahedron]]''', being bounded by tetrahedral [[W:Cell (geometry)|cells]]. The 600-cell's boundary is composed of 600 [[W:Tetrahedron|tetrahedral]] [[W:Cell (mathematics)|cells]] with 20 meeting at each vertex.{{Efn|name=vertex icosahedral pyramid}} Together they form 1200 triangular faces, 720 edges, and 120 vertices. It is the 4-[[W:Four-dimensional space#Dimensional analogy|dimensional analogue]] of the [[W:icosahedron|icosahedron]], since it has five [[W:Tetrahedron|tetrahedra]] meeting at every edge, just as the icosahedron has five [[W:triangle|triangle]]s meeting at every vertex.{{Efn|name=math of dimensional analogy}} Its [[W:dual polytope|dual polytope]] is the [[120-cell|120-cell]]. == Geometry == The 600-cell is the fifth in the sequence of 6 convex regular 4-polytopes (in order of complexity and size at the same radius).{{Efn|name=4-polytopes ordered by size and complexity|group=}} It can be deconstructed into twenty-five overlapping instances of its immediate predecessor the [[24-cell|24-cell]],{{Sfn|Coxeter|1973|loc=§8.51|p=153|ps=; "In fact, the vertices of {3, 3, 5}, each taken 5 times, are the vertices of 25 {3, 4, 3}'s."}} as the 24-cell can be [[24-cell#8-cell|deconstructed]] into three overlapping instances of its predecessor the [[W:Tesseract|tesseract (8-cell)]], and the 8-cell can be [[24-cell#Relationships among interior polytopes|deconstructed]] into two instances of its predecessor the [[16-cell|16-cell]].{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}} 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. The 24-cell's edge length equals its radius, but the 600-cell's edge length is ~0.618 times its radius,{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii), "600-cell" column ₀''R/l'' {{=}} 2𝝓/2}} which is the [[W:golden ratio|golden ratio]]. {{Regular convex 4-polytopes|wiki=W:}} === Coordinates === ==== Unit radius Cartesian coordinates ==== The vertices of a 600-cell of unit radius centered at the origin of 4-space, with edges of length <math>\phi^{-1} \approx 0.618</math> (where <math>\phi = \tfrac12\bigl(1 + \sqrt5~\!\bigr)</math> is the golden ratio), can be given{{Sfn|Coxeter|1973|loc=§8.7 Cartesian coordinates|pp=156-157}} as follows: 8 vertices obtained from :(0, 0, 0, ±1) by permuting coordinates, and 16 vertices of the form: :(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) The remaining 96 vertices are obtained by taking [[W:even permutation|even permutation]]s of :(±{{sfrac|φ|2}}, ±{{sfrac|1|2}}, ±{{sfrac|φ⁻¹|2}}, 0) Note that the first 8 are the vertices of a [[16-cell|16-cell]], the second 16 are the vertices of a [[W:tesseract|tesseract]], and those 24 vertices together are the vertices of a [[24-cell|24-cell]]. The remaining 96 vertices are the vertices of a [[W:snub 24-cell|snub 24-cell]], which can be found by partitioning each of the 96 edges of another 24-cell (dual to the first) in the golden ratio in a consistent manner.{{Sfn|Coxeter|1973|loc=§8.4 The snub {3,4,3}|pp=151-153}} When interpreted as [[#Symmetries|quaternions]],{{Efn|name=quaternions}} these are the unit [[W:icosian|icosian]]s. In the 24-cell, there are [[24-cell#Great squares|squares]], [[24-cell#Great hexagons|hexagons]] and [[24-cell#Great triangles|triangles]] that lie on great circles (in central planes through four or six vertices).{{Efn|name=hypercubic chords}} In the 600-cell there are twenty-five overlapping inscribed 24-cells, with each vertex and square shared by five 24-cells, and each hexagon or triangle shared by two 24-cells.{{Efn|In cases where inscribed 4-polytopes of the same kind occupy disjoint sets of vertices (such as the two 16-cells inscribed in the tesseract, or the three 16-cells inscribed in the 24-cell), their sets of vertex chords, central polygons and cells must likewise be disjoint. In the cases where they share vertices (such as the three tesseracts inscribed in the 24-cell, or the 25 24-cells inscribed in the 600-cell), they also share some vertex chords and central polygons.{{Efn|name=disjoint from 8 and intersects 16}}}} In each 24-cell there are three disjoint 16-cells, so in the 600-cell there are 75 overlapping inscribed 16-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}} Each 16-cell constitutes a distinct orthonormal basis for the choice of a [[16-cell#Coordinates|coordinate reference frame]]. The 60 axes and 75 16-cells of the 600-cell constitute a [[W:Configuration (geometry)|geometric configuration]], which in the language of configurations is written as 60₅75₄ to indicate that each axis belongs to 5 16-cells, and each 16-cell contains 4 axes.{{Sfn|Waegell & Aravind|2009|loc=§3.2 The 75 bases of the 600-cell|pp=3-4|ps=; In the 600-cell the configuration's "points" and "lines" are axes ("rays") and 16-cells ("bases"), respectively.}} Each axis is orthogonal to exactly 15 others, and these are just its companions in the 5 16-cells in which it occurs. ==== Hopf spherical coordinates ==== In the 600-cell there are also great circle [[W:pentagon|pentagon]]s and [[W:decagon|decagon]]s (in central planes through ten vertices).{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020}} Only the decagon edges are visible elements of the 600-cell (because they are the edges of the 600-cell). The edges of the other great circle polygons are interior chords of the 600-cell, which are not shown in any of the 600-cell renderings in this article (except where shown as dashed lines).{{Efn|The 600-cell contains 25 distinct 24-cells, bound to each other by pentagonal rings. Each pentagon links five completely disjoint{{Efn|name=completely disjoint}} 24-cells together, the collective vertices of which are the 120 vertices of the 600-cell. Each 24-point 24-cell contains one fifth of all the vertices in the 120-point 600-cell, and is linked to the other 96 vertices (which comprise a [[#Diminished 600-cells|snub 24-cell]]) by the 600-cell's 144 pentagons. Each of the 25 24-cells intersects each of the 144 great pentagons at just one vertex.{{Efn|Each of the 25 24-cells of the 600-cell contains exactly one vertex of each great pentagon.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|p=438}} Six pentagons intersect at each 600-cell vertex, so each 24-cell intersects all 144 great pentagons.|name=distribution of pentagon vertices in 24-cells}} Five 24-cells meet at each 600-cell vertex,{{Efn|name=five 24-cells at each vertex of 600-cell}} so all 25 24-cells are linked by each great pentagon. The 600-cell can be partitioned into five disjoint 24-cells (10 different ways),{{Efn|name=Schoute's ten ways to get five disjoint 24-cells}} and also into 24 disjoint pentagons (inscribed in the 12 Clifford parallel great decagons of one of the 6 [[#Decagons|decagonal fibrations]]) by choosing a pentagon from the same fibration at each 24-cell vertex.|name=24-cells bound by pentagonal fibers}} By symmetry, an equal number of polygons of each kind pass through each vertex; so it is possible to account for all 120 vertices as the intersection of a set of central polygons of only one kind: decagons, hexagons, pentagons, squares, or triangles. For example, the 120 vertices can be seen as the vertices of 15 pairs of [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} squares which do not share any vertices, or as 100 ''dual pairs'' of non-orthogonal hexagons between which all axis pairs are orthogonal, or as 144 non-orthogonal pentagons six of which intersect at each vertex. This latter pentagonal symmetry of the 600-cell is captured by the set of [[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]]{{Sfn|Zamboj|2021|pp=10-11|loc=§Hopf coordinates}} (𝜉_''i'', 𝜂, 𝜉_''j''){{Efn|name=Hopf coordinates|The [[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]] are triples of three angles: : (𝜉_''i'', 𝜂, 𝜉_''j'') that parameterize the [[W:3-sphere#Hopf coordinates|3-sphere]] by numbering points along its great circles. A Hopf coordinate describes a point as a rotation from a polar point (0, 0, 0).{{Efn|name=Hopf coordinate angles|The angles 𝜉_''i'' and 𝜉_''j'' are angles of rotation in the two [[W:completely orthogonal|completely orthogonal]] invariant planes which characterize [[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The angle 𝜂 is the inclination of both these planes from the polar axis, where 𝜂 ranges from 0 to {{sfrac|𝜋|2}}. The (𝜉_''i'', 0, 𝜉_''j'') coordinates describe the great circles which intersect at the north and south pole ("lines of longitude"). The (𝜉_''i'', {{sfrac|𝜋|2}}, 𝜉_''j'') coordinates describe the great circles orthogonal to longitude ("equators"); there is more than one "equator" great circle in a 4-polytope, as the equator of a 3-sphere is a whole 2-sphere of great circles. The other Hopf coordinates (𝜉_''i'', 0 < 𝜂 < {{sfrac|𝜋|2}}, 𝜉_''j'') describe the great circles (''not'' "lines of latitude") which cross an equator but do not pass through the north or south pole.}} Hopf coordinates are a natural alternative to Cartesian coordinates{{Efn|name=Hopf coordinates conversion|The conversion from Hopf coordinates (𝜉_''i'', 𝜂, 𝜉_''j'') to unit-radius Cartesian coordinates (w, x, y, z) is:<br> : w {{=}} cos 𝜉_''i'' sin 𝜂 : x {{=}} cos 𝜉_''j'' cos 𝜂 : y {{=}} sin 𝜉_''j'' cos 𝜂 : z {{=}} sin 𝜉_''i'' sin 𝜂 The Hopf origin pole (0, 0, 0) is Cartesian (0, 1, 0, 0). The conventional "north pole" of Cartesian standard orientation is (0, 0, 1, 0), which is Hopf ({{sfrac|𝜋|2}}, {{sfrac|𝜋|2}}, {{sfrac|𝜋|2}}). Cartesian (1, 0, 0, 0) is Hopf (0, {{sfrac|𝜋|2}}, 0).}} for framing regular convex 4-polytopes, because the group of [[W:Rotations in 4-dimensional Euclidean space|4-dimensional rotations]], denoted SO(4), generates those polytopes.}} given as: : ({<10}{{sfrac|𝜋|5}}, {≤5}{{sfrac|𝜋|10}}, {<10}{{sfrac|𝜋|5}}) where {<10} is the permutation of the ten digits (0 1 2 3 4 5 6 7 8 9) and {≤5} is the permutation of the six digits (0 1 2 3 4 5). The 𝜉_''i'' and 𝜉_''j'' coordinates range over the 10 vertices of great circle decagons; even and odd digits label the vertices of the two great circle pentagons inscribed in each decagon.{{Efn|There are 600 permutations of these coordinates, but there are only 120 vertices in the 600-cell. These are actually the Hopf coordinates of the vertices of the [[120-cell#Cartesian coordinates|120-cell]], which has 600 vertices and can be seen (two different ways) as a compound of 5 disjoint 600-cells.}} === Structure === ==== Polyhedral sections ==== The mutual distances of the vertices, measured in degrees of arc on the circumscribed [[W:hypersphere|hypersphere]], only have the values 36° = {{sfrac|𝜋|5}}, 60° = {{sfrac|𝜋|3}}, 72° = {{sfrac|2𝜋|5}}, 90° = {{sfrac|𝜋|2}}, 108° = {{sfrac|3𝜋|5}}, 120° = {{sfrac|2𝜋|3}}, 144° = {{sfrac|4𝜋|5}}, and 180° = 𝜋. Departing from an arbitrary vertex V one has at 36° and 144° the 12 vertices of an [[W:icosahedron|icosahedron]],{{Efn|name=vertex icosahedral pyramid}} at 60° and 120° the 20 vertices of a [[W:dodecahedron|dodecahedron]], at 72° and 108° the 12 vertices of a larger icosahedron, at 90° the 30 vertices of an [[W:icosidodecahedron|icosidodecahedron]], and finally at 180° the antipodal vertex of V.{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏⁻¹) beginning with a vertex}}{{Sfn|van Oss|1899|ps=; van Oss does not mention the arc distances between vertices of the 600-cell.}}{{Sfn|Buekenhout & Parker|1998}} These can be seen in the H3 [[W:Coxeter plane|Coxeter plane]] projections with overlapping vertices colored.{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}} :[[File:600-cell-polyhedral levels.png|640px]] These polyhedral sections are ''solids'' in the sense that they are 3-dimensional, but of course all of their vertices lie on the surface of the 600-cell (they are hollow, not solid). Each polyhedron lies in Euclidean 4-dimensional space as a parallel cross section through the 600-cell (a hyperplane). In the curved 3-dimensional space of the 600-cell's boundary surface envelope, the polyhedron surrounds the vertex V the way it surrounds its own center. But its own center is in the interior of the 600-cell, not on its surface. V is not actually at the center of the polyhedron, because it is displaced outward from that hyperplane in the fourth dimension, to the surface of the 600-cell. Thus V is the apex of a [[W:Pyramid (geometry)#Polyhedral pyramid|4-pyramid]] based on the polyhedron. {| class=wikitable !colspan=2|Concentric Hulls |- |align=center|[[Image:Hulls of H4only-orthonormal.png|360px]] |The 600-cell is projected to 3D using an orthonormal basis. The vertices are sorted and tallied by their 3D norm. Generating the increasingly transparent hull of each set of tallied norms shows:<br> <br> 1) two points at the origin<br> 2) two icosahedra<br> 3) two dodecahedra<br> 4) two larger icosahedra<br> 5) and a single icosidodecahedron<br> <br> for a total of 120 vertices. This is the view from ''any'' origin vertex. The 600-cell contains 60 distinct sets of these concentric hulls, one centered on each pair of antipodal vertices. |- |} ==== Golden chords ==== [[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{{Efn|[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.]] The 600-cell geometry is based on the [[24-cell#Hypercubic chords|24-cell]]. The 600-cell rounds out the 24-cell with 2 more great circle polygons (exterior decagon and interior pentagon), adding 4 more chord lengths which alternate with the 24-cell's 4 chord lengths. {{Clear}}|name=hypercubic chords|group=}} with angles of arc. The golden ratio governs{{Efn|name=golden chords|group=}} the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.{{Efn|name=radially golden}}|alt=|400x400px]] {{see also|W:24-cell#Hypercubic chords|label 1=24-cell § Hypercubic chords}} The 120 vertices are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏⁻¹) beginning with a vertex; see column ''a''}} at eight different [[W:Chord (geometry)|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its [[#Geodesics|five great circle polygons]].{{Sfn|Steinbach|1997|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a "fan of chords" diagram like the one here.|p=23|loc=Figure 3}} In ascending order of length, they are: <math>\sqrt{0.382\sim} = \sqrt{2 - \phi} = \phi^{-1} \approx 0.618</math> <math>\sqrt{1}</math> <math>\sqrt{1.382\sim} = \sqrt{3 - \phi} \approx 1.176</math> <math>\sqrt{2}</math> <math>\sqrt{2.618\sim} = \sqrt{1 + \phi} = \phi \approx 1.618</math> <math>\sqrt{3}</math> <math>\sqrt{3.618\sim} = \sqrt{2 + \phi} \approx 1.902</math> <math>\sqrt{4}</math> In the diagram, chord lengths are given as square roots, with a decimal fractional part if necessary, where: <math>\Phi = \phi^{-1} \approx 0.618</math> is the inverse golden ratio, and: <math>\Delta = 1 - \Phi = \Phi^2 \approx 0.382</math> is its square. For example, the 600-cell edge length is: <math>\Phi = \sqrt{0.\Delta} = \sqrt{0.382\sim} \approx 0.618</math> The four [[24-cell#Hypercubic chords|hypercubic chords]] of the 24-cell (<math>\sqrt{1}</math>, <math>\sqrt{2}</math>, <math>\sqrt{3}</math>, <math>\sqrt{4}</math>){{Efn|name=hypercubic chords}} alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of <math>\sqrt{5}</math>.{{Efn|The squares of two of these chord lengths, <math>3 - \phi {{=}} \phi^{-1}\sqrt{5}</math> and <math>2 + \phi {{=}} \phi\sqrt{5}</math>, are [[W:Algebraic conjugate|algebraic conjugate]]s whose product is <math>5</math>.}} The golden chords of the 600-cell exemplify that the [[W:golden ratio|golden ratio]] <math>\phi {{=}} \tfrac{1 + \sqrt{5}}{2} \approx 1.618</math> is a circle ratio related to fifths of <math>\pi</math>. For instance:<br> :<math>\tfrac{\pi}{5} {{=}} \arccos(\tfrac{\phi}{2})</math> is the arc of one 600-cell edge, the <math>\phi^{-1} = \Phi \approx 0.618</math> chord. Reciprocally, in this function discovered by Robert Everest expressing <math>\phi</math> as a function of <math>\pi</math> and the numbers 1, 2, 3 and 5 of the Fibonacci series:<br> : <math>\phi {{=}} 1 - 2 \cos(\tfrac{3\pi}{5})</math> <math>\tfrac{3\pi}{5}</math> is the arc length of the <math>\phi \approx 1.618</math> chord.<ref>{{Cite web|last=Baez|first=John|date=7 March 2017|title=Pi and the Golden Ratio|url=https://johncarlosbaez.wordpress.com/2017/03/07/pi-and-the-golden-ratio/|website=Azimuth|author-link=W:John Carlos Baez|access-date=10 October 2022}}</ref>|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of <math>\sqrt{5}</math>.{{Efn|The 600-cell edges are decagon edges of length <math>\phi^{-1} {{=}} \Phi {{=}} \sqrt{0.\Delta} \approx 0.618</math>, the ''smaller'' golden section of <math>\sqrt{5}</math>; the edges are in the inverse [[W:golden ratio|golden ratio]] <math>\tfrac{1}{\phi} {{=}} \phi^{-1}</math> to the <math>\sqrt{1}</math> hexagon chords (the 24-cell edges). The other fractional-root chords exhibit golden relationships as well. The chord of length <math>\sqrt{3 - \phi} {{=}} \sqrt{1.\Delta}</math> is a pentagon edge. The next fractional-root chord is a decagon diagonal of length <math>\phi {{=}} \sqrt{2.\Phi}</math>, the ''larger'' golden section of <math>\sqrt{5}</math>; it is in the golden ratio{{Efn|name=golden chords|group=}} to the <math>\sqrt{1}</math> chord (and the radius).{{Efn|Notice in the diagram how the <math>\phi</math> chord (the ''larger'' golden section) sums with the adjacent <math>\Phi</math> edge (the ''smaller'' golden section) to <math>\sqrt{5}</math>, as if together they were a <math>\sqrt{5}</math> chord bent to fit inside the <math>\sqrt{4}</math> diameter.}} The last fractional-root chord is the pentagon diagonal of length <math>=\sqrt{2 + \phi} {{=}} \sqrt{3.\Phi}</math>. The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed <math>\sqrt{2 + \phi} / \sqrt{3 - \phi} {{=}} \phi</math>.|name=fractional root chords|group=}} ==== Boundary envelopes ==== [[Image:600-cell.gif|thumb|A 3D projection of a 600-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]. The 3D surface made of 600 tetrahedra is visible.]] The 600-cell ''rounds out'' the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices,{{Efn|name=snub 24-cell|Consider one of the 24-vertex 24-cells inscribed in the 120-vertex 600-cell. The other 96 vertices constitute a [[W:snub 24-cell|snub 24-cell]]. Removing any one 24-cell from the 600-cell produces a snub 24-cell.}} in effect adding twenty-four more overlapping 24-cells inscribed in the 600-cell.{{Efn|The 600-cell contains exactly 25 24-cells, 75 16-cells and 75 8-cells, with each 16-cell and each 8-cell lying in just one 24-cell.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|p=434}}|name=4-polytopes inscribed in the 600-cell}} The new surface thus formed is a tessellation of smaller, more numerous cells{{Efn|Each tetrahedral cell touches, in some manner, 56 other cells. One cell contacts each of the four faces; two cells contact each of the six edges, but not a face; and ten cells contact each of the four vertices, but not a face or edge.|name=tetrahedral cell adjacency}} and faces: 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 [[24-cell#Hypercubic chords|<math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords]]. [[Image:24-cell.gif|thumb|A 3D projection of a [[24-cell|24-cell]] performing a [[24-cell#Simple rotations|simple rotation]]. The 3D surface made of 24 octahedra is visible. It is also present in the 600-cell, but as an invisible interior boundary envelope.]] 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 does not have unit edge-length in a unit-radius coordinate system the way the 24-cell and the tesseract do; unlike those two, the 600-cell is not [[W:Tesseract#Radial equilateral symmetry|radially equilateral]]. Like them it is radially triangular in a special way,{{Efn|All polytopes can be radially triangulated into triangles which meet at their center, each triangle contributing two radii and one edge. There are (at least) three special classes of polytopes which are radially triangular by a special kind of triangle. The ''radially equilateral'' polytopes can be constructed from identical [[W:equilateral triangle|equilateral triangle]]s which all meet at the center.{{Efn|name=radially equilateral|group=}} The ''radially golden'' polytopes can be constructed from identical [[W:Golden triangle (mathematics)|golden triangle]]s which all meet at the center.{{Efn|A [[W:Golden triangle (mathematics)|golden triangle]] is an [[W:Isosceles triangle|isosceles]] [[W:Triangle|triangle]] in which the duplicated side ''a'' is in the [[W:Golden ratio|golden ratio]] to the distinct side ''b'': :<math>\tfrac{a}{b} {{=}} \phi {{=}} \tfrac{1 + \sqrt{5}}{2} \approx 1.618</math> It can be found in a regular [[W:Decagon|decagon]] by connecting any two adjacent vertices to the center, and in the regular [[W:Pentagon|pentagon]] by connecting any two adjacent vertices to the vertex opposite them.<br> The vertex angle is: :<math>\theta = \arccos(\tfrac{\phi}{2}) {{=}} \tfrac{\pi}{5} {{=}} 36^\circ</math> so the base angles are each <math>\tfrac{2\pi}{5} {{=}} 72^\circ</math>. The golden triangle is uniquely identified as the only triangle to have its three angles in 2:2:1 proportions.|name=Golden triangle}} All the [[W:regular polytope|regular polytope]]s are ''radially right'' polytopes which can be constructed, with their various element centers and radii, from identical characteristic [[W:Schläfli orthoscheme|orthoscheme]]s which all meet at the center, subdividing the regular polytope into characteristic [[W:right triangle|right triangle]]s which meet at the center.{{Efn|The [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is the generalization of the [[W:right triangle|right triangle]] to simplex figures of any number of dimensions. Every regular polytope can be radially subdivided into identical [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]]s which meet at its center.{{Efn|name=characteristic orthoscheme}}|name=radially right|group=}}}} but one in which golden triangles rather than equilateral triangles meet at the center.{{Efn|The long radius (center to vertex) of the 600-cell is in the [[W:golden ratio|golden ratio]] to its edge length; thus its radius is <math>\phi</math> if its edge length is 1, and its edge length is <math>\phi^{-1}</math> if its radius is 1.|name=radially golden}} Only a few uniform polytopes have this property, including the four-dimensional 600-cell, the three-dimensional [[W:icosidodecahedron|icosidodecahedron]], and the two-dimensional [[W:Decagon#The golden ratio in decagon|decagon]]. (The icosidodecahedron is the equatorial cross section of the 600-cell, and the decagon is the equatorial cross section of the icosidodecahedron.) '''Radially golden''' polytopes are those which can be constructed, with their radii, from [[W:Golden triangle (mathematics)|golden triangles]].{{Efn|name=Golden triangle}} The boundary envelope of 600 small tetrahedral cells wraps around the twenty-five envelopes of 24 octahedral cells (adding some 4-dimensional space in places between these curved 3-dimensional envelopes). The shape of those interstices must be an [[W:Octahedral pyramid|octahedral 4-pyramid]] of some kind, but in the 600-cell it is [[#Octahedra|not regular]].{{Efn|Beginning with the 16-cell, every regular convex 4-polytope in the unit-radius sequence is inscribed in its successor.{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}} Therefore the successor may be constructed by placing [[W:Pyramid (geometry)#Polyhedral pyramid|4-pyramids]] of some kind on the cells of its predecessor. Between the 16-cell and the tesseract, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract. Between the tesseract and the 24-cell, we have 8 canonical [[W:cubic pyramid|cubic pyramid]]s. But if we place 24 canonical [[W:octahedral pyramid|octahedral pyramid]]s on the 24-cell, we only get another tesseract (of twice the radius and edge length), not the successor 600-cell. Between the 24-cell and the 600-cell there must be 24 smaller, irregular 4-pyramids on a regular octahedral base.|name=truncated irregular octahedral pyramid}} ==== Geodesics ==== The vertex chords of the 600-cell are arranged in [[W:geodesic|geodesic]] [[W:great circle|great circle]] polygons of five kinds: decagons, hexagons, pentagons, squares, and triangles.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=§4 The planes of the 600-cell|pp=437-439}} [[Image:Stereographic polytope 600cell.png|thumb|Cell-centered [[W:stereographic projection|stereographic projection]] of the 600-cell's 72 central decagons onto their great circles. Each great circle is divided into 10 arc-edges at the intersections where 6 great circles cross.]] The <math>\phi^{-1} \approx 0.618</math> edges form 72 flat regular central [[W:decagon|decagon]]s, 6 of which cross at each vertex.{{Efn|name=vertex icosahedral pyramid}} Just as the [[W:icosidodecahedron|icosidodecahedron]] can be partitioned into 6 central decagons (60 edges = 6 × 10), the 600-cell can be partitioned into 72 decagons (720 edges = 72 × 10). The 720 <math>\phi^{-1}</math> edges divide the surface into 1200 triangular faces and 600 tetrahedral cells: a 600-cell. The 720 edges occur in 360 parallel pairs, <math>\sqrt{2 + \phi}</math> apart. As in the decagon and the icosidodecahedron, the edges occur in [[W:Golden triangle (mathematics)|golden triangles]] which meet at the center of the polytope. The 72 great decagons can be divided into 6 sets of 12 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics,{{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; various 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, in the 600-cell, visit all 120 vertices just once. For example, each of the 600 tetrahedra participates in 6 great decagons{{Efn|name=tetrahedron linking 6 decagons}} belonging to 6 discrete [[W:Hopf fibration|Hopf fibration]]s, each filling the whole 600-cell. Each [[#Decagons|fibration]] is a bundle of 12 Clifford parallel decagons which form 20 cell-disjoint intertwining rings of 30 tetrahedral cells,{{Efn|name=Boerdijk–Coxeter helix}} each bounded by three of the 12 great decagons.{{Efn|name=Clifford parallel decagons}}|name=Clifford parallels}} such that only one decagonal great circle in each set passes through each vertex, and the 12 decagons in each set reach all 120 vertices.{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} The <math>\sqrt{1}</math> chords form 200 central hexagons (25 sets of 16, with each hexagon in two sets),{{Efn|1=A 24-cell contains 16 hexagons. In the 600-cell, with 25 24-cells, each 24-cell is disjoint from 8 24-cells and intersects each of the other 16 24-cells in six vertices that form a hexagon.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|p=438}} A 600-cell contains 25・16/2 = 200 such hexagons.|name=disjoint from 8 and intersects 16}} 10 of which cross at each vertex{{Efn|The 10 hexagons which cross at each vertex lie along the 20 short radii of the icosahedral vertex figure.{{Efn|name=vertex icosahedral pyramid}}}} (4 from each of five 24-cells that meet at the vertex, with each hexagon in two of those 24-cells).{{Efn|name=five 24-cells at each vertex of 600-cell}} Each set of 16 hexagons consists of the 96 edges and 24 vertices of one of the 25 overlapping inscribed 24-cells. The <math>\sqrt{1}</math> chords join vertices which are two <math>\phi^{-1}</math> edges apart. Each <math>\sqrt{1}</math> chord is the long diameter of a face-bonded pair of tetrahedral cells (a [[W:triangular bipyramid|triangular bipyramid]]), and passes through the center of the shared face. As there are 1200 faces, there are 1200 <math>\sqrt{1}</math> chords, in 600 parallel pairs, <math>\sqrt{3}</math> apart. The hexagonal planes are non-orthogonal (60 degrees apart) but they occur as 100 ''dual pairs'' in which all 3 axes of one hexagon are orthogonal to all 3 axes of its dual.{{Sfn|Waegell & Aravind|2009|loc=§3.4. The 24-cell: points, lines, and Reye's configuration|p=5|ps=; Here Reye's "points" and "lines" are axes and hexagons, respectively. The dual hexagon ''planes'' are not orthogonal to each other, only their dual axis pairs. Dual hexagon pairs do not occur in individual 24-cells, only between 24-cells in the 600-cell.}} The 200 great hexagons can be divided into 10 sets of 20 non-intersecting Clifford parallel geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 20 hexagons in each set reach all 120 vertices.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} The <math>\sqrt{3 - \phi} \approx 1.176</math> chords form 144 central pentagons, 6 of which cross at each vertex.{{Efn|name=24-cells bound by pentagonal fibers}} The <math>\sqrt{3 - \phi}</math> chords run vertex-to-every-second-vertex in the same planes as the 72 decagons: two pentagons are inscribed in each decagon. They join vertices which are two <math>\phi^{-1}</math> edges apart on a geodesic great circle. The 720 <math>\sqrt{3 - \phi}</math> chords occur in 360 parallel pairs, <math>\phi</math> apart. The <math>\sqrt{2}</math> chords form 450 central squares, 15 of which cross at each vertex (3 from each of the five 24-cells that meet at the vertex). The <math>\sqrt{2}</math> chords join vertices which are three <math>\phi^{-1}</math> edges apart (and two <math>\sqrt{1}</math> chords apart). There are 600 <math>\sqrt{2}</math> chords, in 300 parallel pairs, <math>\sqrt{2}</math> apart. The 450 great squares (225 [[W:Completely orthogonal|completely orthogonal]] pairs) can be divided into 15 sets of 30 non-intersecting Clifford parallel geodesics, such that only one square great circle in each set passes through each vertex, and the 30 squares (15 completely orthogonal pairs) in each set reach all 120 vertices.{{Sfn|Sadoc|2001|p=577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the four-fold screw axis}} The <math>\phi \approx 1.618</math> chords form the legs of 720 central isosceles triangles (72 sets of 10 inscribed in each decagon), 6 of which cross at each vertex. The third edge (base) of each isosceles triangle is of length <math>\sqrt{2 + \phi} \approx 1.902</math>. The <math>\phi</math> chords run vertex-to-every-third-vertex in the same planes as the 72 decagons, joining vertices which are three <math>\phi^{-1}</math> edges apart on a geodesic great circle. There are 720 distinct <math>\phi</math> chords, in 360 parallel pairs, <math>\sqrt{3 - \phi}</math> apart. The <math>\sqrt{3}</math> chords form 400 equilateral central triangles (25 sets of 32, with each triangle in two sets), 10 of which cross at each vertex (4 from each of five [[24-cell#Geodesics|24-cells]], with each triangle in two of the 24-cells). Each set of 32 triangles consists of the 96 <math>\sqrt{3}</math> chords and 24 vertices of one of the 25 overlapping inscribed 24-cells. The <math>\sqrt{3}</math> chords run vertex-to-every-second-vertex in the same planes as the 200 hexagons: two triangles are inscribed in each hexagon. The <math>\sqrt{3}</math> chords join vertices which are four <math>\phi^{-1}</math> edges apart (and two <math>\sqrt{1}</math> chords apart on a geodesic great circle). Each <math>\sqrt{3}</math> chord is the long diameter of two cubic cells in the same 24-cell.{{Efn|The 25 inscribed 24-cells each have 3 inscribed tesseracts, which each have 8 <math>\sqrt{1}</math> cubic cells. The 1200 <math>\sqrt{3}</math> chords are the 4 long diameters of these 600 cubes. The three tesseracts in each 24-cell overlap, and each <math>\sqrt{3}</math> chord is a long diameter of two different cubes, in two different tesseracts, in two different 24-cells. [[24-cell#Relationships among interior polytopes|Each cube belongs to just one tesseract]] in just one 24-cell.|name=600 cubes}} There are 1200 <math>\sqrt{3}</math> chords, in 600 parallel pairs, <math>\sqrt{1}</math> apart. The <math>\sqrt{2 + \phi} \approx 1.902</math> chords (the diagonals of the pentagons) form the legs of 720 central isosceles triangles (144 sets of 5 inscribed in each pentagon), 6 of which cross at each vertex. The third edge (base) of each isosceles triangle is an edge of the pentagon of length <math>\sqrt{3 - \phi} \approx 1.176</math>, so these are [[W:Golden triangle (mathematics)|golden triangles]]. The <math>\sqrt{2 + \phi}</math> chords run vertex-to-every-fourth-vertex in the same planes as the 72 decagons, joining vertices which are four <math>\phi^{-1}</math> edges apart on a geodesic great circle. There are 720 distinct <math>\sqrt{2 + \phi}</math> chords, in 360 parallel pairs, <math>\phi^{-1}</math> apart. The <math>\sqrt{4}</math> chords occur as 60 long diameters (75 sets of 4 orthogonal axes with each set comprising a [[16-cell#Coordinates|16-cell]]), the 120 long radii of the 600-cell. The <math>\sqrt{4}</math> chords join opposite vertices which are five <math>\phi^{-1}</math> edges apart on a geodesic great circle. There are 25 distinct but overlapping sets of 12 diameters, each comprising one of the 25 inscribed 24-cells.{{Efn|name=Schoute's ten ways to get five disjoint 24-cells}} There are 75 distinct but overlapping sets of 4 orthogonal diameters, each comprising one of the 75 inscribed 16-cells. The sum of the squared lengths of all these distinct chords of the 600-cell is 14,400 = 120².{{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}} In this case, <math>(2 - \phi) \cdot 720 + 1 \cdot 1200 + {}\!</math><math>(3 - \phi) \cdot 720 + 2 \cdot 1800 + {}\!</math><math>(1 + \phi)\cdot 720 + 3\cdot 1200 + {}\!</math><math>(2 + \phi) \cdot 720 + 4 \cdot 60</math> is 14,400.}} These are all the central polygons through vertices, but the 600-cell does have one noteworthy great circle that does not pass through any vertices (a 0-gon).{{Efn|Each great decagon central plane is [[W:completely orthogonal|completely orthogonal]] to a great 30-gon{{Efn|A ''[[W:triacontagon|triacontagon]]'' or 30-gon is a thirty-sided polygon. The triacontagon is the largest regular polygon whose interior angle is the sum of the [[W:Interior angle|interior angles]] of smaller polygons: 168° is the sum of the interior angles of the [[W:Equilateral triangle|equilateral triangle]] (60°) and the [[W:Regular pentagon|regular pentagon]] (108°).|name=triacontagon}} central plane which does not intersect any vertices of the 600-cell. The 72 30-gons are each the center axis of a 30-cell [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]],{{Efn|name=Boerdijk–Coxeter helix}} with each segment of the 30-gon passing through a tetrahedron similarly. The 30-gon great circle resides completely in the curved 3-dimensional surface of its 3-sphere;{{Efn|name=0-gon central planes}} its curved segments are not chords. It does not touch any edges or vertices, but it does hit faces. It is the central axis of a spiral skew 30-gram, the [[W:Petrie polygon|Petrie polygon]] of the 600-cell which links all 30 vertices of the 30-cell Boerdijk–Coxeter helix, with three of its edges in each cell.{{Efn|name=Triacontagram}}|name=non-vertex geodesic}} Moreover, 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 600-cell vertices that are helical rather than simply circular; they correspond to isoclinic (diagonal) [[#Rotations|rotations]] rather than simple rotations.{{Efn|name=isoclinic geodesic}} All the geodesic polygons enumerated above lie in central planes of just three kinds, each characterized by a rotation angle: decagon planes ({{sfrac|𝜋|5}} apart), hexagon planes ({{sfrac|𝜋|3}} apart, also in the 25 inscribed 24-cells), and square planes ({{sfrac|𝜋|2}} apart, also in the 75 inscribed 16-cells and the 24-cells). These central planes of the 600-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming an [[W:icosidodecahedron|icosidodecahedron]]. There are 450 great squares 90 degrees apart; 200 great hexagons 60 degrees apart; and 72 great decagons 36 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 are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great decagons are a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in each angle, and ''may'' be the same angle apart in ''both'' angles.{{Efn|The decagonal planes in the 600-cell occur in equi-isoclinic{{Efn|In 4-space no more than 4 great circles may be Clifford parallel{{Efn|name=Clifford parallels}} and all the same angular distance apart.{{Sfn|Lemmens & Seidel|1973}} Such central planes are mutually ''isoclinic'': each pair of planes is separated by two ''equal'' angles, and an isoclinic [[#Rotations|rotation]] by that angle will bring them together. Where three or four such planes are all separated by the ''same'' angle, they are called ''equi-isoclinic''.|name=equi-isoclinic planes}} groups of 3, everywhere that 3 Clifford parallel decagons 36° ({{sfrac|𝝅|5}}) apart form a 30-cell [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]].{{Efn|name=Boerdijk–Coxeter helix}} Also Clifford parallel to those 3 decagons are 3 equi-isoclinic decagons 72° ({{sfrac|2𝝅|5}}) apart, 3 108° ({{sfrac|3𝝅|5}}) apart, and 3 144° ({{sfrac|4𝝅|5}}) apart, for a total of 12 Clifford parallel [[#Decagons|decagons]] (120 vertices) that comprise a discrete Hopf fibration. Because the great decagons lie in isoclinic planes separated by ''two'' equal angles, their corresponding vertices are separated by a combined vector relative to ''both'' angles. Vectors in 4-space may be combined by [[W:Quaternion#Multiplication of basis elements|quaternionic multiplication]], discovered by [[W:William Rowan Hamilton|Hamilton]].{{Sfn|Mamone, Pileio & Levitt|2010|p=1433|loc=§4.1|ps=; A Cartesian 4-coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0). Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector. Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors <small><math>\left(w,x,y,z\right)_1</math></small> and <small><math>\left(w,x,y,z\right)_2</math></small> according to<br> <small><math display=block>\begin{pmatrix} w_2\\ x_2\\ y_2\\ z_2 \end{pmatrix} * \begin{pmatrix} w_1\\ x_1\\ y_1\\ z_1 \end{pmatrix} = \begin{pmatrix} {w_2 w_1 - x_2 x_1 - y_2 y_1 - z_2 z_1}\\ {w_2 x_1 + x_2 w_1 + y_2 z_1 - z_2 y_1}\\ {w_2 y_1 - x_2 z_1 + y_2 w_1 + z_2 x_1}\\ {w_2 z_1 + x_2 y_1 - y_2 x_1 + z_2 w_1} \end{pmatrix} </math></small>}} The corresponding vertices of two great polygons which are 36° ({{sfrac|𝝅|5}}) apart by isoclinic rotation are 60° ({{sfrac|𝝅|3}}) apart in 4-space. The corresponding vertices of two great polygons which are 108° ({{sfrac|3𝝅|5}}) apart by isoclinic rotation are also 60° ({{sfrac|𝝅|3}}) apart in 4-space. The corresponding vertices of two great polygons which are 72° ({{sfrac|2𝝅|5}}) apart by isoclinic rotation are 120° ({{sfrac|2𝝅|3}}) apart in 4-space, and the corresponding vertices of two great polygons which are 144° ({{sfrac|4𝝅|5}}) apart by isoclinic rotation are also 120° ({{sfrac|2𝝅|3}}) apart in 4-space.|name=equi-isoclinic decagons}} Great hexagons may be 60° ({{sfrac|𝝅|3}}) apart in one or ''both'' angles, and may be a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in one or ''both'' angles.{{Efn|The hexagonal planes in the 600-cell occur in equi-isoclinic{{Efn|name=equi-isoclinic planes}} groups of 4, everywhere that 4 Clifford parallel hexagons 60° ({{sfrac|𝝅|3}}) apart form a 24-cell. Also Clifford parallel to those 4 hexagons are 4 equi-isoclinic hexagons 36° ({{sfrac|𝝅|5}}) apart, 4 72° ({{sfrac|2𝝅|5}}) apart, 4 108° ({{sfrac|3𝝅|5}}) apart, and 4 144° ({{sfrac|4𝝅|5}}) apart, for a total of 20 Clifford parallel [[#Hexagons|hexagons]] (120 vertices) that comprise a discrete Hopf fibration.|name=equi-isoclinic hexagons}} Great squares may be 90° ({{sfrac|𝝅|2}}) apart in one or both angles, may be 60° ({{sfrac|𝝅|3}}) apart in one or both angles, and may be a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in one or both angles.{{Efn|The square planes in the 600-cell occur in equi-isoclinic{{Efn|name=equi-isoclinic planes}} groups of 2, everywhere that 2 Clifford parallel squares 90° ({{sfrac|𝝅|2}}) apart form a 16-cell. Also Clifford parallel to those 2 squares are 4 equi-isoclinic groups of 4, where 3 Clifford parallel 16-cells 60° ({{sfrac|𝝅|3}}) apart form a 24-cell. Also Clifford parallel are 4 equi-isoclinic groups of 3: 3 36° ({{sfrac|𝝅|5}}) apart, 3 72° ({{sfrac|2𝝅|5}}) apart, 3 108° ({{sfrac|3𝝅|5}}) apart, and 3 144° ({{sfrac|4𝝅|5}}) apart, for a total of 30 Clifford parallel [[#Squares|squares]] (120 vertices) that comprise a discrete Hopf fibration.|name=equi-isoclinic squares}} Planes which are separated by two equal angles are called ''[[24-cell#Clifford parallel polytopes|isoclinic]]''.{{Efn|name=equi-isoclinic planes}} Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great hexagon and a great decagon ''may'' be isoclinic, but more often they are separated by a {{sfrac|𝝅|3}} (60°) angle ''and'' a multiple (from 1 to 4) of {{sfrac|𝝅|5}} (36°) angle.|name=two angles between central planes}} Each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each great hexagon plane is completely orthogonal to a plane which intersects only two vertices (one <math>\sqrt{4}</math> long diameter): a great [[W:digon|digon]] plane.{{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=digon planes}} Each great decagon plane is completely orthogonal to a plane which intersects ''no'' vertices: a great 0-gon plane.{{Efn|The 600-cell has 72 great 30-gons: 6 sets of 12 Clifford parallel 30-gon central planes, each completely orthogonal to a decagon central plane. Unlike the great circles of the unit-radius 600-cell that pass through its vertices, this 30-gon is not actually a great circle of the unit-radius 3-sphere. Because it passes through face centers rather than vertices, it has a shorter radius and lies on a smaller 3-sphere. Of course, there is also a unit-radius great circle in this central plane completely orthogonal to a decagon central plane, but as a great circle polygon it is a 0-gon, not a 30-gon, because it intersects ''none'' of the points of the 600-cell. In the 600-cell, the great circle polygon completely orthogonal to each great decagon is a 0-gon. |name=0-gon central planes}} ==== Fibrations of great circle polygons ==== Each set of similar great circle polygons (squares or hexagons or decagons) can be divided into bundles of non-intersecting Clifford parallel great circles (of 30 squares or 20 hexagons or 12 decagons).{{Efn|name=Clifford parallels}} Each [[W:fiber bundle|fiber bundle]] of Clifford parallel great circles{{Efn|name=equi-isoclinic planes}} is a discrete [[W:Hopf fibration|Hopf fibration]] which fills the 600-cell, visiting all 120 vertices just once.{{Sfn|Sadoc|2001|pp=575-578|loc=§2 Geometry of the {3,3,5}-polytope in S₃|ps=; Sadoc studied all the Hopf fibrations of the 600-cell into sets of {4}, {6} or {10} great circle fibers on different screw axes, gave their Hopf maps, and fully illustrated the characteristic decagonal cell rings.}} Each discrete Hopf fibration has its 3-dimensional ''base'' which is a distinct polyhedron that acts as a ''map'' or scale model of the fibration.{{Efn|name=Hopf fibration base}} The great circle polygons in each bundle spiral around each other, delineating helical rings of face-bonded cells which nest into each other, pass through each other without intersecting in any cells and exactly fill the 600-cell with their disjoint cell sets. The different fiber bundles with their cell rings each fill the same space (the 600-cell) but their fibers run Clifford parallel in different "directions"; great circle polygons in different fibrations are not Clifford parallel.{{Sfn|Tyrrell & Semple|1971|loc=§4. Isoclinic planes in Euclidean space E₄|pp=6-7}} ===== Decagons ===== [[File:Regular_star_figure_6(5,2).svg|thumb|200px|[[W:Triacontagon#Triacontagram|Triacontagram {30/12}=6{5/2}]] is the [[Schläfli double six|Schläfli double six]] configuration 30₂12₅ characteristic of the H₄ polytopes. The 30 vertex circumference is the skew Petrie polygon.{{Efn|name=Petrie polygons of the 120-cell}} The interior angle between adjacent edges is 36°, also the isoclinic angle between adjacent Clifford parallel decagon planes.{{Efn|name=two angles between central planes}}]] The fibrations of the 600-cell include 6 fibrations of its 72 great decagons: 6 fiber bundles of 12 great decagons.{{Efn|name=Clifford parallel decagons}} The 12 Clifford parallel decagons in each bundle are completely disjoint. Adjacent parallel decagons are spanned by edges of other great decagons. Each fiber bundle{{Efn|name=equi-isoclinic decagons}} delineates [[#Boerdijk–Coxeter helix rings|20 helical rings]] of 30 tetrahedral cells each,{{Efn|name=Boerdijk–Coxeter helix}} with five rings nesting together around each decagon.{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} The Hopf map of this fibration is the [[W:icosahedron|icosahedron]], where each of 12 vertices lifts to a great decagon, and each of 20 triangular faces lifts to a 30-cell ring.{{Efn|name=Hopf fibration base}} Each tetrahedral cell occupies only one of the 20 cell rings in each of the 6 fibrations. The tetrahedral cell contributes each of its 6 edges to a decagon in a different fibration, but contributes that edge to five distinct cell rings in the fibration.{{Efn|name=tetrahedron linking 6 decagons}} The 12 great circles and [[#Boerdijk–Coxeter helix rings|30-cell ring]]s of the 600-cell's 6 characteristic [[W:Hopf fibration|Hopf fibration]]s make the 600-cell a [[W:Configuration (geometry)|geometric configuration]] of 30 "points" and 12 "lines" written as 30₂12₅. It is called the [[Schläfli double six|Schläfli double six]] configuration after [[W:Ludwig Schläfli|Ludwig Schläfli]],{{Sfn|Schläfli|1858|ps=; this paper of Schläfli's describing the [[Schläfli double six|double six configuration]] was one of the only fragments of his discovery of the [[W:Regular polytopes (book)|regular polytopes]] in higher dimensions to be published during his lifetime.{{Sfn|Coxeter|1973|p=211|loc=§11.x Historical remarks|ps=; "The finite group [3^{2, 2, 1}] is isomorphic with the group of incidence-preserving permutations of the 27 lines on the general cubic surface. (For the earliest description of these lines, see Schlafli 2.)".}}}} the Swiss mathematician who discovered the 600-cell and the complete set of regular polytopes in ''n'' dimensions.{{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."}} ===== Hexagons ===== The [[24-cell#Cell rings|fibrations of the 24-cell]] include 4 fibrations of its 16 great hexagons: 4 fiber bundles of 4 great hexagons. The 4 Clifford parallel hexagons in each bundle are completely disjoint. Adjacent parallel hexagons are spanned by edges of other great hexagons. Each fiber bundle delineates 4 helical rings of 6 octahedral cells each, with three rings nesting together around each hexagon. Each octahedral cell occupies only one cell ring in each of the 4 fibrations. The octahedral cell contributes 3 of its 12 edges to 3 different Clifford parallel hexagons in each fibration, but contributes each edge to three distinct cell rings in the fibration. The 600-cell contains 25 24-cells, and can be seen (10 different ways) as a compound of 5 disjoint 24-cells.{{Efn|name=24-cells bound by pentagonal fibers}} It has 10 fibrations of its 200 great hexagons: 10 fiber bundles of 20 great hexagons. The 20 Clifford parallel hexagons in each bundle are completely disjoint. Adjacent parallel hexagons are spanned by edges of great decagons.{{Efn|name=equi-isoclinic hexagons}} Each fiber bundle delineates 20 helical rings of 6 octahedral cells each, with three rings nesting together around each hexagon. The Hopf map of this fibration is the [[W:dodecahedron|dodecahedron]], where the 20 vertices each lift to a bundle of great hexagons.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} Each octahedral cell occupies only one of the 20 6-octahedron rings in each of the 10 fibrations. The 20 6-octahedron rings belong to 5 disjoint 24-cells of 4 6-octahedron rings each; each hexagonal fibration of the 600-cell consists of 5 disjoint 24-cells. ===== Squares ===== The [[16-cell#Helical construction|fibrations of the 16-cell]] include 3 fibrations of its 6 great squares: 3 fiber bundles of 2 great squares. The 2 Clifford parallel squares in each bundle are completely disjoint. Adjacent parallel squares are spanned by edges of other great squares. Each fiber bundle delineates 2 helical rings of 8 tetrahedral cells each. Each tetrahedral cell occupies only one cell ring in each of the 3 fibrations. The tetrahedral cell contributes each of its 6 edges to a different square (contributing two opposite non-intersecting edges to each of the 3 fibrations), but contributes each edge to both of the two distinct cell rings in the fibration. The 600-cell contains 75 16-cells, and can be seen (10 different ways) as a compound of 15 disjoint 16-cells. It has 15 fibrations of its 450 great squares: 15 fiber bundles of 30 great squares. The 30 Clifford parallel squares in each bundle are completely disjoint. Adjacent parallel squares are spanned by edges of great decagons.{{Efn|name=equi-isoclinic squares}} Each fiber bundle delineates 30 cell-disjoint helical rings of 8 tetrahedral cells each.{{Efn|These are the <math>\sqrt{2}</math> tetrahedral cells of the 75 inscribed 16-cells, ''not'' the <math>\phi^{-1}</math> tetrahedral cells of the 600-cell.|name=two different tetrahelixes}} The Hopf map of this fibration is the [[W:icosidodecahedron|icosidodecahedron]], where the 30 vertices each lift to a bundle of great squares.{{Sfn|Sadoc|2001|p=577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the four-fold screw axis}} Each tetrahedral cell occupies only one of the 30 8-tetrahedron rings in each of the 15 fibrations. ===== Clifford parallel cell rings ===== The densely packed helical cell rings{{Sfn|Coxeter|1970|ps=, 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]].{{Efn|name=orthoscheme ring}} He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s 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 the regular 4-polytopes with tetrahedral cells (5-cell, 16-cell, 600-cell) have twisted cell rings, and the others (whose cells have opposing faces) do not.{{Efn|name=directly congruent versus twisted cell rings}} Separately, he categorized cell rings by whether they form their honeycombs in hyperbolic or Euclidean space, the latter being those found in the 4-polytopes which can tile 4-space by translation to form Euclidean honeycombs (16-cell, 8-cell, 24-cell).}}{{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 decompositions composed of meridian and equatorial cell rings with illustrations.}}{{Sfn|Sadoc|2001|pp=575-578|loc=§2 Geometry of the {3,3,5}-polytope in S₃|ps=; Sadoc studied all the Hopf fibrations of the 600-cell into sets of {4}, {6} or {10} great circle fibers on different screw axes, gave their Hopf maps, and fully illustrated the characteristic decagonal cell rings.}} of fibrations are cell-disjoint, but they share vertices, edges and faces. Each fibration of the 600-cell can be seen as a dense packing of cell rings with the corresponding faces of adjacent cell rings face-bonded to each other.{{Efn|name=fibrations are distinguished only by rotations}} The same fibration can also be seen as a minimal ''sparse'' arrangement of fewer ''completely disjoint'' cell rings that do not touch at all.{{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}} The fibrations of great decagons can be seen (five different ways) as 4 completely disjoint 30-cell rings with spaces separating them, rather than as 20 face-bonded cell rings, by leaving out all but one cell ring of the five that meet at each decagon.{{Sfn|Sadoc|2001|loc=§2.6 The {3, 3, 5} polytope: a set of four helices|p=578}} The five different ways you can do this are equivalent, in that all five correspond to the same discrete fibration (in the same sense that the 6 decagonal fibrations are equivalent, in that all 6 cover the same 600-cell). The 4 cell rings still constitute the complete fibration: they include all 12 Clifford parallel decagons, which visit all 120 vertices.{{Efn|The only way to partition the 120 vertices of the 600-cell into 4 completely disjoint 30-vertex, 30-cell rings{{Efn|name=Boerdijk–Coxeter helix}} is by partitioning each of 15 completely disjoint 16-cells similarly into 4 symmetric parts: 4 antipodal vertex pairs lying on the 4 orthogonal axes of the 16-cell. The 600-cell contains 75 distinct 16-cells which can be partitioned into sets of 15 completely disjoint 16-cells. In any set of 4 completely disjoint 30-cell rings, there is a set of 15 completely disjoint 16-cells, with one axis of each 16-cell in each 30-cell ring.|name=fifteen 16-cells partitioned among four 30-cell rings}} This subset of 4 of 20 cell rings is dimensionally analogous{{Efn|One might ask whether dimensional analogy "always works", or if it is perhaps "just guesswork" that might sometimes be incapable of producing a correct dimensionally analogous figure, especially when reasoning from a lower to a higher dimension. Apparently dimensional analogy in both directions has firm mathematical foundations. Dechant{{Sfn|Dechant|2021|loc=§1. Introduction}} derived the 4D symmetry groups from their 3D symmetry group counterparts by induction, demonstrating that there is nothing in 4D symmetry that is not already inherent in 3D symmetry. He showed that neither 4D symmetry nor 3D symmetry is more fundamental than the other, as either can be derived from the other. This is true whether dimensional analogies are computed using Coxeter group theory, or Clifford geometric algebra. These two rather different kinds of mathematics contribute complementary geometric insights. Another profound example of dimensional analogy mathematics is the [[W:Hopf fibration|Hopf fibration]], a mapping between points on the 2-sphere and disjoint (Clifford parallel) great circles on the 3-sphere.|name=math of dimensional analogy}} to the subset of 12 of 72 decagons, in that both are sets of completely disjoint [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]] which visit all 120 vertices.{{Efn|Unlike their bounding decagons, the 20 cell rings themselves are ''not'' all Clifford parallel to each other, because only completely disjoint polytopes are Clifford parallel.{{Efn|name=completely disjoint}} The 20 cell rings have 5 different subsets of 4 Clifford parallel cell rings. Each cell ring is bounded by 3 Clifford parallel great decagons, so each subset of 4 Clifford parallel cell rings is bounded by a total of 12 Clifford parallel great decagons (a discrete Hopf fibration). In fact each of the 5 different subsets of 4 cell rings is bounded by the ''same'' 12 Clifford parallel great decagons (the same Hopf fibration); there are 5 different ways to see the same 12 decagons as a set of 4 cell rings (and equivalently, just one way to see them as a single set of 20 cell rings).}} The subset of 4 of 20 cell rings is one of 5 fibrations ''within'' the fibration of 12 of 72 decagons: a fibration of a fibration. All the fibrations have this two level structure with ''subfibrations''. The fibrations of the 24-cell's great hexagons can be seen (three different ways) as 2 completely disjoint 6-cell rings with spaces separating them, rather than as 4 face-bonded cell rings, by leaving out all but one cell ring of the three that meet at each hexagon. Therefore each of the 10 fibrations of the 600-cell's great hexagons can be seen as 2 completely disjoint octahedral cell rings. The fibrations of the 16-cell's great squares can be seen (two different ways) as a single 8-tetrahedral-cell ring with an adjacent cell-ring-sized empty space, rather than as 2 face-bonded cell rings, by leaving out one of the two cell rings that meet at each square. Therefore each of the 15 fibrations of the 600-cell's great squares can be seen as a single tetrahedral cell ring.{{Efn|name=two different tetrahelixes}} The sparse constructions of the 600-cell's fibrations correspond to lower-symmetry decompositions of the 600-cell, 24-cell or [[16-cell#Helical construction|16-cell]] with cells of different colors to distinguish the cell rings from the spaces between them.{{Efn|Note that the differently colored helices of cells are different cell rings (or ring-shaped holes) in the same fibration, ''not'' the different fibrations of the 4-polytope. Each fibration is the entire 4-polytope.}} The particular lower-symmetry form of the 600-cell corresponding to the sparse construction of the great decagon fibrations is dimensionally analogous{{Efn|name=math of dimensional analogy}} to the [[W:Icosahedron#Pyritohedral symmetry|snub tetrahedron]] form of the icosahedron (which is the ''base''{{Efn|Each [[W:Hopf fibration|Hopf fibration]] of the 3-sphere into Clifford parallel great circle fibers has a map (called its ''base'') which is an ordinary [[W:2-sphere#Dimensionality|2-sphere]].{{Sfn|Zamboj|2021}} On this map each great circle fiber appears as a single point. The base of a great decagon fibration of the 600-cell is the [[W:icosahedron|icosahedron]], in which each vertex represents one of the 12 great decagons.{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} To a toplogist the base is not necessarily any part of the thing it maps: the base icosahedron is not expected to be a cell or interior feature of the 600-cell, it is merely the dimensionally analogous sphere,{{Efn|name=math of dimensional analogy}} useful for reasoning about the fibration. But in fact the 600-cell does have [[#Icosahedra|icosahedra]] in it: 120 icosahedral [[W:Vertex figure|vertex figure]]s,{{Efn|name=vertex icosahedral pyramid}} any of which can be seen as its base: a 3-dimensional 1:10 scale model of the whole 4-dimensional 600-cell. Each 3-dimensional vertex icosahedron is ''lifted'' to the 4-dimensional 600-cell by a 720 degree [[24-cell#Isoclinic rotations|isoclinic rotation]],{{Efn|name=isoclinic geodesic}} which takes each of its 4 disjoint triangular faces in a circuit around one of 4 disjoint 30-vertex [[#Boerdijk–Coxeter helix rings|rings of 30 tetrahedral cells]] (each [[W:Braid|braid]]ed of 3 Clifford parallel great decagons), and so visits all 120 vertices of the 600-cell, ''generating'' the 600-cell. Since the 12 decagonal great circles (of the 4 rings) are Clifford parallel [[#Decagons|decagons of the same fibration]], we can see geometrically how the icosahedron works as a map of a Hopf fibration of the entire 600-cell, and how the Hopf fibration's characteristic isoclinic rotation generates the 600-cell, since the Hopf fibration is an expression of an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic symmetry]].{{Efn|Sadoc studied twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space, as the the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack perfectly in 4-space without exhibiting any torsion, although their packing in 3-space was imperfect, "frustrated" by their torsion. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [circular] 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]]....{{Efn|name=Petrie polygon of a honeycomb}} 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=dense fabric of pole-circles}} 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>|name=Sadoc frustration}}|name=Hopf fibration base}} of these fibrations on the 2-sphere). Each of the 20 [[#Boerdijk–Coxeter helix rings|Boerdijk-Coxeter cell rings]]{{Efn|name=Boerdijk–Coxeter helix}} is ''lifted'' from a corresponding ''face'' of the icosahedron.{{Efn|The 4 {{Background color|red}} faces of the [[W:Icosahedron#Pyritohedral symmetry|snub tetrahedron]] correspond to the 4 completely disjoint cell rings of the sparse construction of the fibration (its ''subfibration''). The red faces are centered on the vertices of an inscribed tetrahedron, and lie in the center of the larger faces of an inscribing tetrahedron.}} === Constructions === The 600-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, the 120-cell, and the polygons {7} and above.{{Sfn|Coxeter|1973|loc=Table VI (iii): 𝐈𝐈 = {3,3,5}|p=303}} Consequently, there are numerous ways to construct or deconstruct the 600-cell, but none of them are trivial. The construction of the 600-cell from its regular predecessor the 24-cell can be difficult to visualize. ==== Gosset's construction ==== [[W:Thorold Gosset|Thorold Gosset]] discovered the [[W:Semiregular polytope|semiregular 4-polytopes]], including the [[W:Snub 24-cell|snub 24-cell]] with 96 vertices, which falls between the 24-cell and the 600-cell in the sequence of convex 4-polytopes of increasing size and complexity in the same radius. Gosset's construction of the 600-cell from the 24-cell is in two steps, using the snub 24-cell as an intermediate form. In the first, more complex step (described [[W:Snub 24-cell#Constructions|elsewhere]]) the snub 24-cell is constructed by a special snub truncation of a 24-cell at the [[W:Golden ratio|golden sections]] of its edges.{{Sfn|Coxeter|1973|loc=§8.4 The snub {3,4,3}|pp=151-153}} In the second step the 600-cell is constructed in a straightforward manner by adding 4-pyramids (vertices) to facets of the snub 24-cell.{{Sfn|Coxeter|1973|loc=§8.5 Gosset's construction for {3,3,5}|p=153}} The snub 24-cell is a diminished 600-cell from which 24 vertices (and the cluster of 20 tetrahedral cells around each) have been truncated,{{Efn|name=snub 24-cell}} leaving a "flat" icosahedral cell in place of each removed icosahedral pyramid.{{Efn|name=vertex icosahedral pyramid}} The snub 24-cell thus has 24 icosahedral cells and the remaining 120 tetrahedral cells. The second step of Gosset's construction of the 600-cell is simply the reverse of this diminishing: an icosahedral pyramid of 20 tetrahedral cells is placed on each icosahedral cell. Constructing the unit-radius 600-cell from its precursor the unit-radius 24-cell by Gosset's method actually requires ''three'' steps. The 24-cell precursor to the snub-24 cell is ''not'' of the same radius: it is larger, since the snub-24 cell is its truncation. Starting with the unit-radius 24-cell, the first step is to reciprocate it around its [[W:Midsphere|midsphere]] to construct its outer [[W:Dual polyhedra#Canonical duals|canonical dual]]: a larger 24-cell, since the 24-cell is self-dual. That larger 24-cell can then be snub truncated into a unit-radius snub 24-cell. ==== Cell clusters ==== Since it is so indirect, Gosset's construction may not help us very much to directly visualize how the 600 tetrahedral cells fit together into a curved 3-dimensional [[#Boundary envelopes|surface envelope]], or how they lie on the underlying surface envelope of the 24-cell's octahedral cells. For that it is helpful to build up the 600-cell directly from clusters of tetrahedral cells.{{Efn|name=tetrahedral cell adjacency}} Most of us have difficulty [[#Visualization|visualizing]] the 600-cell ''from the outside'' in 4-space, or recognizing an [[#3D projections|outside view]] of the 600-cell due to our total lack of sensory experience in 4-dimensional spaces,{{Sfn|Borovik|2006|ps=; "The environment which directed the evolution of our brain never provided our ancestors with four-dimensional experiences.... [Nevertheless] we humans are blessed with a remarkable piece of mathematical software for image processing hardwired into our brains. Coxeter made full use of it, and expected the reader to use it.... Visualization is one of the most powerful interiorization techniques. It anchors mathematical concepts and ideas into one of the most powerful parts of our brain, the visual processing module. Coxeter Theory [of polytopes generated by] finite reflection groups allow[s] an approach to their study based on a systematic reduction of complex geometric configurations to much simpler two- and three-dimensional special cases."}} but we should be able to visualize the surface envelope of 600 cells ''from the inside'' because that volume is a 3-dimensional honeycomb{{Sfn|Coxeter|1970|loc=§9. The 120-cell and the 600-cell|p=19}} that we could actually "walk around in" and explore.{{Sfn|Miyazaki|1990|ps=; Miyazaki showed that the surface envelope of the 600-cell can be realized architecturally in our ordinary 3-dimensional space as physical buildings (geodesic domes).}} In these exercises of building the 600-cell up from cell clusters, we are entirely within a 3-dimensional space, albeit a strangely small, [[W:Elliptic geometry#Hyperspherical model|closed curved space]], in which we can go a mere ten edge lengths away in a straight line in any direction and return to our starting point. ===== Icosahedra ===== [[File:Uniform polyhedron-43-h01.svg|thumb|A regular icosahedron colored in [[W:Regular icosahedron#Symmetries|snub octahedron]] symmetry.{{Efn|Because the octahedron can be [[W:Snub (geometry)|snub truncated]] yielding an icosahedron,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7}} another name for the icosahedron is [[W:Regular icosahedron#Symmetries|snub octahedron]]. This term refers specifically to a [[W:Icosahedron#Pyritohedral symmetry|lower symmetry]] arrangement of the icosahedron's faces (with 8 faces of one color and 12 of another).|name=snub octahedron}} Icosahedra in the 600-cell are face bonded to each other at the yellow faces, and to clusters of 5 tetrahedral cells at the blue faces. The apex of the [[W:Icosahedral pyramid|icosahedral pyramid]] (not visible) is a 13th 600-cell vertex inside the icosahedron (but above its hyperplane).|alt=|200x200px]] [[File:5-cell net.png|thumb|A cluster of 5 tetrahedral cells: four cells face-bonded around a fifth cell (not visible). The four cells lie in different hyperplanes.|alt=|200x200px]] The [[W:Vertex figure|vertex figure]] of the 600-cell is the [[W:Icosahedron|icosahedron]].{{Efn|In the curved 3-dimensional space of the 600-cell's boundary surface, at each vertex one finds the twelve nearest other vertices surrounding the vertex the way an icosahedron's vertices surround its center. Twelve 600-cell edges converge at the icosahedron's center, where they appear to form six straight lines which cross there. However, the center is actually displaced in the 4th dimension (radially outward from the center of the 600-cell), out of the hyperplane defined by the icosahedron's vertices. Thus the vertex icosahedron is actually a canonical [[W:Icosahedral pyramid|icosahedral pyramid]],{{Efn|name=120 overlapping icosahedral pyramids}} composed of 20 regular tetrahedra on a regular icosahedron base, and the vertex is its apex.{{Efn|name=radially equilateral icosahedral pyramid}}|name=vertex icosahedral pyramid|group=}} Twenty tetrahedral cells meet at each vertex, forming an [[W:Icosahedral pyramid|icosahedral pyramid]] whose apex is the vertex, surrounded by its base icosahedron. It is remarkable that twenty regular tetrahedra fit inside a regular icosahedral pyramid in 4-space. In 3-space, twenty triangular pyramids fit inside a regular icosahedron around its center but they are ''not'' regular tetrahedra, because the regular icosahedron's radius is not the same as its edge length.{{Efn|In Euclidean 3-space, the icosahedron is not [[W:Cuboctahedron#Radial equilateral symmetry|radially equilateral like the cuboctahedron]]. The icosahedron's radii are shorter than its edge length. But in the [[W:3-sphere|spherical 3-space]] of the 600-cell's surface the center of a regular icosahedron is lifted orthogonally out of its 3-space hyperplane: remarkably, just far enough to make its radii the same length as its edges. As a figure in Euclidean 4-space, this radially equilateral spherical icosahedron is an [[W:Icosahedral pyramid|icosahedral pyramid]]. In 4-space the 12 edges radiating from its apex are not actually its radii: the apex of the icosahedral pyramid is not its center, just one of its vertices. But in curved 3-space the 12 edges radiating symmetrically from the apex ''are'' radii, so the icosahedron is radially equilateral ''in that spherical space''. In Euclidean 4-space there are only two radially equilateral figures: 24 edges radiating symmetrically from a central point make the [[24-cell#Tetrahedral constructions|radially equilateral 24-cell]], and a symmetrical subset of 16 of those edges make the [[W:tesseract#Radial equilateral symmetry|radially equilateral tesseract]].|name=radially equilateral icosahedral pyramid}} The 600-cell has a [[W:Dihedral angle|dihedral angle]] of {{nowrap|{{sfrac|𝜋|3}} + arccos(−{{sfrac|1|4}}) ≈ 164.4775°}}.{{Sfn|Coxeter|1973|p=293|ps=; 164°29'}} An entire 600-cell can be assembled from 24 such icosahedral pyramids (bonded face-to-face at 8 of the 20 faces of the icosahedron, colored yellow in the illustration), plus 24 clusters of 5 tetrahedral cells (four cells face-bonded around one) which fill the voids remaining between the icosahedra. Each icosahedron is face-bonded to each adjacent cluster of 5 cells by two blue faces that share an edge (which is also one of the six edges of the central tetrahedron of the five). Six clusters of 5 cells surround each icosahedron, and six icosahedra surround each cluster of 5 cells. Five tetrahedral cells surround each icosahedron edge: two from inside the icosahedral pyramid, and three from outside it.{{Efn|An icosahedron edge between two blue faces is surrounded by two blue-faced icosahedral pyramid cells and 3 cells from an adjacent cluster of 5 cells (one of which is the central tetrahedron of the five)}} The apexes of the 24 icosahedral pyramids are the vertices of a 24-cell inscribed in the 600-cell. The other 96 vertices (the vertices of the icosahedra) are the vertices of an inscribed [[W:Snub 24-cell|snub 24-cell]], which has exactly the same [[W:Snub 24-cell#Structure|structure]] of icosahedra and tetrahedra described here, except that the icosahedra are not 4-pyramids filled by tetrahedral cells; they are only "flat" 3-dimensional icosahedral cells, because the central apical vertex is missing. The 24-cell edges joining icosahedral pyramid apex vertices run through the centers of the yellow faces. Coloring the icosahedra with 8 yellow and 12 blue faces can be done in 5 distinct ways.{{Efn|The pentagonal pyramids around each vertex of the "[[W:Regular icosahedron#Symmetries|snub octahedron]]" icosahedron all look the same, with two yellow and three blue faces. Each pentagon has five distinct rotational orientations. Rotating any pentagonal pyramid rotates all of them, so the five rotational positions are the only five different ways to arrange the colors.}} Thus each icosahedral pyramid's apex vertex is a vertex of 5 distinct 24-cells,{{Efn|Five 24-cells meet at each icosahedral pyramid apex{{Efn|name=vertex icosahedral pyramid}} of the 600-cell. Each 24-cell shares not just one vertex but 6 vertices (one of its four hexagonal central planes) with each of the other four 24-cells.{{Efn|name=disjoint from 8 and intersects 16}}|name=five 24-cells at each vertex of 600-cell}} and the 120 vertices comprise 25 (not 5) 24-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}} The icosahedra are face-bonded into geodesic "straight lines" by their opposite yellow faces, bent in the fourth dimension into a ring of 6 icosahedral pyramids. Their apexes are the vertices of a [[24-cell#Great hexagons|great circle hexagon]]. This hexagonal geodesic traverses a ring of 12 tetrahedral cells, alternately bonded face-to-face and vertex-to-vertex. The long diameter of each face-bonded pair of tetrahedra (each [[W:Triangular bipyramid|triangular bipyramid]]) is a hexagon edge (a 24-cell edge). There are 4 rings of 6 icosahedral pyramids intersecting at each apex-vertex, just as there are 4 cell-disjoint interlocking [[24-cell#Cell rings|rings of 6 octahedra]] in the 24-cell (a [[#Hexagons|hexagonal fibration]]).{{Efn|There is a vertex icosahedron{{Efn|name=vertex icosahedral pyramid}} inside each 24-cell octahedral central section (not inside a <math>\sqrt{1}</math>octahedral cell, but in the larger <math>\sqrt{2}</math> octahedron that lies in a central hyperplane), and a larger icosahedron inside each 24-cell cuboctahedron. The two different-sized icosahedra are the second and fourth [[#Polyhedral sections|sections of the 600-cell (beginning with a vertex)]]. The octahedron and the cuboctahedron are the central sections of the 24-cell (beginning with a vertex and beginning with a cell, respectively).{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections}} The cuboctahedron, large icosahedron, octahedron, and small icosahedron nest like [[W:Russian dolls|Russian dolls]] and are related by a helical contraction.{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids}} The contraction begins with the square faces of the cuboctahedron folding inward along their diagonals to form pairs of triangles.{{Efn|Notice that the contraction is chiral, since there are two choices of diagonal on which to begin folding the square faces.}} The 12 vertices of the cuboctahedron move toward each other to the points where they form a regular icosahedron (the large icosahedron); they move slightly closer together until they form a [[W:Jessen's icosahedron|Jessen's icosahedron]]; they continue to spiral toward each other until they merge into the 6 vertices of the octahedron;{{Sfn|Itoh & Nara|2021|loc=§4. From the 24-cell onto an octahedron|ps=; "This article addresses the 24-cell and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the [[W:Jitterbug transformation|Jitterbug]] by [[W:Buckminster Fuller|Buckminster Fuller]]."}} and they continue moving along the same helical paths, separating again into the 12 vertices of the snub octahedron (the small icosahedron).{{Efn|name=snub octahedron}} The geometry of this sequence of transformations{{Efn|These transformations are not among the orthogonal transformations of the Coxeter groups generated by reflections.{{Efn|name=transformations}} They are transformations of the [[W:Tetrahedral symmetry#Pyritohedral symmetry|pyritohedral 3D symmetry group]], the unique polyhedral point group that is neither a rotation group nor a reflection group.{{Sfn|Koca et. al.|2016|loc=4. Pyritohedral Group and Related Polyhedra|p=145|ps=; see Table 1.}}}} in [[W:3-sphere|S³]] is similar to the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]] and the [[W:Tensegrity#Tensegrity icosahedra|tensegrity icosahedron]] in [[W:Three-dimensional space|R³]]. The twisting, expansive-contractive transformations between these polyhedra were named [[Kinematics of the cuboctahedron#Jitterbug transformations|Jitterbug transformations]] by [[W:Buckminster Fuller|Buckminster Fuller]].<ref>{{cite journal | last = Verheyen | first = H. F. | doi = 10.1016/0898-1221(89)90160-0 | issue = 1–3 | journal = [[W:Computers and Mathematics with Applications|Computers and Mathematics with Applications]] | mr = 0994201 | pages = 203–250 | title = The complete set of Jitterbug transformers and the analysis of their motion | volume = 17 | year = 1989| doi-access = free }}</ref>}} The tetrahedral cells are face-bonded into [[W:Boerdijk-Coxeter helix|triple helices]], bent in the fourth dimension into [[#Boerdijk–Coxeter helix rings|rings of 30 tetrahedral cells]].{{Efn|Since tetrahedra{{Efn|name=tetrahedron linking 6 decagons}} do not have opposing faces, the only way they can be stacked face-to-face in a straight line is in the form of a twisted chain called a [[W:Boerdijk-Coxeter helix|Boerdijk-Coxeter helix]]. This is a Clifford parallel{{Efn|name=Clifford parallels}} triple helix as shown in the [[#Boerdijk–Coxeter helix rings|illustration]]. In the 600-cell we find them bent in the fourth dimension into geodesic rings. Each ring has 30 cells and touches 30 vertices. The cells are each face-bonded to two adjacent cells, but one of the six edges of each tetrahedron belongs only to that cell, and these 30 edges form 3 Clifford parallel great decagons which spiral around each other.{{Efn|name=Clifford parallel decagons}} 5 30-cell rings meet at and spiral around each decagon (as 5 tetrahedra meet at each edge). A bundle of 20 such cell-disjoint rings fills the entire 600-cell, thus constituting a discrete [[W:Hopf fibration|Hopf fibration]]. There are 6 distinct such Hopf fibrations, covering the same space but running in different "directions".|name=Boerdijk–Coxeter helix}} The three helices are geodesic "straight lines" of 10 edges: [[#Hopf spherical coordinates|great circle decagons]] which run Clifford parallel{{Efn|name=Clifford parallels}} to each other. Each tetrahedron, having six edges, participates in six different decagons{{Efn|The six great decagons which pass by each tetrahedral cell along its edges do not all intersect with each other, because the 6 edges of the tetrahedron do not all share a vertex. Each decagon intersects four of the others (at 60 degrees), but just misses one of the others as they run past each other (at 90 degrees) along the opposite and perpendicular [[W:Skew lines|skew edges]] of the tetrahedron. Each tetrahedron links three pairs of decagons which do ''not'' intersect at a vertex of the tetrahedron. However, none of the six decagons are Clifford parallel;{{Efn|name=Clifford parallels}} each belongs to a different [[W:Hopf fibration|Hopf fiber bundle]] of 12. Only one of the tetrahedron's six edges may be part of a helix in any one [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]].{{Efn|name=Boerdijk–Coxeter helix}} Incidentally, this footnote is one of a tetrahedron of four footnotes about Clifford parallel decagons{{Efn|name=Clifford parallel decagons}} that all reference each other.|name=tetrahedron linking 6 decagons}} and thereby in all 6 of the [[#Decagons|decagonal fibrations of the 600-cell]]. The partitioning of the 600-cell into clusters of 20 cells and clusters of 5 cells is artificial, since all the cells are the same. One can begin by picking out an icosahedral pyramid cluster centered at any arbitrarily chosen vertex, so there are 120 overlapping icosahedra in the 600-cell.{{Efn|The 120-point 600-cell has 120 overlapping icosahedral pyramids.{{Efn|name=vertex icosahedral pyramid}}|name=120 overlapping icosahedral pyramids}} Their 120 apexes are each a vertex of five 24-vertex 24-cells, so there are 5*120/24 = 25 overlapping 24-cells.{{Efn|name=24-cells bound by pentagonal fibers}} ===== Octahedra ===== There is another useful way to partition the 600-cell surface, into 24 clusters of 25 tetrahedral cells, which reveals more structure{{Sfn|Coxeter|1973|p=299|loc=Table V: (iv) Simplified sections of {3,3,5} ... beginning with a cell}} and a direct construction of the 600-cell from its predecessor the 24-cell. Begin with any one of the clusters of 5 cells (above), and consider its central cell to be the center object of a new larger cluster of tetrahedral cells. The central cell is the first section of the 600-cell beginning with a cell. By surrounding it with more tetrahedral cells, we can reach the deeper sections beginning with a cell. First, note that a cluster of 5 cells consists of 4 overlapping pairs of face-bonded tetrahedra ([[W:Triangular dipyramid|triangular dipyramid]]s) whose long diameter is a 24-cell edge (a hexagon edge) of length <math>\sqrt{1}</math> Six more triangular dipyramids fit into the concavities on the surface of the cluster of 5,{{Efn|These 12 cells are edge-bonded to the central cell, face-bonded to the exterior faces of the cluster of 5, and face-bonded to each other in pairs. They are blue-faced cells in the 6 different icosahedral pyramids surrounding the cluster of 5.}} so the exterior chords connecting its 4 apical vertices are also 24-cell edges of length <math>\sqrt{1}</math>. They form a tetrahedron of edge length <math>\sqrt{1}</math>, which is the second section of the 600-cell beginning with a cell.{{Efn|The <math>\sqrt{1}</math> tetrahedron has a volume of 9 <math>\phi^{-1}</math> tetrahedral cells. In the curved 3-dimensional volume of the 600 cells, it encloses the cluster of 5 cells, which do not entirely fill it. The 6 dipyramids (12 cells) which fit into the concavities of the cluster of 5 cells overfill it: only one third of each dipyramid lies within the <math>\sqrt{1}</math> tetrahedron. The dipyramids contribute one-third of each of 12 cells to it, a volume equivalent to 4 cells.|name=}} There are 600 of these <math>\sqrt{1}</math> tetrahedral sections in the 600-cell. With the six triangular dipyamids fit into the concavities, there are 12 new cells and 6 new vertices in addition to the 5 cells and 8 vertices of the original cluster. The 6 new vertices form the third section of the 600-cell beginning with a cell, an octahedron of edge length <math>\sqrt{1}</math>, obviously the cell of a 24-cell.{{Efn|The 600-cell also contains 600 ''octahedra''. The first section of the 600-cell beginning with a cell is tetrahedral, and the third section is octahedral. These internal octahedra are not ''cells'' of the 600-cell because they are not volumetrically disjoint, but they are each a cell of one of the 25 internal 24-cells. The 600-cell also contains 600 cubes, each a cell of one of its 75 internal 8-cell tesseracts.{{Efn|name=600 cubes}}|name=600 octahedra}} As partially filled so far (by 17 tetrahedral cells), this <math>\sqrt{1}</math> octahedron has concave faces into which a short triangular pyramid fits; it has the same volume as a regular tetrahedral cell but an irregular tetrahedral shape.{{Efn|Each <math>\sqrt{1}</math> edge of the octahedral cell is the long diameter of another tetrahedral dipyramid (two more face-bonded tetrahedral cells). In the 24-cell, three octahedral cells surround each edge, so one third of the dipyramid lies inside each octahedron, split between two adjacent concave faces. Each concave face is filled by one-sixth of each of the three dipyramids that surround its three edges, so it has the same volume as one tetrahedral cell.}} Each octahedron surrounds 1 + 4 + 12 + 8 = 25 tetrahedral cells: 17 regular tetrahedral cells plus 8 volumetrically equivalent tetrahedral cells each consisting of 6 one-sixth fragments from 6 different regular tetrahedral cells that each span three adjacent octahedral cells.{{Efn|A <math>\sqrt{1}</math> octahedral cell (of any 24-cell inscribed in the 600-cell) has six vertices which all lie in the same hyperplane: they bound an octahedral section (a flat three-dimensional slice) of the 600-cell. The same <math>\sqrt{1}</math> octahedron filled by 25 tetrahedral cells has a total of 14 vertices lying in three parallel three-dimensional sections of the 600-cell: the 6-point <math>\sqrt{1}</math> octahedral section, a 4-point <math>\sqrt{1}</math> tetrahedral section, and a 4-point <math>\phi^{-1}</math> tetrahedral section. In the curved three-dimensional space of the 600-cell's surface, the <math>\sqrt{1}</math> octahedron surrounds the <math>\sqrt{1}</math> tetrahedron which surrounds the <math>\phi^{-1}</math> tetrahedron, as three concentric hulls. This 14-vertex 4-polytope is a 4-pyramid with a regular octahedron base: not a canonical [[W:Octahedral pyramid|octahedral pyramid]] with one apex (which has only 7 vertices) but an irregular truncated octahedral pyramid. Because its base is a regular octahedron which is a 24-cell octahedral cell, this 4-pyramid ''lies on'' the surface of the 24-cell.}} Thus the unit-radius 600-cell may be constructed directly from its predecessor,{{Efn||name=truncated irregular octahedral pyramid}} the unit-radius 24-cell, by placing on each of its octahedral facets a truncated{{Efn|The apex of a canonical <math>\sqrt{1}</math> [[W:Octahedral pyramid|octahedral pyramid]] has been truncated into a regular tetrahedral cell with shorter <math>\phi^{-1}</math> edges, replacing the apex with four vertices. The truncation has also created another four vertices (arranged as a <math>\sqrt{1}</math> tetrahedron in a hyperplane between the octahedral base and the apex tetrahedral cell), and linked these eight new vertices with <math>\phi^{-1}</math> edges. The truncated pyramid thus has eight 'apex' vertices above the hyperplane of its octahedral base, rather than just one apex: 14 vertices in all. The original pyramid had flat sides: the five geodesic routes from any base vertex to the opposite base vertex ran along two <math>\sqrt{1}</math> edges (and just one of those routes ran through the single apex). The truncated pyramid has rounded sides: five geodesic routes from any base vertex to the opposite base vertex run along three <math>\phi^{-1}</math> edges (and pass through two 'apexes').}} irregular octahedral pyramid of 14 vertices{{Efn|The uniform 4-polytopes which this 14-vertex, 25-cell irregular 4-polytope most closely resembles may be the 10-vertex, 10-cell [[W:Rectified 5-cell|rectified 5-cell]] and its dual (it has characteristics of both).}} constructed (in the above manner) from 25 regular tetrahedral cells of edge length <math>\phi^{-1}</math>. ===== Union of two tori ===== There is yet another useful way to partition the 600-cell surface into clusters of tetrahedral cells, which reveals more structure{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5}|ps=; "Let us now proceed to a toroidal decomposition of the {3, 3, 5} polytope."}} and the [[#Decagons|decagonal fibrations]] of the 600-cell. An entire 600-cell can be assembled around two rings of 5 icosahedral pyramids, bonded vertex-to-vertex into two geodesic "straight lines". [[File:100 tets.jpg|thumb|100 tetrahedra in a 10×10 array forming a [[W:Clifford torus|Clifford torus]] boundary in the 600 cell.{{Efn|name=why 100}} Its opposite edges are identified, forming a [[W:Duocylinder|duocylinder]].]] The [[120-cell|120-cell]] can be decomposed into [[120-cell#Intertwining rings|two disjoint tori]]. Since it is the dual of the 600-cell, this same dual tori structure exists in the 600-cell, although it is somewhat more complex. The 10-cell geodesic path in the 120-cell corresponds to the 10-vertex decagon path in the 600-cell.{{Sfn|Coxeter|1970|loc=§9. The 120-cell and the 600-cell|pp=19-23}} Start by assembling five tetrahedra around a common edge. This structure looks somewhat like an angular "flying saucer". Stack ten of these, vertex to vertex, "pancake" style. Fill in the annular ring between each pair of "flying saucers" with 10 tetrahedra to form an icosahedron. You can view this as five vertex-stacked [[W:Icosahedral pyramids|icosahedral pyramids]], with the five extra annular ring gaps also filled in.{{Efn|The annular ring gaps between icosahedra are filled by a ring of 10 face-bonded tetrahedra that all meet at the vertex where the two icosahedra meet. This 10-cell ring is shaped like a [[W:Pentagonal antiprism|pentagonal antiprism]] which has been hollowed out like a bowl on both its top and bottom sides, so that it has zero thickness at its center. This center vertex, like all the other vertices of the 600-cell, is itself the apex of an icosahedral pyramid where 20 tetrahedra meet.{{Efn|name=120 overlapping icosahedral pyramids}} Therefore the annular ring of 10 tetrahedra is itself an equatorial ring of an icosahedral pyramid, containing 10 of the 20 cells of its icosahedral pyramid.|name=annular ring}} The surface is the same as that of ten stacked [[W:pentagonal antiprism|pentagonal antiprism]]s: a triangular-faced column with a pentagonal cross-section.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5}, Fig. 2. A five fold symmetry column|ps=; in caption (sic) dodecagons should be decagons.}} Bent into a columnar ring this is a torus consisting of 150 cells, ten edges long, with 100 exposed triangular faces,{{Efn|The 100-face surface of the triangular-faced 150-cell column could be scissors-cut lengthwise along a 10 edge path and peeled and laid flat as a 10×10 parallelogram of triangles.|name=triangles 10×10}} 150 exposed edges, and 50 exposed vertices. Stack another tetrahedron on each exposed face. This will give you a somewhat bumpy torus of 250 cells with 50 raised vertices, 50 valley vertices, and 100 valley edges.{{Efn|Because the 100-face surface of the 150-cell torus is alternately convex and concave, 100 tetrahedra stack on it in face-bonded pairs, as 50 [[W:Triangular bipyramid|triangular bipyramid]]s which share one raised vertex and bury one formerly exposed valley edge. The triangular bipyramids are vertex-bonded to each other in 5 parallel lines of 5 bipyramids (10 tetrahedra) each, which run straight up and down the outside surface of the 150-cell column.}} The valleys are 10 edge long closed paths and correspond to other instances of the 10-vertex decagon path mentioned above (great circle decagons). These decagons spiral around the center core decagon,{{Efn|5 decagons spiral clockwise and 5 spiral counterclockwise, intersecting each other at the 50 valley vertices.}} but mathematically they are all equivalent (they all lie in central planes). Build a second identical torus of 250 cells that interlinks with the first. This accounts for 500 cells. These two tori mate together with the valley vertices touching the raised vertices, leaving 100 tetrahedral voids that are filled with the remaining 100 tetrahedra that mate at the valley edges. This latter set of 100 tetrahedra are on the exact boundary of the [[W:Duocylinder|duocylinder]] and form a [[W:Clifford torus|Clifford torus]].{{Efn|A [[W:Clifford torus|Clifford torus]] is the [[W:Hopf fibration|Hopf fiber bundle]] of a distinct [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] of a rigid [[W:3-sphere|3-sphere]], involving all of its points. The [[W:SO(4)#Visualization of 4D rotations|torus embedded in 4-space]], like the double rotation, is the [[W:Cartesian product|Cartesian product]] of two [[W:Completely orthogonal|completely orthogonal]] [[W:Great circle|great circle]]s. It is a filled [[W:Doughnut|doughnut]] not a ring doughnut; there is no hole in the 3-sphere except the [[W:4-ball (mathematics)|4-ball]] it encloses. A regular 4-polytope has a distinct number of characteristic Clifford tori, because it has a distinct number of characteristic rotational symmetries. Each forms a discrete fibration that reaches all the discrete points once each, in an isoclinic rotation with a distinct set of pairs of completely orthogonal invariant planes.|name=Clifford torus}} They can be "unrolled" into a square 10×10 array. Incidentally this structure forms one tetrahedral layer in the [[W:Tetrahedral-octahedral honeycomb|tetrahedral-octahedral honeycomb]]. There are exactly 50 "egg crate" recesses and peaks on both sides that mate with the 250 cell tori.{{Efn|How can a bumpy "egg crate" square of 100 tetrahedra lie on the smooth surface of the Clifford torus?{{Efn|name=Clifford torus}} How can a flat 10x10 square represent the 120-vertex 600-cell (where are the other 20 vertices)? In the isoclinic rotation of the 600-cell in [[#Decagons|great decagon invariant planes]], the Clifford torus is a smooth [[W:Clifford torus|Euclidean 2-surface]] which intersects the mid-edges of exactly 100 tetrahedral cells. Edges are what tetrahedra have 6 of. The mid-edges are not vertices of the 600-cell, but they are all 600 vertices of its equal-radius dual polytope, the 120-cell. The 120-cell has 5 disjoint 600-cells inscribed in it, two different ways. This distinct smooth Clifford torus (this rotation) is a discrete fibration of the 120-cell in 60 decagon invariant planes, and a discrete fibration of the 600-cell in 12 decagon invariant planes.|name=why 100}} In this case into each recess, instead of an octahedron as in the honeycomb, fits a [[W:Triangular bipyramid|triangular bipyramid]] composed of two tetrahedra. This decomposition of the 600-cell has [[W:Coxeter notation|symmetry]] [10,2⁺,10], order 400, the same symmetry as the [[W:Grand antiprism|grand antiprism]].{{Sfn|Dechant|2021|pp=20-22|loc=§7. The Grand Antiprism and H₂ × H₂}} The grand antiprism is just the 600-cell with the two above 150-cell tori removed, leaving only the single middle layer of 300 tetrahedra, dimensionally analogous{{Efn|name=math of dimensional analogy}} to the 10-face belt of an icosahedron with the 5 top and 5 bottom faces removed (a [[W:Pentagonal antiprism|pentagonal antiprism]]).{{Efn|The same 10-face belt of an icosahedral pyramid is an annular ring of 10 tetrahedra around the apex.{{Efn|name=annular ring}}}} The two 150-cell tori each contain 6 Clifford parallel great decagons (five around one), and the two tori are Clifford parallel to each other, so together they constitute a complete [[#Clifford parallel cell rings|fibration of 12 decagons]] that reaches all 120 vertices, despite filling only half the 600-cell with cells. ===== Boerdijk–Coxeter helix rings ===== The 600-cell can also be partitioned into 20 cell-disjoint intertwining rings of 30 cells,{{Sfn|Sadoc|2001|pp=577-578|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries}} each ten edges long, forming a discrete [[W:Hopf fibration|Hopf fibration]] which fills the entire 600-cell.{{Sfn|Zamboj|2021|pp=6-12|loc=§2 Mathematical background}} Each ring of 30 face-bonded tetrahedra is a cylindrical [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]] bent into a ring in the fourth dimension. {| class="wikitable" width="600" |[[File:600-cell tet ring.png|200px]]<br>A single 30-tetrahedron [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]] ring within the 600-cell, seen in stereographic projection.{{Efn|name=Boerdijk–Coxeter helix}} |[[File:600-cell Coxeter helix-ring.png|200px]]<br>A 30-tetrahedron ring can be seen along the perimeter of this 30-gonal orthogonal projection of the 600-cell.{{Efn|name=non-vertex geodesic}} |[[File:Regular_star_polygon_30-11.svg|200px]]<br>The 30-cell ring as a {30/11} polygram of 30 edges wound into a helix that twists around its axis 11 times. This projection along the axis of the 30-cell cylinder shows the 30 vertices 12° apart around the cylinder's circular cross section, with the edges connecting every 11th vertex on the circle.{{Efn|The 30 vertices and 30 edges of the 30-cell ring lie on a [[W:Skew polygon|skew]] {30/11} [[W:Star polygon|star polygon]] with a [[W:Winding number|winding number]] of 11 called a [[W:Triacontagon#Triacontagram|triacontagram₁₁]], a continuous tight corkscrew [[W:Helix|helix]] bent into a loop of 30 edges (the {{Background color|magenta|magenta}} edges in the [[#Boerdijk–Coxeter helix rings|triple helix illustration]]), which [[W:Density (polytope)#Polygons|winds]] 11 times around itself in the course of a single revolution around the 600-cell, accompanied by a single 360 degree twist of the 30-cell ring.{{Sfn|Sadoc|2001|pp=577-578|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries}} The same 30-cell ring can also be [[W:Density (polytope)|characterized]] as the [[W:Petrie polygon|Petrie polygon]] of the 600-cell.{{Efn|name=Petrie polygon in 30-cell ring}}|name=Triacontagram}} |- |colspan=3|[[File:Coxeter_helix_edges.png|625px]]<br>The 30-vertex, 30-tetrahedron [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]] ring, cut and laid out flat in 3-dimensional space. Three {{Background color|cyan|cyan}} Clifford parallel great decagons bound the ring.{{Efn|name=Clifford parallel decagons}} They are bridged by a skew 30-gram helix of 30 {{Background color|magenta|magenta}} edges linking all 30 vertices: the [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] of the 600-cell.{{Efn|name=Petrie polygon in 30-cell ring}} The 15 {{Background color|orange|orange}} edges and 15 {{Background color|yellow|yellow}} edges form separate 15-gram helices, the edge-paths of ''isoclines''. |} The 30-cell ring is the 3-dimensional space occupied by the 30 vertices of three {{Background color|cyan|cyan}} Clifford parallel great decagons that lie adjacent to each other, 36° = {{sfrac|𝜋|5}} = one 600-cell edge length apart at all their vertex pairs.{{Efn|name=triple-helix of three central decagonal planes}} The 30 {{Background color|magenta|magenta}} edges joining these vertex pairs form a helical [[W:Triacontagon#Triacontagram|triacontagram]], a skew 30-gram spiral of 30 edge-bonded triangular faces, that is the [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] of the 600-cell.{{Efn|The 600-cell's [[W:Petrie polygon|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|triacontagon {30}]]. It can be [[#Decagons|seen in orthogonal projection as the circumference]] of a [[W:Triacontagon#Triacontagram|triacontagram {30/3}=3{10}]] helix which zig-zags 60° left and right, bridging the space between the 3 Clifford parallel great decagons of the 30-cell ring. In the completely orthogonal plane it projects to the regular [[W:Triacontagon#Triacontagram|triacontagram {30/11}]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii)|ps=; ''600-cell h₁ h₂''.}}|name=Petrie polygon in 30-cell ring}} The dual of the 30-cell ring (the skew 30-gon made by connecting its cell centers) is the [[W:Skew polygon#Regular skew polygons in four dimensions|Petrie polygon]] of the [[120-cell|120-cell]], the 600-cell's [[W:Dual polytope|dual polytope]].{{Efn|The [[W:Skew polygon#Regular skew polygons in four dimensions|regular skew 30-gon]] is the [[W:Petrie polygon|Petrie polygon]] of the 600-cell and its dual the [[120-cell|120-cell]]. The Petrie polygons of the 120-cell occur in the 600-cell as duals of the 30-cell [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix rings]]: connecting their 30 cell centers together produces the Petrie polygons of the dual 120-cell, as noticed by Rolfdieter Frank (circa 2001). Thus he discovered that the vertex set of the 120-cell partitions into 20 non-intersecting Petrie polygons. This set of 20 disjoint Clifford parallel skew polygons is a discrete [[W:Hopf fibration|Hopf fibration]] of the 120-cell (just as their 20 dual 30-cell rings are a discrete [[#Decagons|fibration]] of the 600-cell).|name=Petrie polygons of the 120-cell}} The central axis of the 30-cell ring is a great 30-gon geodesic that passes through the center of 30 faces, but does not intersect any vertices.{{Efn|name=non-vertex geodesic}} The 15 {{Background color|orange|orange}} edges and 15 {{Background color|yellow|yellow}} edges form separate 15-gram helices. Each orange or yellow edge crosses between two {{Background color|cyan|cyan}} great decagons. Successive orange or yellow edges of these 15-gram helices do not lie on the same great circle; they lie in different central planes inclined at 36° = {{sfrac|𝝅|5}} to each other.{{Efn|name=two angles between central planes}} Each 15-gram helix is noteworthy as the edge-path of an [[#Rotations on polygram isoclines|isocline]], the [[W:Geodesic|geodesic]] path of an isoclinic [[#Rotations|rotation]].{{Efn|name=isoclinic geodesic}} The isocline is a circular curve which intersects every ''second'' vertex of the 15-gram, missing the vertex in between. A single isocline runs twice around each orange (or yellow) 15-gram through every other vertex, hitting half the vertices on the first loop and the other half of them on the second loop. The two connected loops forms a single [[W:Möbius loop|Möbius loop]], a skew {15/2} [[W:Pentadecagram|pentadecagram]]. The pentadecagram is not shown in these illustrations (but [[#Decagons and pentadecagrams|see below]]), because its edges are invisible chords between vertices which are two orange (or two yellow) edges apart, and no chords are shown in these illustrations. Although the 30 vertices of the 30-cell ring do not lie in one great 30-gon central plane,{{Efn|The 30 vertices of the [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple-helix ring]] lie in 3 decagonal central planes which intersect only at one point (the center of the 600-cell), even though they are not completely orthogonal or orthogonal at all: they are {{sfrac|{{pi}}|5}} apart.{{Efn|name=two angles between central planes}} Their decagonal great circles are Clifford parallel: one 600-cell edge-length apart at every point.{{Efn|name=Clifford parallels}} They are ordinary 2-dimensional great circles, ''not'' helices, but they are [[W:link (knot theory)|linked]] Clifford parallel circles.|name=triple-helix of three central decagonal planes}} these invisible [[#Decagons and pentadecagrams|pentadecagram isoclines]] are true geodesic circles of a special kind, that wind through all four dimensions rather than lying in a 2-dimensional plane as an ordinary geodesic great circle does.{{Efn|name=4-dimensional great circles}} Five of these 30-cell [[W:Helix|helices]] nest together and spiral around each of the 10-vertex decagon paths, forming the 150-cell torus described in the [[#Union of two tori|grand antiprism decomposition]] above.{{Sfn|Dechant|2021|pp=20-22|loc=§7. The Grand Antiprism and H₂ × H₂}} Thus ''every'' great decagon is the center core decagon of a 150-cell torus.{{Efn|The 20 30-cell rings are [[W:Chiral|chiral]] objects; they either spiral clockwise (right) or counterclockwise (left). The 150-cell torus (formed by five cell-disjoint 30-cell rings of the same chirality surrounding a great decagon) is not itself a chiral object, since it can be decomposed into either five parallel left-handed rings or five parallel right-handed rings. Unlike the 20-cell rings, the 150-cell tori are directly congruent with no [[W:Torsion of a curve|torsion]], like the octahedral [[24-cell#6-cell rings|6-cell rings of the 24-cell]]. Each great decagon has five left-handed 30-cell rings surrounding it, and also five right-handed 30-cell rings surrounding it; but left-handed and right-handed 30-cell rings are not cell-disjoint and belong to different distinct rotations: the left and right rotations of the same fibration. In either distinct isoclinic rotation (left or right), the vertices of the 600-cell move along the axial [[#Decagons and pentadecagrams|15-gram isoclines]] of 20 left 30-cell rings or 20 right 30-cell rings. Thus the great decagons, the 30-cell rings, and the 150-cell tori all occur as sets of Clifford parallel interlinked circles,{{Efn|name=Clifford parallels}} although the exact way they nest together, avoid intersecting each other, and pass through each other to form a [[W:Hopf link|Hopf link]] is not identical for these three different kinds of [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]], in part because the linked pairs are variously of no inherent chirality (the decagons), the same chirality (the 30-cell rings), or no net torsion and both left and right interior organization (the 150-cell tori) but tracing the same chirality of interior organization in any distinct left or right rotation.|name=chirality of cell rings}} The 600-cell may be decomposed into 20 30-cell rings, or into two 150-cell tori and 10 30-cell rings, but not into four 150-cell tori of this kind.{{Efn|A point on the icosahedron Hopf map{{Efn|name=Hopf fibration base}} of the 600-cell's decagonal fibration lifts to a great decagon; a triangular face lifts to a 30-cell ring; and a pentagonal pyramid of 5 faces lifts to a 150-cell torus.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5}, Fig. 2. A five fold symmetry column|ps=; in caption (sic) dodecagons should be decagons.}} In the [[#Union of two tori|grand antiprism decomposition]], two completely disjoint 150-cell tori are lifted from antipodal pentagons, leaving an equatorial ring of 10 icosahedron faces between them: a Petrie decagon of 10 triangles, which lift to 10 30-cell rings. The two completely disjoint 150-cell tori contain 12 disjoint (Clifford parallel) decagons and all 120 vertices, so they comprise a complete Hopf fibration; there is no room for more 150-cell tori of this kind. To get a decomposition of the 600-cell into four 150-cell tori of this kind, the icosahedral map would have to be decomposed into four pentagons, centered at the vertices of an inscribed tetrahedron, and the icosahedron cannot be decomposed that way.}} The 600-cell ''can'' be decomposed into four 150-cell tori of a different kind.{{Efn|Sadoc describes the decomposition of the 600-cell into four tori.{{Sfn|Sadoc|2001|loc=§2.6 The {3, 3, 5} polytope: a set of four helices|p=578}} It is the same [[#Decagons|fibration of 12 great decagons and 20 30-cell rings]], seen as a [[#Clifford parallel cell rings|fibration of four completely disjoint 30-cell rings]]{{Efn|name=completely disjoint}} with spaces between them, which still encompasses all 12 decagons and all 120 vertices. If we look closely at the spaces between the four disjoint 30-cell rings, we ''can'' discern four 150-cell rings of 5 30-cell rings each. But these 150-cell rings do not have 5 30-cell rings around a common decagon axis, and 6 decagons each. Their axis is a 30-cell ring, not a decagon, and they contain only 3 decagons each. To construct them, on each of the four completely disjoint 30-cell rings, face-bond three more 30-cell rings to the exterior faces, making four stellated ("bumpy") rings containing four 30-cell rings (120 cells) each. Collectively they contain 16 of the 20 30-cell rings: there are still four 30-cell ring "holes" left to fill in the 600-cell. To do that, fill some of the surface concavities of each 120-tetrahedron ring by wrapping a fifth 30-cell ring around its circumference, completely orthogonal to the axial 30-cell ring you started with. The result is four 150-cell tori, of 5 30-cell rings each, each having two completely orthogonal 30-cell ring axes, either of which can be seen as either an axis or a circumference: it is both. On the icosahedron Hopf map,{{Efn|name=Hopf fibration base}} the four 30-cell rings lift from a star of four icosahedron faces (three faces edge-bonded around one). The fifth 30-cell ring lifts from a fifth face edge-bonded to the star, a sort of "extra flap" like the sixth square flap of the [[W:Cube#Orthogonal projections|net of a cube]] before you fold it up into a cube. It does not matter which of the six possible adjacent faces you choose as the flap, but the choice determines the choice for all four 150-cell rings. There are six choices because there are six decagonal fibrations; this is when you fix which fibration you are taking. Thus ''every'' 30-cell ring is the center core of a 150-cell ring.}} ==== Radial golden triangles ==== The 600-cell can be constructed radially from 720 [[W:Golden triangle (mathematics)|golden triangle]]s of edge lengths <math>1, 1, \phi^{-1}</math> which meet at the center of the 4-polytope, each contributing two <math>\sqrt{1}</math> radii and a <math>\phi^{-1}</math> edge. They form 1200 triangular pyramids with their apexes at the center: irregular tetrahedra with equilateral <math>\phi^{-1}</math> bases (the faces of the 600-cell). These form 600 tetrahedral pyramids with their apexes at the center: irregular 5-cells with regular <math>\phi^{-1}</math> tetrahedron bases (the cells of the 600-cell). ==== Characteristic orthoscheme ==== {| class="wikitable floatright" !colspan=6|Characteristics of the 600-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "600-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>phi{-1} \approx 0.618</math></small> |align=center|<small>36°</small> |align=center|<small><math>\tfrac{\pi}{5}</math></small> |align=center|<small>164°29′</small> |align=center|<small><math>\pi-2\psi</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{2}{3\phi^2}} \approx 0.505</math></small> |align=center|<small>22°15′20″</small> |align=center|<small><math>\tfrac{\pi}{3} - \eta</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</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}{2\phi^2}} \approx 0.437</math></small> |align=center|<small>18°</small> |align=center|<small><math>\tfrac{\pi}{10}</math></small> |align=center|<small>36°</small> |align=center|<small><math>\tfrac{\pi}{5}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{6\phi^2}} \approx 0.252</math></small> |align=center|<small>17°44′40″</small> |align=center|<small><math>\eta - \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{3}{4\phi^2}} \approx 0.535</math></small> |align=center|<small>22°15′20″</small> |align=center|<small><math>\tfrac{\pi}{3} - \eta</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\phi^2}} \approx 0.309</math></small> |align=center|<small>18°</small> |align=center|<small><math>\tfrac{\pi}{10}</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}{12\phi^2}} \approx 0.178</math></small> |align=center|<small>17°44′40″</small> |align=center|<small><math>\eta - \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>\tfrac12\sqrt{2 + \phi} \approx 0.951</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{\phi^2}{3}} \approx 0.934</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{\phi^4}{8}} \approx 0.926</math></small> |align=center| |align=center| |align=center| |align=center| |- | | | | | |- !align=right|<small><math>\eta</math></small> |align=center| |align=center|<small>37°44′40″</small> |align=center|<small><math>\tfrac{\text{arc sec }4}{2}</math></small> |align=center| |align=center| |} Every regular 4-polytope has its characteristic 4-orthoscheme, an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{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}} The '''characteristic 5-cell of the regular 600-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|3|node|5|node}}, which can be read as a list of the dihedral angles between its mirror facets. It is an irregular [[W:Pyramid (mathematics)#Polyhedral pyramid|tetrahedral pyramid]] based on the [[W:Tetrahedron#Orthoschemes|characteristic tetrahedron of the regular tetrahedron]]. The regular 600-cell is subdivided by its symmetry hyperplanes into 14400 instances of its characteristic 5-cell that all meet at its center.{{Efn|‟The Petrie polygons of the Platonic solid <small><math>\{p, q\}</math></small> correspond to equatorial polygons of the truncation <small><math>\{\tfrac{p}{q}\}</math></small> and to ''equators'' of the simplicially subdivided spherical tessellation <small><math>\{p, q\}</math></small>. This "[[W:Schläfli orthoscheme#Characteristic simplex of the general regular polytope|simplicial subdivision]]" is the arrangement of <small><math>g = g_{p, q}</math></small> right-angled spherical triangles into which the sphere is decomposed by the planes of symmetry of the solid. The great circles that lie in these planes were formerly called "lines of symmetry", but perhaps a more vivid name is ''reflecting circles''. The analogous simplicial subdivision of the spherical honeycomb <small><math>\{p, q, r\}</math></small> consists of the <small><math>g = g_{p, q, r}</math></small> tetrahedra '''0123''' into which a hypersphere (in Euclidean 4-space) is decomposed by the hyperplanes of symmetry of the polytope <small><math>\{p, q, r\}</math></small>. The great spheres which lie in these hyperplanes are naturally called ''reflecting spheres''. Since the orthoscheme has no obtuse angles, it entirely contains the arc that measures the absolutely shortest distance 𝝅/''h'' [between the] 2''h'' tetrahedra [that] are strung like beads on a necklace, or like a "rotating ring of tetrahedra" ... whose opposite edges are generators of a helicoid. The two opposite edges of each tetrahedron are related by a screw-displacement.{{Efn|name=transformations}} Hence the total number of spheres is 2''h''.”{{Sfn|Coxeter|1973|pp=227−233|loc=§12.7 A necklace of tetrahedral beads}}|name=orthoscheme ring}} 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 600-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of a 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 600-cell has unit radius and edge length <small><math>\ell = \phi^{-1} \approx 0.618</math></small>, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{2}{3\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{2\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6\phi^2}}</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{3}{4\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{12\phi^2}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the regular tetrahedron), plus <small><math>1</math></small>, <small><math>\tfrac12\sqrt{2 + \phi}</math></small>, <small><math>\sqrt{\tfrac{\phi^2}{3}}</math></small>, <small><math>\sqrt{\tfrac{\phi^4}{8}}</math></small> (edges which are the characteristic radii of the 600-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{2\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{\phi^4}{8}}</math></small>, first from a 600-cell vertex to a 600-cell edge center, then turning 90° to a 600-cell face center, then turning 90° to a 600-cell tetrahedral cell center, then turning 90° to the 600-cell center. ==== Reflections ==== The 600-cell can be 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}}{{Sfn|Dechant|2017|pp=410-419|loc=§6. The Coxeter Plane; see p. 416, Table 1. Summary of the factorisations of the Coxeter versors of the 4D root systems|ps=; "Coxeter (reflection) groups in the Clifford framework ... afford a uniquely simple prescription for reflections. Via the Cartan-Dieudonné theorem, performing two reflections successively generates a rotation, which in Clifford algebra is described by a spinor that is simply the geometric product of the two vectors generating the reflections."}} so for example an ''n''-dimensional reflection is an (''n''+1)-dimensional half-turn.{{Sfn|Coxeter|1973|loc=§12-34|p=220}} A full isoclinic revolution of the 600-cell in decagonal invariant planes takes ''each'' of the 120 vertices through 15 vertices and back to itself, on a skew pentadecagram₂ geodesic [[#Decagons and pentadecagrams|isocline]] of circumference 5𝝅 that winds around the 3-sphere, as each great decagon rotates (like a wheel) and also tilts sideways (like a coin flipping) with the completely orthogonal plane.{{Efn|name=one true 5𝝅 circle}} Any set of four orthogonal pairs of antipodal vertices (the 8 vertices of one of the 75 inscribed 16-cells){{Efn|name=fifteen 16-cells partitioned among four 30-cell rings}} performing such an orbit visits 15 * 8 = 120 distinct vertices and [[24-cell#Clifford parallel polytopes|generates the 600-cell]] sequentially in one full isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 120 vertices simultaneously by reflection.{{Efn|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). 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> 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). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations,{{Efn|name=double rotation}} 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}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} ==== Weyl orbits ==== Another construction method uses [[#Symmetries|quaternions]] and the [[W:Icosahedral symmetry|icosahedral symmetry]] of [[W:Weyl group|Weyl group]] orbits <math>O(\Lambda)=W(H_4)=I</math> of order 120.{{Sfn|Koca, Al-Ajmi, & Ozdes Koca|2011|loc=6. Dual of the snub 24-cell|pp=986-988}} The following are the orbits of weights of D4 under the Weyl group W(D4): : O(0100) : T = {±1,±e1,±e2,±e3,(±1±e1±e2±e3)/2} : O(1000) : V1 : O(0010) : V2 : O(0001) : V3 [[File:120Cell-SimpleRoots-Quaternion-Tp.png|600px]] With quaternions <math>(p,q)</math> where <math>\bar p</math> is the conjugate of <math>p</math> and <math>[p,q]:r\rightarrow r'=prq</math> and <math>[p,q]^*:r\rightarrow r''=p\bar rq</math>, then the [[W:Coxeter group|Coxeter group]] <math>W(H_4)=\lbrace[p,\bar p] \oplus [p,\bar p]^*\rbrace </math> is the symmetry group of the 600-cell and the [[120-cell|120-cell]] of order 14400. Given <math>p \in T</math> such that <math>\bar p=\pm p^4, \bar p^2=\pm p^3, \bar p^3=\pm p^2, \bar p^4=\pm p</math> and <math>p^\dagger</math> as an exchange of <math>-1/\varphi \leftrightarrow \varphi</math> within <math>p</math>, we can construct: * the [[W:Snub 24-cell|snub 24-cell]] <math>S=\sum_{i=1}^4\oplus p^i T</math> * the 600-cell <math>I=T+S=\sum_{i=0}^4\oplus p^i T</math> * the [[120-cell|120-cell]] <math>J=\sum_{i,j=0}^4\oplus p^i\bar p^{\dagger j}T'</math> === 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)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 600-cell has 14,400 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝛨₄.{{Efn|name=distinct rotations}}}} about a fixed point in 4-dimensional Euclidean space.{{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 [[W:Completely orthogonal|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|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}} 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), 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, similar but not identical to two simple rotations through the ''same'' angle.{{Efn|A point under isoclinic rotation traverses the diagonal{{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]].{{Efn|name=isoclinic geodesic}} 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=isoclinic rotation to non-adjacent vertices}} For example, when the unit-radius 600-cell rotates isoclinically 36 degrees in a decagon invariant plane and 36 degrees in its completely orthogonal invariant plane,{{Efn|name=non-vertex geodesic}} each vertex is displaced to another vertex <math>\sqrt{1}</math> (60°) distant, moving <math>\sqrt{1/4} {{=}} 1/2</math> (half the <math>\sqrt{1}</math> overall displacement) in four orthogonal directions.|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.{{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 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-space analogues{{Efn|name=math of dimensional analogy}} of 2-dimensional great circles in 3-space (great 1-spheres).|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true 5𝝅 circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles. These '''[[#Rotations on polygram isoclines|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 [[W:Chiral|chiral]] pairs as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{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 5𝝅 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}} 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 of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}|name=identical rotations}} The 600-cell is generated by [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]]{{Efn|name=isoclinic geodesic}} of the 24-cell by 36° = {{sfrac|𝜋|5}} (the arc of one 600-cell edge length).{{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{{Efn|name=isoclinic 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:William Kingdon Clifford|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 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 rotates sideways.{{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}} ''All'' central polygons (of every kind) rotate by the same angle (though not all do so invariantly), and are also displaced sideways by the same angle to a Clifford parallel polygon (of the same kind).|name=Clifford displacement}} ==== Twenty-five 24-cells ==== There are 25 inscribed 24-cells in the 600-cell.{{sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020}}{{Efn|The 600-cell has 7200 distinct rotational displacements, each with its invariant rotation plane. The 7200 distinct central planes can be grouped into sets of Clifford parallel invariant rotation planes of 25 distinct ''isoclinic'' rotations, and are usually given as those sets.{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2}}|name=distinct rotations}} Therefore there are also 25 inscribed snub 24-cells, 75 inscribed tesseracts and 75 inscribed 16-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}} The 8-vertex 16-cell has 4 long diameters inclined at 90° = {{sfrac|𝜋|2}} to each other, often taken as the 4 orthogonal axes or [[16-cell#Coordinates|basis]] of the coordinate system.{{Efn|name=Six orthogonal planes of the Cartesian basis}} The 24-vertex 24-cell has 12 long diameters inclined at 60° = {{sfrac|𝜋|3}} to each other: 3 disjoint sets of 4 orthogonal axes, each set comprising the diameters of one of 3 inscribed 16-cells, isoclinically rotated by {{sfrac|𝜋|3}} with respect to each other.{{Efn|The three 16-cells in the 24-cell are rotated by 60° ({{sfrac|𝜋|3}}) isoclinically with respect to each other. Because an isoclinic rotation is a rotation in two completely orthogonal planes at the same time, this means their corresponding vertices are 120° ({{sfrac|2𝜋|3}}) apart. In a unit-radius 4-polytope, vertices 120° apart are joined by a <math>\sqrt{3}</math> chord.|name=120° apart}} The 120-vertex 600-cell has 60 long diameters: ''not just'' 5 disjoint sets of 12 diameters, each comprising one of 5 inscribed 24-cells (as we might suspect by analogy), but 25 distinct but overlapping sets of 12 diameters, each comprising one of 25 inscribed 24-cells.{{Sfn|Waegell & Aravind|2009|loc=§3. The 600-cell|pp=2-5}} There ''are'' 5 disjoint 24-cells in the 600-cell, but not ''just'' 5: there are 10 different ways to partition the 600-cell into 5 disjoint 24-cells.{{Efn|name=Schoute's ten ways to get five disjoint 24-cells|[[W:Pieter Hendrik Schoute|Schoute]] was the first to state (a century ago) that there are exactly ten ways to partition the 120 vertices of the 600-cell into five disjoint 24-cells. The 25 24-cells can be placed in a 5 x 5 array such that each row and each column of the array partitions the 120 vertices of the 600-cell into five disjoint 24-cells. The rows and columns of the array are the only ten such partitions of the 600-cell.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|p=434}}}} Like the 16-cells and 8-cells inscribed in the 24-cell, the 25 24-cells inscribed in the 600-cell are mutually [[24-cell#Clifford parallel polytopes|isoclinic polytopes]]. The rotational distance between inscribed 24-cells is always {{sfrac|𝜋|5}} in each invariant plane of rotation.{{Efn|There is a single invariant plane in each simple rotation, and a completely orthogonal fixed plane. There are an infinite number of pairs of [[W:Completely orthogonal|completely orthogonal]] invariant planes in each isoclinic rotation, all rotating through the same angle;{{Efn|name=dense fabric of pole-circles}} nonetheless, not all [[#Geodesics|central planes]] are [[24-cell#Isoclinic rotations|invariant planes of rotation]]. The invariant planes of an isoclinic rotation constitute a [[#Fibrations of great circle polygons|fibration]] of the entire 4-polytope.{{Sfn|Kim|Rote|2016|loc=§8.2 Equivalence of an Invariant Family and a Hopf Bundle|pp=13-14}} In every isoclinic rotation of the 600-cell taking vertices to vertices either 12 Clifford parallel great [[#Decagons|decagons]], ''or'' 20 Clifford parallel great [[#Hexagons|hexagons]] ''or'' 30 Clifford parallel great [[#Squares|squares]] are invariant planes of rotation.|name=isoclinic invariant planes}} Five 24-cells are disjoint because they are Clifford parallel: their corresponding vertices are {{sfrac|𝜋|5}} apart on two non-intersecting Clifford parallel{{Efn|name=Clifford parallels}} decagonal great circles (as well as {{sfrac|𝜋|5}} apart on the same decagonal great circle).{{Efn|Two Clifford parallel{{Efn|name=Clifford parallels}} great decagons don't intersect, but their corresponding vertices are linked by one edge of another decagon. The two parallel decagons and the ten linking edges form a double helix ring. Three decagons can also be parallel (decagons come in parallel [[W:Hopf fibration|fiber bundles]] of 12) and three of them may form a triple helix ring. If the ring is cut and laid out flat in 3-space, it is a [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]]{{Efn|name=Boerdijk–Coxeter helix}} 30 tetrahedra{{Efn|name=tetrahedron linking 6 decagons}} long. The three Clifford parallel decagons can be seen as the {{Background color|cyan}} edges in the [[#Boerdijk–Coxeter helix rings|triple helix illustration]]. Each {{Background color|magenta}} edge is one edge of another decagon linking two parallel decagons.|name=Clifford parallel decagons}} An isoclinic rotation of decagonal planes by {{sfrac|𝜋|5}} takes each 24-cell to a disjoint 24-cell (just as an [[24-cell#Clifford parallel polytopes|isoclinic rotation of hexagonal planes]] by {{sfrac|𝜋|3}} takes each 16-cell to a disjoint 16-cell).{{Efn|name=isoclinic geodesic displaces every central polytope}} Each isoclinic rotation occurs in two chiral forms: there are 4 disjoint 24-cells to the ''left'' of each 24-cell, and another 4 disjoint 24-cells to its ''right''.{{Efn|A ''disjoint'' 24-cell reached by an isoclinic rotation is not any of the four adjacent 24-cells; the double rotation{{Efn|name=identical rotations}} takes it past (not through) the adjacent 24-cell it rotates toward,{{Efn|Five 24-cells meet at each vertex of the 600-cell,{{Efn|name=five 24-cells at each vertex of 600-cell}} so there are four different directions in which the vertices can move to rotate the 24-cell (or all the 24-cells at once in an [[24-cell#Isoclinic rotations|isoclinic rotation]]{{Efn|name=isoclinic geodesic displaces every central polytope}}) directly toward an adjacent 24-cell.|name=four directions toward another 24-cell}} and left or right to a more distant 24-cell from which it is completely disjoint.{{Efn|name=completely disjoint}} The four directions reach 8 different 24-cells{{Efn|name=disjoint from 8 and intersects 16}} because in an isoclinic rotation each vertex moves in a spiral along two completely orthogonal great circles at once. Four paths are right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, and four are 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}}|name=rotations to 8 disjoint 24-cells}} The left and right rotations reach different 24-cells; therefore each 24-cell belongs to two different sets of five disjoint 24-cells. All [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]] are isoclinic, but not all isoclinic polytopes are Clifford parallels (completely disjoint objects).{{Efn|All isoclinic ''polygons'' are Clifford parallels (completely disjoint).{{Efn||name=completely disjoint}} Polyhedra (3-polytopes) and polychora (4-polytopes) may be isoclinic and ''not'' disjoint, if all of their corresponding central polygons are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same object, shared). For example, the 24-cell, 600-cell and 120-cell contain pairs of inscribed tesseracts (8-cells) which are isoclinically rotated by {{sfrac|𝜋|3}} with respect to each other, yet are not disjoint: they share a [[16-cell#Octahedral dipyramid|16-cell]] (8 vertices, 6 great squares and 4 octahedral central hyperplanes), and some corresponding pairs of their great squares are cocellular (intersecting) rather than Clifford parallel (disjoint).|name=isoclinic and not disjoint}} Each 24-cell is isoclinic ''and'' Clifford parallel to 8 others, and isoclinic but ''not'' Clifford parallel to 16 others.{{Efn|name=disjoint from 8 and intersects 16}} With each of the 16 it shares 6 vertices: a hexagonal central plane.{{Efn|name=five 24-cells at each vertex of 600-cell}} Non-disjoint 24-cells are related by a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] by {{sfrac|𝜋|5}} in an invariant plane intersecting only two vertices of the 600-cell,{{Efn|name=digon planes}} a rotation in which the completely orthogonal [[24-cell#Simple rotations|fixed plane]] is their common hexagonal central plane. They are also related by an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in which both planes rotate by {{sfrac|𝜋|5}}.{{Efn|In the 600-cell, there is a [[24-cell#Simple rotations|simple rotation]] which will take any vertex ''directly'' to any other vertex, also moving most or all of the other vertices but leaving 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 decagon, a great hexagon, a great square or a great [[W:Digon|digon]],{{Efn|name=digon planes}} and the completely orthogonal fixed plane intersects 0 vertices (a 30-gon),{{Efn|name=non-vertex geodesic}} 2 vertices (a digon), 4 vertices (a square) or 6 vertices (a hexagon) respectively. Two ''non-disjoint'' 24-cells are related by a [[24-cell#Simple rotations|simple rotation]] through {{sfrac|𝜋|5}} of the digon central plane completely orthogonal to their common hexagonal central plane. In this simple rotation, the hexagon does not move. The two ''non-disjoint'' 24-cells are also related by an isoclinic rotation in which the shared hexagonal plane ''does'' move.{{Efn|name=rotations to 16 non-disjoint 24-cells}}|name=direct simple rotations}} There are two kinds of {{sfrac|𝜋|5}} isoclinic rotations which take each 24-cell to another 24-cell.{{Efn|Any isoclinic rotation by {{sfrac|𝜋|5}} in decagonal invariant planes{{Efn|Any isoclinic rotation in a decagonal invariant plane is an isoclinic rotation in 24 invariant planes: 12 Clifford parallel decagonal planes,{{Efn|name=isoclinic invariant planes}} and the 12 Clifford parallel 30-gon planes completely orthogonal to each of those decagonal planes.{{Efn|name=non-vertex geodesic}} As the invariant planes rotate in two completely orthogonal directions at once,{{Efn|name=helical geodesic}} all points in the planes move with them (stay in their planes and rotate with them), describing helical isoclines{{Efn|name=isoclinic geodesic}} through 4-space. Note however that in a ''discrete'' decagonal fibration of the 600-cell (where 120 vertices are the only points considered), the 12 30-gon planes contain ''no'' points.}} takes ''every'' [[#Geodesics|central polygon]], [[#Clifford parallel cell rings|geodesic cell ring]] or inscribed 4-polytope{{Efn|name=4-polytopes inscribed in the 600-cell}} in the 600-cell to a [[24-cell#Clifford parallel polytopes|Clifford parallel polytope]] {{sfrac|𝜋|5}} away.|name=isoclinic geodesic displaces every central polytope}} ''Disjoint'' 24-cells are related by a {{sfrac|𝜋|5}} isoclinic rotation of an entire [[#Decagons|fibration of 12 Clifford parallel ''decagonal'' invariant planes]]. (There are 6 such sets of fibers, and a right or a left isoclinic rotation possible with each set, so there are 12 such distinct rotations.){{Efn|name=rotations to 8 disjoint 24-cells}} ''Non-disjoint'' 24-cells are related by a {{sfrac|𝜋|5}} isoclinic rotation of an entire [[#Hexagons|fibration of 20 Clifford parallel ''hexagonal'' invariant planes]].{{Efn|Notice the apparent incongruity of rotating ''hexagons'' by {{sfrac|𝜋|5}}, since only their opposite vertices are an integral multiple of {{sfrac|𝜋|5}} apart. However, [[#Icosahedra|recall]] that 600-cell vertices which are one hexagon edge apart are exactly two decagon edges and two tetrahedral cells (one triangular dipyramid) apart. The hexagons have their own [[#Hexagons|10 discrete fibrations]] and [[#Clifford parallel cell rings|cell rings]], not Clifford parallel to the [[#Decagons|decagonal fibrations]] but also by fives{{Efn|name=24-cells bound by pentagonal fibers}} in that five 24-cells meet at each vertex, each pair sharing a hexagon.{{Efn|name=five 24-cells at each vertex of 600-cell}} Each hexagon rotates ''non-invariantly'' by {{sfrac|𝜋|5}} in a [[#Hexagons and hexagrams|hexagonal isoclinic rotation]] between ''non-disjoint'' 24-cells.{{Efn|name=rotations to 16 non-disjoint 24-cells}} Conversely, in all [[#Decagons and pentadecagrams|{{sfrac|𝜋|5}} isoclinic rotations in ''decagonal'' invariant planes]], all the vertices travel along isoclines{{Efn|name=isoclinic geodesic}} which follow the edges of ''hexagons''.|name=apparent incongruity}} (There are 10 such sets of fibers, so there are 20 such distinct rotations.){{Efn|At each vertex, a 600-cell has four adjacent (non-disjoint){{Efn||name=completely disjoint}} 24-cells that can each be reached by a simple rotation in that direction.{{Efn|name=four directions toward another 24-cell}} Each 24-cell has 4 great hexagons crossing at each of its vertices, one of which it shares with each of the adjacent 24-cells; in a simple rotation that hexagonal plane remains fixed (its vertices do not move) as the 600-cell rotates ''around'' the common hexagonal plane. The 24-cell has 16 great hexagons altogether, so it is adjacent (non-disjoint) to 16 other 24-cells.{{Efn|name=disjoint from 8 and intersects 16}} In addition to being reachable by a simple rotation, each of the 16 can also be reached by an isoclinic rotation in which the shared hexagonal plane is ''not'' fixed: it rotates (non-invariantly) through {{sfrac|𝜋|5}}. The double rotation reaches an adjacent 24-cell ''directly'' as if indirectly by two successive simple rotations:{{Efn|name=double rotation}} first to one of the ''other'' adjacent 24-cells, and then to the destination 24-cell (adjacent to both of them).|name=rotations to 16 non-disjoint 24-cells}} On the other hand, each of the 10 sets of five ''disjoint'' 24-cells is Clifford parallel because its corresponding great ''hexagons'' are Clifford parallel. (24-cells do not have great decagons.) The 16 great hexagons in each 24-cell can be divided into 4 sets of 4 non-intersecting Clifford parallel [[24-cell#Geodesics|geodesics]], each set of which covers all 24 vertices of the 24-cell. The 200 great hexagons in the 600-cell can be divided into 10 sets of 20 non-intersecting Clifford parallel [[#Geodesics|geodesics]], each set of which covers all 120 vertices and constitutes a discrete [[#Hexagons|hexagonal fibration]]. Each of the 10 sets of 20 disjoint hexagons can be divided into five sets of 4 disjoint hexagons, each set of 4 covering a disjoint 24-cell. Similarly, the corresponding great ''squares'' of disjoint 24-cells are Clifford parallel. ==== Rotations on polygram isoclines ==== The regular convex 4-polytopes each have their characteristic kind of right (and left) [[W:Isoclinic rotation|isoclinic rotation]], corresponding to their characteristic kind of discrete [[W:Hopf fibration|Hopf fibration]] of great circles.{{Efn|The poles of the invariant axis of a rotating 2-sphere are dimensionally analogous to the pair of invariant planes of a rotating 3-sphere. The poles of the rotating 2-sphere are dimensionally analogous to linked great circles on the 3-sphere. By dimensional analogy, each 1D point in 3D lifts to a 2D line in 4D, in this case a circle.{{Efn|name=Hopf fibration base}} The two antipodal rotation poles lift to a pair of circular Hopf fibers which are not merely Clifford parallel and interlinked,{{Efn|name=Clifford parallels}} but also [[W:Completely orthogonal|completely orthogonal]]. ''The invariant great circles of the 4D rotation are its poles.'' In the case of an isoclinic rotation, there is not merely one such pair of 2D poles (completely orthogonal Hopf great circle fibers), there are many such pairs: a finite number of circle-pairs if the 3-sphere fibration is discrete (e.g. a regular polytope with a finite number of vertices), or else an infinite number of orthogonal circle-pairs, entirely filling the 3-sphere. Every point in the curved 3-space of the 3-sphere lies on ''one'' such circle (never on two, since the completely orthogonal circles, like all the Clifford parallel Hopf great circle fibers, do not intersect). Where a 2D rotation has one pole, and a 3D rotation of a 2-sphere has 2 poles, ''an isoclinic 4D rotation of a 3-sphere has nothing but poles'', an infinite number of them. In a discrete 4-polytope, all the Clifford parallel invariant great polygons of the rotation are poles, and they fill the 4-polytope, passing through every vertex just once. ''In one full revolution of such a rotation, every point in the space loops exactly once through its pole-circle.''{{Efn|Consider the statement: ''In one full revolution of an isoclinic rotation, every point in the space loops exactly once through its great circle Hopf fiber.'' It can be found in the literature, expressed in the mathematical language of the Hopf fibration,{{Sfn|Kim|Rote|2016|loc= 8 The Construction of Hopf Fibrations|pp=12-16|ps=; see Theorem 13.}} but as a plain language statement of Euclidean geometry, how exactly should we visualize it? It paints a clear picture of all the great circles of a Hopf fibration rotating as rigid wheels, in parallel. That is a correct visualization, except for the fact that points moving under isoclinic rotation traverse an invariant great circle only in the sense that they stay on that circle as the whole circle itself is tilting sideways, rotating in parallel with the completely orthogonal great circle.{{Efn|name=isoclinic geodesic}} With respect to the stationary reference frame, the points move diagonally on a helical isocline, they do not move on a planar great circle.{{Efn|name=helical geodesic}} Each helical isocline is itself a kind of circle, but it is not a planar great circle of the [[W:Hopf fibration|Hopf fibration]]: it is a special kind of geodesic circle whose circumference is greater than 2𝝅''r'', and it is not pictured explicitly at all by the plain statement we are trying to visualize.{{Efn|name=isocline circumference.}} We cannot easily visualize this statement about the Hopf great circles in a stationary reference frame. The statement does ''not'' simply mean that in an isoclinic rotation every point on a stationary Hopf great circle loops through its stationary great circle. Rather, it means that every point on every Hopf great circle loops through its great circle ''as every great circle itself is moving orthogonally'', flipping like a coin in the plane completely orthogonal to its own plane (at any instant, because of course the completely orthogonal plane is moving too). This simultaneous ''twisting'' rotation in two completely orthogonal planes is a double rotation; if the angle of rotation in the two completely orthogonal planes is exactly the same, it is isoclinic. An isoclinic rotation takes each rigid planar Hopf great circle to the stationary position of another Hopf great circle, while simultaneously each Hopf great circle also rotates like a wheel. This fibration of doubly rotating rigid wheels is undoubtably hard to visualize. In any graphical animation (whether actually rendered or merely imagined) it will be difficult to track the motions of the different rotating wheels, because Clifford parallel circles are not parallel in the ordinary sense, and every great circle is moving in a different direction at any one instant. There is one more way in which this simple statement belies the full complexity of the isoclinic motion. While it is true that every point loops through its Hopf great circle exactly once ''in a full isoclinic revolution, every vertex moves more than 360 degrees,'' as measured in the stationary reference frame. In any distinct isoclinic rotation, all the vertices move the same angular distance in the stationary reference frame in one full revolution, but each distinct pair of left-right isoclinic rotations corresponds to a unique Hopf fibration,{{Sfn|Kim|Rote|2016|loc=§8.2 Equivalence of an Invariant Family and a Hopf Bundle|pp=13-14}} and the characteristic distance moved is different for each kind of Hopf fibration. For example, in the [[24-cell#Isoclinic rotations|isoclinic rotation of a great hexagon fibration of the 24-cell]], each vertex moves 720 degrees in the stationary reference frame (2 times the distance it moves within its moving Hopf great circle);{{Efn|name=4𝝅 rotation}} but in the [[#Decagons and pentadecagrams|isoclinic rotation of a great decagon fibration of the 600-cell]], each vertex moves 900 degrees in the stationary reference frame (2.5 times its great circle distance).}} The circles are arranged with a surprising symmetry, so that ''each pole-circle links with every other pole-circle'', like a maximally dense fabric of 4D [[W:Chain mail|chain mail]] in which all the circles are linked through each other, but no two circles ever intersect.|name=dense fabric of pole-circles}} For example, the 600-cell can be fibrated six different ways into a set of Clifford parallel [[#Decagons|great decagons]], so the 600-cell has six distinct right (and left) isoclinic rotations in which those great decagon planes are [[24-cell#Isoclinic rotations|invariant planes of rotation]]. We say these isoclinic rotations are ''characteristic'' of the 600-cell because the 600-cell's edges lie in their invariant planes. These rotations only emerge in the 600-cell, although they are also found in larger regular polytopes (the 120-cell) which contain inscribed instances of the 600-cell. Just as the [[#Geodesics|geodesic]] ''polygons'' (decagons or hexagons or squares) in the 600-cell's central planes form [[#Fibrations of great circle polygons|fiber bundles of Clifford parallel ''great circles'']], the corresponding geodesic [[W:Skew polygon|skew]] ''[[W:Polygram (geometry)|polygrams]]'' (which trace the paths on the [[W:Clifford torus|Clifford torus]] of vertices under isoclinic rotation){{Sfn|Perez-Gracia & Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} form [[W:Fiber bundle|fiber bundle]]s of Clifford parallel ''isoclines'': helical circles which wind through all four dimensions.{{Efn|name=isoclinic geodesic}} Since isoclinic rotations are [[W:Chiral|chiral]], occurring in left-handed and right-handed forms, each polygon fiber bundle has corresponding left and right polygram fiber bundles.{{Sfn|Kim|Rote|2016|p=12-16|loc=§8 The Construction of Hopf Fibrations; see §8.3}} All the fiber bundles are aspects of the same discrete [[W:Hopf fibration|Hopf fibration]], because the fibration is the various expressions of the same distinct left-right pair of isoclinic rotations. Cell rings are another expression of the Hopf fibration. Each discrete fibration has a set of cell-disjoint cell rings that tesselates the 4-polytope.{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. In isoclinic rotations, one set of cell rings (one fibration) is distinguished as the unique container of that distinct left-right pair of rotations and its isoclines.|name=fibrations are distinguished only by rotations}} The isoclines in each chiral bundle spiral around each other: they are axial geodesics of the rings of face-bonded cells. The [[#Clifford parallel cell rings|Clifford parallel cell rings]] of the fibration nest into each other, pass through each other without intersecting in any cells, and exactly fill the 600-cell with their disjoint cell sets. Isoclinic rotations rotate a rigid object's vertices along parallel paths, each vertex circling within two orthogonal moving great circles, the way a [[W:Loom|loom]] weaves a piece of fabric from two orthogonal sets of parallel fibers. A bundle of Clifford parallel great circle polygons and a corresponding bundle of Clifford parallel skew polygram isoclines are the [[W:Warp and woof|warp and woof]] of the same distinct left or right isoclinic rotation, which takes Clifford parallel great circle polygons to each other, flipping them like coins and rotating them through a Clifford parallel set of central planes. Meanwhile, because the polygons are also rotating individually like wheels, vertices are displaced along helical Clifford parallel isoclines (the chords of which form the skew polygram), through vertices which lie in successive Clifford parallel polygons.{{Efn|name=helical geodesic}} In the 600-cell, each family of isoclinic skew polygrams (moving vertex paths in the decagon {10}, hexagon {6}, or square {4} great polygon rotations) can be divided into bundles of non-intersecting Clifford parallel polygram isoclines.{{Sfn|Perez-Gracia & Thomas|2017|loc=§1. Introduction|ps=; "This article [will] derive a spectral decomposition of isoclinic rotations and explicit formulas in matrix and Clifford algebra for the computation of Cayley's [isoclinic] factorization."{{Efn|name=double rotation}}}} The isocline bundles occur in pairs of ''left'' and ''right'' chirality; the isoclines in each rotation act as [[W:Chiral|chiral]] objects, as does each distinct isoclinic rotation itself.{{Efn|The fibration's [[#Clifford parallel cell rings|Clifford parallel cell rings]] may or may not be [[W:Chiral|chiral]] objects, depending upon whether the 4-polytope's cells have opposing faces or not. The characteristic cell rings of the 16-cell and 600-cell (with tetrahedral cells) are chiral: they twist either clockwise or counterclockwise. Isoclines acting with either left or right chirality (not both) run through cell rings of this kind, though each fibration contains both left and right cell rings.{{Efn|Each isocline has no inherent chirality but can act as either a left or right isocline; it is shared by a distinct left rotation and a distinct right rotation of different fibrations.|name=isoclines have no inherent chirality}} The characteristic cell rings of the 8-cell tesseract, 24-cell and 120-cell (with cubical, octahedral, and dodecahedral cells respectively) are directly congruent, not chiral: there is only one kind of characteristic cell ring in each of these 4-polytopes, and it is not twisted (it has no [[W:Torsion of a curve|torsion]]). Pairs of left-handed and right-handed isoclines run through cell rings of this kind. The left and right isoclines are enantiomorphously congruent (mirror images) of each other. Note that all these 4-polytopes (except the 16-cell) contain fibrations of their inscribed [[#Geometry|predecessors]]' characteristic cell rings in addition to their own characteristic fibrations, so the 600-cell contains both chiral and directly congruent cell rings.|name=directly congruent versus twisted cell rings}} Each fibration contains an equal number of left and right isoclines, in two disjoint bundles, which trace the paths of the 600-cell's vertices during the fibration's left or right isoclinic rotation respectively. Each left or right fiber bundle of isoclines ''by itself'' constitutes a discrete Hopf fibration which fills the entire 600-cell, visiting all 120 vertices just once. It is a ''different bundle of fibers'' than the bundle of Clifford parallel polygon great circles, but the two fiber bundles describe the ''same discrete fibration'' because they enumerate those 120 vertices together in the same distinct right (or left) isoclinic rotation, by their intersection as a fabric of cross-woven parallel fibers. Each isoclinic rotation involves pairs of completely orthogonal invariant central planes of rotation, which both rotate through the same angle. There are two ways they can do this: by both rotating in the "same" direction, or by rotating 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). The right polygram and right isoclinic rotation conventionally correspond to invariant pairs rotating in the same direction; the left polygram and left isoclinic rotation correspond to pairs rotating in opposite directions.{{Sfn|Perez-Gracia & Thomas|2017|loc=§5. A useful mapping|pp=12−13}} Left and right isoclines are different paths that go to different places. In addition, each distinct isoclinic rotation (left or right) can be performed in a positive or negative direction along the circular parallel fibers. A fiber bundle of Clifford parallel isoclines is the set of helical vertex circles described by a distinct left or right isoclinic rotation. Each moving vertex travels along an isocline contained within a (moving) cell ring. While the left and right isoclinic rotations each double-rotate the same set of Clifford parallel invariant [[24-cell#Planes of rotation|planes of rotation]], they step through different sets of great circle polygons because left and right isoclinic rotations hit alternate vertices of the great circle {2p} polygon (where p is a prime ≤ 5).{{Efn|name={2p} isoclinic rotations}} The left and right rotation share the same Hopf bundle of {2p} polygon fibers, which is ''both'' a left and right bundle, but they have different bundles of {p} polygons{{Sfn|Kim|Rote|2016|p=14|loc=§8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle}} because the discrete fibers are opposing left and right {p} polygons inscribed in the {2p} polygon.{{Efn|Each discrete fibration of a regular convex 4-polytope is characterized by a unique left-right pair of isoclinic rotations and a unique bundle of great circle {2p} polygons (0 ≤ p ≤ 5) in the invariant planes of that pair of rotations. Each distinct rotation has a unique bundle of left (or right) {p} polygons inscribed in the {2p} polygons, and a unique bundle of skew {2p} polygrams which are its discrete left (or right) isoclines. The {p} polygons weave the {2p} polygrams into a bundle, and vice versa.}} A [[24-cell#Simple rotations|simple rotation]] is direct and local, taking some vertices to adjacent vertices along great circles, and some central planes to other central planes within the same hyperplane. (The 600-cell has four orthogonal [[#Polyhedral sections|central hyperplanes]], each of which is an icosidodecahedron.) In a simple rotation, there is just a single pair of completely orthogonal invariant central planes of rotation; it does not constitute a fibration. An [[24-cell#Isoclinic rotations|isoclinic rotation]] is diagonal and global, taking ''all'' the vertices to ''non-adjacent'' vertices{{Efn|Isoclinic rotations take each vertex to a non-adjacent vertex. In the characteristic isoclinic rotations of the 5-cell, 16-cell, 24-cell and 600-cell, the non-adjacent vertex is the opposite vertex of a neighboring cell. In the 8-cell it is three zig-zag edge-lengths away in the same cell: the opposite vertex of a cube. In the 120-cell it is four zig-zag edges away in the same cell: the opposite vertex of a dodecahedron. |name=isoclinic rotation to non-adjacent vertices}} along diagonal isoclines, and ''all'' the central plane polygons to Clifford parallel polygons (of the same kind). A left-right pair of isoclinic rotations constitutes a discrete fibration. All the Clifford parallel central planes of the fibration are invariant planes of rotation, separated by ''two'' equal angles and lying in different hyperplanes.{{Efn|name=two angles between central planes}} The diagonal isocline{{Efn|name=isoclinic 4-dimensional diagonal}} is a shorter route between the non-adjacent vertices than the multiple simple routes between them available along edges: it is the ''shortest route'' on the 3-sphere, the [[W:Geodesic|geodesic]]. ==== Decagons and pentadecagrams ==== The [[#Fibrations of great circle polygons|fibrations of the 600-cell]] include 6 [[#Decagons|fibrations of its 72 great decagons]]: 6 fiber bundles of 12 great decagons,{{Efn|name=Clifford parallel decagons}} each delineating [[#Boerdijk–Coxeter helix rings|20 chiral cell rings]] of 30 tetrahedral cells each,{{Efn|name=Boerdijk–Coxeter helix}} with three great decagons bounding each cell ring, and five cell rings nesting together around each decagon. The 12 Clifford parallel decagons in each bundle are completely disjoint. Adjacent parallel decagons are spanned by edges of other great decagons.{{Efn|name=equi-isoclinic decagons}} Each fibration corresponds to a distinct left (and right) isoclinic rotation of the 600-cell in 12 great decagon invariant planes on 5𝝅 isoclines. The bundle of 12 Clifford parallel decagon fibers is divided into a bundle of 12 left pentagon fibers and a bundle of 12 right pentagon fibers, with each left-right pair of pentagons inscribed in a decagon.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16}} 12 great polygons comprise a fiber bundle covering all 120 vertices in a discrete [[W:Hopf fibration|Hopf fibration]]. There are 20 cell-disjoint 30-cell rings in the fibration, but only 4 completely disjoint 30-cell rings.{{Efn|name=completely disjoint}} The 600-cell has six such discrete [[#Decagons|decagonal fibrations]], and each is the domain (container) of a unique left-right pair of isoclinic rotations (left and right fiber bundles of 12 great pentagons).{{Efn|There are six congruent decagonal fibrations of the 600-cell. Choosing one decagonal fibration means choosing a bundle of 12 directly congruent Clifford parallel decagonal great circles, and a cell-disjoint set of 20 directly congruent 30-cell rings which tesselate the 600-cell. The fibration and its great circles are not chiral, but it has distinct left and right expressions in a left-right pair of isoclinic rotations. In the right (left) rotation the vertices move along a right (left) Hopf fiber bundle of Clifford parallel isoclines and intersect a right (left) Hopf fiber bundle of Clifford parallel great pentagons. The 30-cell rings are the only chiral objects, other than the ''bundles'' of isoclines or pentagons.{{Sfn|Kim|Rote|2016|p=14|loc=§8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle}} A right (left) pentagon bundle contains 12 great pentagons, inscribed in the 12 Clifford parallel great [[#Decagons|decagons]]. A right (left) isocline bundle contains 20 Clifford parallel pentadecagrams, one in each 30-cell ring.|name=decagonal fibration of chiral bundles}} Each great decagon belongs to just one fibration,{{Sfn|Kim|Rote|2016|p=14|loc=§8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle}} but each 30-cell ring belongs to 5 of the six fibrations (and is completely disjoint from 1 other fibration). The 600-cell contains 72 great decagons, divided among six fibrations, each of which is a set of 20 cell-disjoint 30-cell rings (4 completely disjoint 30-cell rings), but the 600-cell has only 20 distinct 30-cell rings altogether. Each 30-cell ring contains 3 of the 12 Clifford parallel decagons in each of 5 fibrations, and 30 of the 120 vertices. In these ''decagonal'' isoclinic rotations, vertices travel along isoclines which follow the edges of ''hexagons'',{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} advancing a pythagorean distance of one hexagon edge in each double 36°×36° rotational unit.{{Efn||name=apparent incongruity}} In an isoclinic rotation, each successive hexagon edge travelled lies in a different great hexagon, so the isocline describes a skew polygram, not a polygon. In a 60°×60° isoclinic rotation (as in the [[24-cell#Isoclinic rotations|24-cell's characteristic hexagonal rotation]], and [[#Hexagons and hexagrams|below in the ''hexagonal'' rotations of the 600-cell]]) this polygram is a <s>[[W:Hexagram|hexagram]]</s>: the isoclinic rotation follows a 6-edge circular path, just as a simple hexagonal rotation does, although it takes ''two'' revolutions to enumerate all the vertices in it, because the isocline is a double loop through every other vertex, and its chords are <math>\sqrt{3}</math> chords of the hexagon instead of <math>\sqrt{1}</math> hexagon edges.{{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. 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 <math>\sqrt{3}</math> longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|name=4𝝅 rotation}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) although all such circles of the same circumference are directly or enantiomorphously congruent.{{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.|name=one true 4𝝅 circle}} But in the 600-cell's 36°×36° characteristic ''decagonal'' rotation, successive great hexagons are closer together and more numerous, and the isocline polygram formed by their 15 hexagon ''edges'' is a pentadecagram (15-gram).{{Efn|name=one true 5𝝅 circle}} It is not only not the same period as the hexagon or the simple decagonal rotation, it is not even an integer multiple of the period of the hexagon, or the decagon, or either's simple rotation. Only the compound {30/4}=2{15/2} triacontagram (30-gram), which is two 15-grams rotating in parallel (a black and a white), is a multiple of them all, and so constitutes the rotational unit of the decagonal isoclinic rotation.{{Efn|The analogous relationships among three kinds of {2p} isoclinic rotations, in [[#Fibrations of great circle polygons|Clifford parallel bundles of {4}, {6} or {10} great polygon invariant planes]] respectively, are at the heart of the complex nested relationship among the [[#Geometry|regular convex 4-polytopes]].{{Efn|name=4-polytopes ordered by size and complexity}} In the <math>\sqrt{1}</math> [[#Hexagons and hexagrams|hexagon {6} rotations characteristic of the 24-cell]], the [[#Rotations on polygram isoclines|isocline chords (polygram edges)]] are simply <math>\sqrt{3}</math> chords of the great hexagon, so the [[24-cell#Simple rotations|simple {6} hexagon rotation]] and the [[24-cell#Isoclinic rotations|isoclinic <s>{6/2} hexagram</s> rotation]] both rotate circles of 6 vertices. The <s>hexagram</s> isocline, a special kind of great circle, has a circumference of 4𝝅 compared to the hexagon 2𝝅 great circle.{{Efn|name=one true 4𝝅 circle}} The invariant central plane completely orthogonal to each {6} great hexagon is a {2} great digon,{{Efn|name=digon planes}} so an [[#Hexagons and hexagrams|isoclinic {6} rotation of <s>hexagrams</s>]] is also a {2} rotation of ''axes''.{{Efn|name=direct simple rotations}} In the <math>\sqrt{2}</math> [[#Squares and octagrams|square {4} rotations characteristic of the 16-cell]], the isocline polygram is an [[16-cell#Helical construction|octagram]], and the isocline's chords are its <math>\sqrt{2}</math> edges and its <math>\sqrt{4}</math> diameters, so the isocline is a circle of circumference 4𝝅. In an isoclinic rotation, the eight vertices of the {8/3} octagram change places, each making one complete revolution through 720° as the isocline [[W:Winding number|winds]] ''three'' times around the 3-sphere. The invariant central plane completely orthogonal to each {4} great square is another {4} great square <math>\sqrt{4}</math> distant, so a ''right'' {4} rotation of squares is also a ''left'' {4} rotation of squares. The 16-cell's [[W:Dural polytope|dual polytope]] the [[W:8-cell|8-cell tesseract]] inherits the same simple {4} and isoclinic {8/3} rotations, but its characteristic isoclinic rotation takes place in completely orthogonal invariant planes which contain a {4} great ''rectangle'' or a {2} great digon (from its successor the 24-cell). In the 8-cell this is a rotation of <math>\sqrt{1}</math> × <math>\sqrt{3}</math> great rectangles, and also a rotation of <math>\sqrt{4}</math> axes, but it is the same isoclinic rotation as the 24-cell's characteristic rotation of {6} great hexagons (in which the great rectangles are inscribed), as a consequence of the unique circumstance that [[24-cell#Geometry|the 8-cell and 24-cell have the same edge length]]. In the <math>\phi^{-1}</math> [[#Decagons|decagon {10} rotations characteristic of the 600-cell]], the isocline ''chords'' are <math>\sqrt{1}</math> hexagon ''edges'', the isocline polygram is a pentadecagram, and the isocline has a circumference of 5𝝅.{{Efn|name=one true 5𝝅 circle}} The [[#Decagons and pentadecagrams|isoclinic {15/2} pentadecagram rotation]] rotates a circle of {15} vertices in the same time as the simple decagon rotation of {10} vertices. The invariant central plane completely orthogonal to each {10} great decagon is a {0} great 0-gon,{{Efn|name=0-gon central planes}} so a {10} rotation of decagons is also a {0} rotation of planes containing no vertices. The 600-cell's dual polytope the [[120-cell#Chords|120-cell inherits]] the same simple {10} and isoclinic {15/2} rotations, but its characteristic isoclinic rotation takes place in completely orthogonal invariant planes which contain {2} great [[W:Digon|digon]]s (from its successor the 5-cell).{{Efn|120 regular 5-cells are inscribed in the 120-cell. The [[5-cell#Geodesics and rotations|5-cell has digon central planes]], no two of which are orthogonal. It has 10 digon central planes, where each vertex pair is an edge, not an axis. The 5-cell is self-dual, so by reciprocation the 120-cell can be inscribed in a regular 5-cell of larger radius. Therefore the finite sequence of 6 regular 4-polytopes{{Efn|name=4-polytopes ordered by size and complexity}} nested like [[W:Russian dolls|Russian dolls]] can also be seen as an infinite sequence.|name=infinite inscribed sequence}} This is a rotation of [[120-cell#Chords|irregular great hexagons]] {6} of two alternating edge lengths (analogous to the tesseract's great rectangles), where the two different-length edges are three 120-cell edges and three [[5-cell#Boerdijk–Coxeter helix|5-cell edges]].|name={2p} isoclinic rotations}} In the 30-cell ring, the non-adjacent vertices linked by isoclinic rotations are two edge-lengths apart, with three other vertices of the ring lying between them.{{Efn|In the 30-cell ring, each isocline runs from a vertex to a non-adjacent vertex in the third shell of vertices surrounding it. Three other vertices between these two vertices can be seen in the 30-cell ring, two adjacent in the first [[#Polyhedral sections|surrounding shell]], and one in the second surrounding shell.}} The two non-adjacent vertices are linked by a <math>\sqrt{1}</math> chord of the isocline which is a great hexagon edge (a 24-cell edge). The <math>\sqrt{1}</math> chords of the 30-cell ring (without the <math>\phi^{-1}</math> 600-cell edges) form a skew [[W:Triacontagram|triacontagram]]_{{30/4}=2{15/2}} which contains 2 disjoint {15/2} Möbius double loops, a left-right pair of [[W:Pentadecagram|pentadecagram]]₂ isoclines. Each left (or right) bundle of 12 pentagon fibers is crossed by a left (or right) bundle of 8 Clifford parallel pentadecagram fibers. Each distinct 30-cell ring has 2 double-loop pentadecagram isoclines running through its even or odd (black or white) vertices respectively.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 600 cells (and the 120 vertices) of the 600-cell into two disjoint subsets of 300 cells (and 60 vertices), even and odd (or black and white), which shift places among themselves on black or white isoclines, in a manner dimensionally analogous{{Efn|name=math of dimensional analogy}} to the way the [[W:Bishop (chess)|bishops]]' diagonal moves restrict them to the white or the black squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 600 cells (and 120 vertices) into black and white in the same way.{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}} 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.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} ''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]], '''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. (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.) Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''isoclinic rotations''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''pairs of Clifford parallel great polygon planes''',{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} '''[[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 [[#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 [[24-cell#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 part of left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves.{{Efn|name=isoclines have no inherent chirality}} 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}} The black and white subsets are also divided among black and white invariant great circle polygons of the isoclinic rotation. In a discrete rotation (as of a 4-polytope with a finite number of vertices) the black and white subsets correspond to sets of inscribed great polygons {p} in invariant great circle polygons {2p}. For example, in the 600-cell a black and a white great pentagon {5} are inscribed in an invariant great decagon {10} of the characteristic decagonal isoclinic rotation. Importantly, a black and white pair of polygons {p} of the same distinct isoclinic rotation are never inscribed in the same {2p} polygon; there is always a black and a white {p} polygon inscribed in each invariant {2p} polygon, but they belong to distinct isoclinic rotations: the left and right rotation of the same fibraton, which share the same set of invariant planes. Black (white) isoclines intersect only black (white) great {p} polygons, so each vertex is either black or white.|name=black and white}} The pentadecagram helices have no inherent chirality, but each acts as either a left or right isocline in any distinct isoclinic rotation.{{Efn|name=isoclines have no inherent chirality}} The 2 pentadecagram fibers belong to the left and right fiber bundles of 5 different fibrations. At each vertex, there are six great decagons and six pentadecagram isoclines (six black or six white) that cross at the vertex.{{Efn|Each axis of the 600-cell touches a left isocline of each fibration at one end and a right isocline of the fibration at the other end. Each 30-cell ring's axial isocline passes through only one of the two antipodal vertices of each of the 30 (of 60) 600-cell axes that the isocline's 30-vertex, 30-cell ring touches (at only one end).}} Eight pentadecagram isoclines (four black and four white) comprise a unique right (or left) fiber bundle of isoclines covering all 120 vertices in the distinct right (or left) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of 12 pentagons and 8 pentadecagram isoclines. There are only 20 distinct black isoclines and 20 distinct white isoclines in the 600-cell. Each distinct isocline belongs to 5 fiber bundles. {| class="wikitable" width="450" !colspan=4|Three sets of 30-cell ring chords from the same [[W:Orthogonal projection|orthogonal projection]] viewpoint |- ![[W:Pentadecagon#Pentadecagram|Pentadecagram {15/2}]] ![[W:Triacontagon#Triacontagram|Triacontagram {30/4}=2{15/2}]] ![[W:Triacontagon#Triacontagram|Triacontagram {30/6}=6{5}]] |- |colspan=2 align=center|All edges are [[W:Pentadecagram|pentadecagram]] isocline chords of length <math>\sqrt{1}</math>, which are also [[24-cell#Great hexagons|great hexagon]] edges of 24-cells inscribed in the 600-cell. |colspan=1 align=center|Only [[#Golden chords|great pentagon edges]] of length <math>\sqrt{3 - \phi} \approx 1.176</math>. |- |[[File:Regular_star_polygon_15-2.svg|200px]] |[[File:Regular_star_figure_2(15,2).svg|200px]] |[[File:Regular_star_figure_6(5,1).svg|200px]] |- |valign=top|A single black (or white) isocline is a Möbius double loop skew pentadecagram {15/2} of circumference 5𝝅.{{Efn|name=one true 5𝝅 circle}} The <math>\sqrt{1}</math> chords are 24-cell edges (hexagon edges) from different inscribed 24-cells. These chords are invisible (not shown) in the [[#Boerdijk–Coxeter helix rings|30-cell ring illustration]], where they join opposite vertices of two face-bonded tetrahedral cells that are two orange edges apart or two yellow edges apart. |valign=top|The 30-cell ring as a skew compound of two disjoint pentadecagram {15/2} isoclines (a black-white pair, shown here as orange-yellow).{{Efn|name=black and white}} The <math>\sqrt{1}</math> chords of the isoclines link every 4th vertex of the 30-cell ring in a straight chord under two orange edges or two yellow edges. The doubly-curved isocline is the geodesic (shortest path) between those vertices; they are also two edges apart by three different angled paths along the edges of the face-bonded tetrahedra. |valign=top|Each pentadecagram isocline (at left) intersects all six great pentagons (above) in two or three vertices. The pentagons lie on flat 2𝝅 great circles in the decagon invariant planes of rotation. The pentadecagrams are ''not'' flat: they are helical 5𝝅 isocline circles whose 15 chords lie in successive great ''hexagon'' planes inclined at 𝝅/5 = 36° to each other. The isocline circle is said to be twisting either left or right with the rotation, but all such pentadecagrams are directly or enantiomorphously congruent. |- |colspan=3|No 600-cell edges appear in these illustrations, only [[#Hopf spherical coordinates|invisible interior chords of the 600-cell]]. In this article, they should all properly be drawn as dashed lines. |} Two 15-gram double-loop isoclines are axial to each 30-cell ring. The 30-cell rings are chiral; each fibration contains 10 right (clockwise-spiraling) rings and 10 left (counterclockwise spiraling) rings. The right and left isoclines in each 3-cell ring are enantiomorphously congruent (mirror images).{{Efn|The chord-path of an isocline 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 [[#Boerdijk–Coxeter helix 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 two 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,{{Efn|An isoclinic rotation by 36° is two simple rotations by 36° at the same time.{{Efn|The composition of two simple 36° rotations in a pair of completely orthogonal invariant planes is a 36° isoclinic rotation in ''twelve'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of twelve simple rotations, and all 120 vertices rotate in invariant decagon planes, versus just 10 vertices in a simple rotation.}} It moves all the vertices 60° at the same time, in various different directions. Fifteen successive diagonal rotational increments, of 36°×36° each, move each vertex 900° through 15 vertices on a Möbius double loop of circumference 5𝝅 called an ''isocline'', winding around the 600-cell and back to its point of origin, in one-and-one-half the time (15 rotational increments) that it would take a simple rotation to take the vertex once around the 600-cell on an ordinary {10} great circle (in 10 rotational increments).{{Efn|name=double threaded}} The helical double loop 5𝝅 isocline is just a special kind of ''single'' full circle, of 1.5 the period (15 chords instead of 10) as the simple great circle. The isocline is ''one'' true circle, as perfectly round and geodesic as the simple great circle, even through its chords are φ longer, its circumference is 5𝝅 instead of 2𝝅, it circles through four dimensions instead of two, and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly or enantiomorphously congruent.{{Efn|name=isocline circumference.}} 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=isoclinic geodesic}}|name=one true 5𝝅 circle}} with no net [[W:Torsion of a curve|torsion]].{{Efn|name=Sadoc frustration}} An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).|name=Clifford polygon}} Each acts as a left (or right) isocline a left (or right) rotation, but has no inherent chirality.{{Efn|name=isoclines have no inherent chirality}} The fibration's 20 left and 20 right 15-grams altogether contain 120 disjoint open pentagrams (60 left and 60 right), the open ends of which are adjacent 600-cell vertices (one <math>\phi^{-1}</math> edge-length apart). The 30 chords joining the isocline's 30 vertices are <math>\sqrt{1}</math> hexagon edges (24-cell edges), connecting 600-cell vertices which are ''two'' 600-cell <math>\phi^{-1}</math> edges apart on a decagon great circle. {{Efn|Because the 600-cell's [[#Decagons and pentadecagrams|helical pentadecagram₂ geodesic]] is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself after each revolution, without ever reversing its direction of rotation (left or right). The 30-vertex isoclinic path follows a Möbius double loop, forming a single continuous 15-vertex loop traversed in two revolutions. The Möbius helix is a geodesic "straight line" or ''[[#Decagons and pentadecagrams|isocline]]''. The isocline connects the vertices of a lower frequency (longer wavelength) skew polygram than the Petrie polygon. The Petrie triacontagon has <math>\phi^{-1}</math> edges; the isoclinic pentadecagram₂ has <math>\sqrt{1}</math> edges which join vertices which are two <math\phi^{-1}</math> edges apart. Each <math>\sqrt{1}</math> edge belongs to a different [[#Hexagons|great hexagon]], and successive <math>\sqrt{1}</math> edges belong to different 24-cells, as the isoclinic rotation takes hexagons to Clifford parallel hexagons and passes through successive Clifford parallel 24-cells.|name=double threaded}} These isocline chords are both hexa''gon'' edges and penta''gram'' edges. The 20 Clifford parallel isoclines (30-cell ring axes) of each left (or right) isocline bundle do not intersect each other. Either distinct decagonal isoclinic rotation (left or right) rotates all 120 vertices (and all 600 cells), but pentadecagram isoclines and pentagons are connected such that vertices alternate as 60 black and 60 white vertices (and 300 black and 300 white cells), like the black and white squares of the [[W:Chessboard|chessboard]].{{Efn|name=isoclinic chessboard}} In the course of the rotation, the vertices on a left (or right) isocline rotate within the same 15-vertex black (or white) isocline, and the cells rotate within the same black (or white) 30-cell ring. ==== Hexagons and <s>hexagrams</s> ==== [[File:Regular_star_figure_2(10,3).svg|thumb|[[W:Icosagon#Related polygons|Icosagram {20/6}{{=}}2{10/3}]] contains 2 disjoint {10/3} decagrams (red and orange) which connect vertices 3 apart on the {10} and 6 apart on the {20}. In the 600-cell the edges are great pentagon edges spanning 72°.]]The [[#Fibrations of great circle polygons|fibrations of the 600-cell]] include 10 [[#Hexagons|fibrations of its 200 great hexagons]]: 10 fiber bundles of 20 great hexagons. The 20 Clifford parallel hexagons in each bundle are completely disjoint. Adjacent parallel hexagons are spanned by edges of great decagons.{{Efn|name=equi-isoclinic hexagons}} Each fibration corresponds to a distinct left (and right) isoclinic rotation of the 600-cell in 20 great hexagon invariant planes on 4𝝅 isoclines. Each fiber bundle delineates 20 disjoint directly congruent [[24-cell#6-cell rings|cell rings of 6 octahedral cells]] each, with three cell rings nesting together around each hexagon. The bundle of 20 Clifford parallel hexagon fibers is divided into a bundle of 20 black <math>\sqrt{3}</math> [[24-cell#Great triangles|great triangle]] fibers and a bundle of 20 white great triangle fibers, with a black and a white triangle inscribed in each hexagon and 6 black and 6 white triangles in each 6-octahedron ring. The black or white triangles are joined by three intersecting black or white isoclines, each of which is a special kind of helical great circle{{Efn|name=one true 4𝝅 circle}} through the corresponding vertices in 10 Clifford parallel black (or white) great triangles. The 10 <math>\sqrt{3 - \phi}</math> chords of each isocline form a skew [[W:Decagon#decagram|decagram {10/3}]], 10 great pentagon edges joined end-to-end in a helical loop, [[W:Winding number|winding]] 3 times around the 600-cell through all four dimensions rather than lying flat in a central plane. Each pair of black and white isoclines (intersecting antipodal great hexagon vertices) forms a compound 20-gon [[W:Icosagon#Related polygons|icosagram {20/6}{{=}}2{10/3}]]. Notice the relation between the [[24-cell#Helical dodecagrams and their isoclines|24-cell's characteristic rotation in great hexagon invariant planes]] (on <s>hexagram</s> isoclines), and the 600-cell's own version of the rotation of great hexagon planes (on decagram isoclines). They are exactly the same isoclinic rotation: they have the same isocline. They have different numbers of the same isocline because the 600-cell contains multiple 24-cells, and the 600-cell's <math>\sqrt{3 - \phi}</math> isocline chord is shorter than the 24-cell's <math>\sqrt{3}</math> isocline chord because the isocline intersects more vertices in the 600-cell (10) than it does in the 24-cell (6), but both Clifford polygrams have a 4𝝅 circumference.{{Efn|The 24-cell rotates hexagons on <s>[[24-cell#Helical dodecagrams and their isoclines|hexagrams]]</s>, while the 600-cell rotates hexagons on decagrams, but these are discrete instances of the same kind of isoclinic rotation in hexagon invariant planes. In particular, their directly or enantiomorphously congruent isoclines are all a geodesic circle of circumference 4𝝅.{{Efn|All 3-sphere isoclines{{Efn|name=isoclinic geodesic}} 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 2𝝅''r''; simple rotations take place on these isoclines. Double rotations have isoclines of more than 2𝝅''r'' circumference, because their circle does not close in a single revolution. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic (equi-angled) double 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''r'' circumference. The 600-cell edge-rotates on isoclines of 5𝝅''r'' circumference.|name=isocline circumference.}}|name=4𝝅 rotation}} They have different isocline polygrams only because the isocline curve intersects more vertices in the 600-cell than it does in the 24-cell.{{Efn|The 600-cell's helical {20/6}{{=}}2{10/3} [[W:20-gon|icosagram]] is a compound of the 24-cell's <s>helical {6/2} hexagram</s>, which is inscribed within it just as the 24-cell is inscribed in the 600-cell.}} ==== Squares and octagrams ==== [[File:Regular_star_polygon_24-5.svg|thumb|The Clifford polygon of the 600-cell's isoclinic rotation in great square invariant planes is a skew regular [[W:24-gon#Related polygons|{24/5} 24-gram]], with <math>\phi</math> edges that connect vertices 5 apart on the 24-vertex circumference, which is a unique 24-cell (<math>\sqrt{1}</math> edges not shown).]]The [[#Fibrations of great circle polygons|fibrations of the 600-cell]] include 15 [[#Squares|fibrations of its 450 great squares]]: 15 fiber bundles of 30 great squares. The 30 Clifford parallel squares in each bundle are completely disjoint. Adjacent parallel squares are spanned by edges of great decagons.{{Efn|name=equi-isoclinic squares}} Each fibration corresponds to a distinct left (and right) isoclinic rotation of the 600-cell in 30 great square invariant planes (15 completely orthogonal pairs) on 4𝝅 isoclines. Each fiber bundle delineates 30 chiral [[16-cell#Helical construction|cell rings of 8 tetrahedral cells]] each,{{Efn|name=two different tetrahelixes}} with a left and right cell ring nesting together to fill each of the 15 disjoint 16-cells inscribed in the 600-cell. Axial to each 8-tetrahedron ring is a special kind of helical great circle, an isocline.{{Efn|name=isoclinic geodesic}} In a left (or right) isoclinic rotation of the 600-cell in great square invariant planes, all the vertices circulate on one of 15 Clifford parallel isoclines. The 30 Clifford parallel squares in each bundle are joined by four Clifford parallel 24-gram isoclines (one through each vertex), each of which intersects one vertex in 24 of the 30 squares, and all 24 vertices of just one of the 600-cell's 25 24-cells. Each isocline is a 24-gram circuit intersecting all 25 24-cells, 24 of them just once and one of them 24 times. The 24 vertices in each 24-gram isocline comprise a unique 24-cell; there are 25 such distinct isoclines in the 600-cell. Each isocline is a skew {24/5} 24-gram, 24 <math>\phi</math> chords joined end-to-end in a helical loop, winding 5 times around one 24-cell through all four dimensions rather than lying flat in a central plane. Adjacent vertices of the 24-cell are one <math>\sqrt{1}</math> chord apart, and 5 <math>\phi</math> chords apart on its isocline. A left (or right) isoclinic rotation through 720° takes each 24-cell to and through every other 24-cell. Notice the relations between the [[16-cell#Helical construction|16-cell's rotation in just 2 completely orthogonal great square planes]], the [[24-cell#Helical octagrams and their isoclines|24-cell's rotation in 6 Clifford parallel great squares]], and this rotation of the 600-cell in 30 Clifford parallel great squares. These three rotations are the same rotation, taking place on exactly the same kind of isocline circles, which happen to intersect more vertices in the 600-cell (24) than they do in the 16-cell (8).{{Efn|The 16-cell rotates squares on [[16-cell#Helical construction|{8/3} octagrams]], the 24-cell rotates squares on [[24-cell#Helical octagrams and their isoclines|{24/9}=3{8/3} octagrams]], and the 600 rotates squares on {24/5} 24-grams, but these are discrete instances of the same kind of isoclinic rotation in great square invariant planes. In particular, their directly or enantiomorphously congruent isoclines are all exactly the same geodesic circle of circumference 4𝝅. They have different isocline polygrams only because the isocline curve intersects more vertices in the 600-cell than it does in the 24-cell or the 16-cell. The 600-cell's helical {24/5} 24-gram is a compound of the 24-cell's helical {24/9} octagram, which is inscribed within the 600-cell just as the 16-cell's helical {8/3} octagram is inscribed within the 24-cell.}} In the 16-cell's rotation the distance between vertices on an isocline curve is the <math>\sqrt{4}</math> diameter. In the 600-cell vertices are closer together, and its <math>\phi</math> chord is the distance between adjacent vertices on the same isocline, but all these isoclines have a 4𝝅 circumference.{{Efn|name=isocline circumference.}} === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 600-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 600-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element. <math>\begin{bmatrix}\begin{matrix}120 & 12 & 30 & 20 \\ 2 & 720 & 5 & 5 \\ 3 & 3 & 1200 & 2 \\ 4 & 6 & 4 & 600 \end{matrix}\end{bmatrix}</math> Here is the configuration expanded with ''k''-face elements and ''k''-figures. The diagonal element counts are the ratio of the full [[W:Coxeter group|Coxeter group]] order, 14400, divided by the order of the subgroup with mirror removal. {| class=wikitable !H₄||{{Coxeter–Dynkin diagram|node_1|3|node|3|node|5|node}} ! [[W:k-face|''k''-face]]||f_''k''||f₀ || f₁||f₂||f₃||[[W:Vertex figure|''k''-fig]] !Notes |- align=right |H₃ || {{Coxeter–Dynkin diagram|node_x|2|node|3|node|5|node}} ||( ) !f₀ || 120 || 12 || 30 || 20 ||[[W:icosahedron|{3,5}]] || H₄/H₃ = 14400/120 = 120 |- align=right |A₁H₂ ||{{Coxeter–Dynkin diagram|node_1|2|node_x|2|node|5|node}} ||{ } !f₁ || 2 || 720 || 5 || 5 || [[W:pentagon|{5}]] || H₄/H₂A₁ = 14400/10/2 = 720 |- align=right |A₂A₁ ||{{Coxeter–Dynkin diagram|node_1|3|node|2|node_x|2|node}} ||[[W:equilateral triangle|{3}]] !f₂ || 3 || 3 || 1200 || 2 || { } || H₄/A₂A₁ = 14400/6/2 = 1200 |- align=right |A₃ ||{{Coxeter–Dynkin diagram|node_1|3|node|3|node|2|node_x}} ||[[W:tetrahedron|{3,3}]] !f₃ || 4 || 6 || 4 || 600|| ( ) || H₄/A₃ = 14400/24 = 600 |} == Symmetries == The [[W:Icosian|icosian]]s are a specific set of Hamiltonian [[W:Quaternion|quaternion]]s with the same symmetry as the 600-cell.{{Sfn|van Ittersum|2020|loc=§4.3|pp=80-95}} The icosians lie in the ''golden field'', (''a'' + ''b''{{radic|5}}) + (''c'' + ''d''{{radic|5}})'''i''' + (''e'' + ''f''{{radic|5}})'''j''' + (''g'' + ''h''{{radic|5}})'''k''', where the eight variables are [[W:Rational number|rational number]]s.{{Sfn|Steinbach|1997|p=24}} The finite sums of the 120 [[W:Icosian#Unit icosians|unit icosians]] are called the [[W:Icosian#Icosian ring|icosian ring]]. When interpreted as quaternions,{{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 120 vertices of the 600-cell form a [[W:group (mathematics)|group]] under quaternionic multiplication. This group is often called the [[W:Binary icosahedral group|binary icosahedral group]] and denoted by ''2I'' as it is the double cover of the ordinary [[W:Icosahedral group|icosahedral group]] ''I''.{{Sfn|Stillwell|2001|loc=The Poincaré Homology Sphere|pp=22-23}} It occurs twice in the rotational symmetry group ''RSG'' of the 600-cell as an [[W:Invariant subgroup|invariant subgroup]], namely as the subgroup ''2I_L'' of quaternion left-multiplications and as the subgroup ''2I_R'' of quaternion right-multiplications. Each rotational symmetry of the 600-cell is generated by specific elements of ''2I_L'' and ''2I_R''; the pair of opposite elements generate the same element of ''RSG''. The [[W:Center of a group|centre]] of ''RSG'' consists of the non-rotation ''Id'' and the central inversion ''−Id''. We have the isomorphism ''RSG ≅ (2I_L × 2I_R) / {Id, -Id}''. The order of ''RSG'' equals {{sfrac|120 × 120|2}} = 7200. The [[W:Quaternion algebra|quaternion algebra]] as a tool for the treatment of 3D and 4D rotations, and as a road to the full understanding of the theory of [[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]], is described by Mebius.{{Sfn|Mebius|1994|p=1|loc="''[[W:Quaternion algebra|Quaternion algebra]]'' is the tool ''par excellence'' for the treatment of three- and four- dimensional (3D and 4D) rotations. Obviously only 3D and by implication 2D rotations have an everyday practical meaning, but the [[W:Rotations in 4-dimensional Euclidean space|theory of 4D rotations]] turns out to offer the easiest road to the representation of 3D rotations by quaternions."}} The binary icosahedral group is [[W:Isomorphic|isomorphic]] to [[W:special linear group|SL(2,5)]]. The full [[W:Symmetry group|symmetry group]] of the 600-cell is the [[W:H4 (mathematics)|Coxeter group H₄]].{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=§2 The Labeling of H₄}} This is a [[W:Group (mathematics)|group]] of order 14400. It consists of 7200 [[W:Rotation (mathematics)|rotations]] and 7200 rotation-reflections. The rotations form an [[W:Invariant subgroup|invariant subgroup]] of the full symmetry group. The rotational symmetry group was first described by S.L. van Oss.{{Sfn|van Oss|1899||pp=1-18}} The H₄ group and its Clifford algebra construction from 3-dimensional symmetry groups by induction is described by Dechant.{{Sfn|Dechant|2021|loc=Abstract|ps=; "[E]very 3D root system allows the construction of a corresponding 4D root system via an 'induction theorem'. In this paper, we look at the icosahedral case of H3 → H4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem ... shed[ding] light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries ... including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes.... This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework."}} == Visualization == The symmetries of the 3-D surface of the 600-cell are somewhat difficult to visualize due to both the large number of tetrahedral cells,{{Efn||name=tetrahedral cell adjacency}} and the fact that the tetrahedron has no opposing faces or vertices.{{Efn|name=directly congruent versus twisted cell rings}} One can start by realizing the 600-cell is the dual of the 120-cell. One may also notice that the 600-cell also contains the vertices of a dodecahedron,{{Sfn|Coxeter|1973|loc=Table VI (iii): 𝐈𝐈 = {3,3,5}|p=303}} which with some effort can be seen in most of the below perspective projections. === 2D projections === The H3 [[W:Decagon|decagon]]al projection shows the plane of the [[W:van Oss polygon|van Oss polygon]]. {| class="wikitable" width=600 |+ [[W:Orthographic projection|Orthographic projection]]s by [[W:Coxeter plane|Coxeter plane]]s{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}} |- align=center !H₄ ! - !F₄ |- align=center |[[File:600-cell graph H4.svg|200px]]<br>[30]<br>(Red=1) |[[File:600-cell t0 p20.svg|200px]]<br>[20]<br>(Red=1) |[[File:600-cell t0 F4.svg|200px]]<br>[12]<br>(Red=1) |- align=center !H₃ !A₂ / B₃ / D₄ !A₃ / B₂ |- align=center |[[File:600-cell t0 H3.svg|200px]]<br>[10]<br>(Red=1,orange=5,yellow=10) |[[File:600-cell t0 A2.svg|200px]]<br>[6]<br>(Red=1,orange=3,yellow=6) |[[File:600-cell t0.svg|200px]]<br>[4]<br>(Red=1,orange=2,yellow=4) |} === 3D projections === A three-dimensional model of the 600-cell, in the collection of the [[W:Institut Henri Poincaré|Institut Henri Poincaré]], was photographed in 1934–1935 by [[W:Man Ray|Man Ray]], and formed part of two of his later "Shakesperean Equation" paintings.<ref>{{citation|title=Man Ray Human Equations: A journey from mathematics to Shakespeare|publisher=Hatje Cantz|editor1-first=Wendy A.|editor1-last=Grossman|editor2-first=Edouard|editor2-last=Sebline|year=2015}}. See in particular ''mathematical object mo-6.2'', p.&nbsp;58; ''Antony and Cleopatra'', SE-6, p.&nbsp;59; ''mathematical object mo-9'', p.&nbsp;64; ''Merchant of Venice'', SE-9, p.&nbsp;65, and "The Hexacosichoron", Philip Ordning, p.&nbsp;96.</ref> {| class=wikitable !colspan=2|Vertex-first projection |- |[[Image:600cell-perspective-vertex-first-multilayer-01.png|320px]] |This image shows a vertex-first perspective projection of the 600-cell into 3D. The 600-cell is scaled to a vertex-center radius of 1, and the 4D viewpoint is placed 5 units away. Then the following enhancements are applied: * The 20 tetrahedra meeting at the vertex closest to the 4D viewpoint are rendered in solid color. Their icosahedral arrangement is clearly shown. * The tetrahedra immediately adjoining these 20 cells are rendered in transparent yellow. * The remaining cells are rendered in edge-outline. * Cells facing away from the 4D viewpoint (those lying on the "far side" of the 600-cell) have been culled, to reduce visual clutter in the final image. |- !colspan=2|Cell-first projection |- |[[Image:600cell-perspective-cell-first-multilayer-02.png|320px]] |This image shows the 600-cell in cell-first perspective projection into 3D. Again, the 600-cell to a vertex-center radius of 1 and the 4D viewpoint is placed 5 units away. The following enhancements are then applied: * The nearest cell to the 4d viewpoint is rendered in solid color, lying at the center of the projection image. * The cells surrounding it (sharing at least 1 vertex) are rendered in transparent yellow. * The remaining cells are rendered in edge-outline. * Cells facing away from the 4D viewpoint have been culled for clarity. This particular viewpoint shows a nice outline of 5 tetrahedra sharing an edge, towards the front of the 3D image. |} === Animations=== {| class=wikitable width=540 !colspan=1|Coxeter section views |- |align=center|[[File:Cell120-OmniTruncated-Sections.webm|300px]]<br>Sections of an omnitrucated 4D 600/120-cell 97 frames (=48x2 L/R+1 Center) shown in 4D to 3D [[W:Flatland|Flatland]]er views. The center section is highlighted by also showing it as a combined set of convex hulls. |} == Diminished 600-cells == The [[W:Snub 24-cell|snub 24-cell]] may be obtained from the 600-cell by removing the vertices of an inscribed [[24-cell|24-cell]] and taking the [[W:Convex hull|convex hull]] of the remaining vertices.{{Sfn|Dechant|2021|pp=22-24|loc=§8. Snub 24-cell}} This process is a ''[[W:Diminishment (geometry)|diminishing]]'' of the 600-cell. The [[W:Grand antiprism|grand antiprism]] may be obtained by another diminishing of the 600-cell: removing 20 vertices that lie on two mutually orthogonal rings and taking the convex hull of the remaining vertices.{{Sfn|Dechant|2021|pp=20-22|loc=§7. The Grand Antiprism and H₂ × H₂}} A bi-24-diminished 600-cell, with all [[W:Tridiminished icosahedron|tridiminished icosahedron]] cells has 48 vertices removed, leaving 72 of 120 vertices of the 600-cell. The dual of a bi-24-diminished 600-cell, is a tri-24-diminished 600-cell, with 48 vertices and 72 hexahedron cells. There are a total of 314,248,344 diminishings of the 600-cell by non-adjacent vertices. All of these consist of regular tetrahedral and icosahedral cells.<ref>{{Cite journal|last1=Sikiric|first1=Mathieu|last2=Myrvold|first2=Wendy|date=2007|title=The special cuts of 600-cell|journal=Beiträge zur Algebra und Geometrie|volume=49|issue=1|arxiv=0708.3443}}</ref> {| class="wikitable collapsible" !colspan=12|Diminished 600-cells |- !Name !Tri-24-diminished 600-cell !Bi-24-diminished 600-cell ![[W:Snub 24-cell|Snub 24-cell]]<br>(24-diminished 600-cell) ![[W:Grand antiprism|Grand antiprism]]<br>(20-diminished 600-cell) !600-cell |- align=center !Vertices |48 |72 |96 |100 |120 |- align=center !Vertex figure<br>(Symmetry) |[[File:Dual tridiminished icosahedron.png|120px]]<br>dual of tridiminished icosahedron<br>([3], order 6) |[[File:Biicositetradiminished 600-cell vertex figure.png|120px]]<br>[[W:Hexahedron|tetragonal antiwedge]]<br>([2]⁺, order 2) |[[File:Snub 24-cell verf.png|120px]]<br>[[W:tridiminished icosahedron|tridiminished icosahedron]]<br>([3], order 6) |[[File:Grand antiprism verf.png|120px]]<br>[[W:Edge-contracted icosahedron|bidiminished icosahedron]]<br>([2], order 4) |[[File:600-cell verf.svg|120px]]<br>[[W:Icosahedron|icosahedron]]<br>([5,3], order 120) |- align=center !Symmetry |colspan=2|Order 144 (48×3 or 72×2) |[3⁺,4,3]<br>Order 576 (96×6) |[10,2⁺,10]<br>Order 400 (100×4) |[5,3,3]<br>Order 14400 (120×120) |- align=center !Net |[[File:Triicositetradiminished hexacosichoron net.png|100px]] |[[File:Biicositetradiminished hexacosichoron net.png|100px]] |[[File:Snub 24-cell-net.png|100px]] |[[File:Grand antiprism net.png|100px]] |[[File:600-cell net.png|100px]] |- align=center !Ortho<br>H₄ plane |[[File:Tridiminished 600-cell H4 Coxeter plane.svg|120px]] |[[File:bidex ortho-30-gon.png|120px]] |[[File:Snub 24-cell ortho30-gon.png|120px]] |[[File:Grand antiprism ortho-30-gon.png|120px]] |[[File:600-cell graph H4.svg|120px]] |- align=center !Ortho<br>F₄ plane |[[File:Tridiminished 600-cell F4 Coxeter plane.svg|120px]] |[[File:Bidex ortho 12-gon.png|120px]] |[[File:24-cell h01 F4.svg|120px]] |[[File:GrandAntiPrism-2D-F4.svg|120px]] |[[File:600-cell t0 F4.svg|120px]] |} == Related polytopes and honeycombs == The 600-cell is one of 15 regular and uniform polytopes with the same H₄ symmetry [3,3,5]:{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020}} {{H4_family}} It is similar to three [[W:Regular 4-polytope|regular 4-polytope]]s: the [[5-cell|5-cell]] {3,3,3}, [[16-cell|16-cell]] {3,3,4} of Euclidean 4-space, and the [[W:Order-6 tetrahedral honeycomb|order-6 tetrahedral honeycomb]] {3,3,6} of hyperbolic space. All of these have [[W:Tetrahedron|tetrahedral]] cells. {{Tetrahedral cell tessellations}} This 4-polytope is a part of a sequence of 4-polytope and honeycombs with [[W:Icosahedron|icosahedron]] vertex figures: {{Icosahedral vertex figure tessellations}} The [[W:regular complex polytope|regular complex polygons]] ₃{5}₃, {{Coxeter–Dynkin diagram|3node_1|5|3node}} and ₅{3}₅, {{Coxeter–Dynkin diagram|5node_1|3|5node}}, in <math>\mathbb{C}^2</math> have a real representation as ''600-cell'' in 4-dimensional space. Both have 120 vertices, and 120 edges. The first has [[W:Complex reflection group|complex reflection group]] ₃[5]₃, order 360, and the second has symmetry ₅[3]₅, order 600.{{Sfn|Coxeter|1991|pp=48-49}} {| class="wikitable collapsed collapsible" !colspan=3| Regular complex polytope in orthogonal projection of H₄ Coxeter plane{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}} |- align=center |[[File:600-cell graph H4.svg|240px]]<br>{3,3,5}<br>Order 14400 |[[File:Complex polygon 3-5-3.png|240px]]<br>₃{5}₃<br>Order 360 |[[File:Complex polygon 5-3-5.png|240px]]<br>₅{3}₅<br>Order 600 |} == See also == * [[W:600-cell|Wikipedia:600-cell]], the article this article is an expanded version of * [[24-cell|24-cell]], the predecessor 4-polytope on which the 600-cell is based * [[120-cell|120-cell]], the dual 4-polytope to the 600-cell, and its successor * [[W:Uniform 4-polytope#The H4 family|Uniform 4-polytope family with [5,3,3] symmetry]] * [[W:Regular 4-polytope|Regular 4-polytope]] * [[W:Polytope|Polytope]] == Notes == {{Regular convex 4-polytopes Notelist|wiki=W:}} == Citations == {{Regular convex 4-polytopes Reflist|wiki=W:}} == References == {{Refbegin}} * {{Citation | last=Schläfli | first=Ludwig | author-link=W:Ludwig Schläfli |editor-first=Arthur | editor-last=Cayley | editor-link=W:Arthur Cayley | title=An attempt to determine the twenty-seven lines upon a surface of the third order, and to derive such surfaces in species, in reference to the reality of the lines upon the surface | url=http://resolver.sub.uni-goettingen.de/purl?PPN600494829_0002 | year=1858 | journal=Quarterly Journal of Pure and Applied Mathematics | volume=2 | pages=55–65, 110–120 }} {{Regular convex 4-polytopes Refs|wiki=W:}} * {{Cite journal | first1=F. | last1=Buekenhout | first2=M. | last2=Parker | title=The number of nets of the regular convex polytopes in dimension <= 4 | journal=[[W:Discrete Mathematics (journal)|Discrete Mathematics]] | volume=186 | issue=1–3 | date=15 May 1998 | pages=69–94| doi=10.1016/S0012-365X(97)00225-2 | doi-access=free | ref={{SfnRef|Buekenhout & Parker|1998}} }} * {{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}} }} * [http://www.polytope.de Four-dimensional Archimedean Polytopes] (German), Marco Möller, 2004 PhD dissertation [http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf] {{Webarchive|url=https://web.archive.org/web/20050322235615/http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf |date=2005-03-22 }} * {{Cite journal | last=Oss | first=Salomon Levi van | title=Das regelmässige Sechshundertzell und seine selbstdeckenden Bewegungen | journal=Verhandelingen der Koninklijke (Nederlandse) Akademie van Wetenschappen, Sectie 1 (Afdeeling Natuurkunde) | volume=7 | issue=1 | pages=1–18 | place=Amsterdam | year=1899 | url=https://books.google.com/books?id=AfQ3AQAAMAAJ&pg=PA3 | ref={{SfnRef|van Oss|1899}} }} {{Refend}} == External links == * [https://bendwavy.org/klitzing/incmats/ex.htm ex], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes] * [https://polytope.miraheze.org/wiki/Hexacosichoron Hexacosichoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki] * [http://hi.gher.space/wiki/Hydrochoron Hydrochoron], at [http://hi.gher.space/wiki/Main_Page Higher space] * [https://www.qfbox.info/4d/600-cell The 600-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space] [[Category:Geometry]] [[Category:Polyscheme]] 01j4cqde1fcay4lz6s50vgutrcx41lt 2812062 2812061 2026-05-29T21:10:22Z Dc.samizdat 2856930 /* Golden chords */ 2812062 wikitext text/x-wiki {{Short description|Four-dimensional analog of the icosahedron}} {{Polyscheme|radius=an '''expanded version''' of|active=is the focus of active research}} {{Infobox 4-polytope | Name=600-cell| Image_File=Schlegel_wireframe_600-cell_vertex-centered.png| Image_Caption=[[W:Schlegel diagram|Schlegel diagram]], vertex-centered<br>(vertices and edges)| Type=[[W:Convex regular 4-polytope|Convex regular 4-polytope]]| Last=[[W:Rectified 600-cell|34]]| Index=35| Next=[[W:Truncated 120-cell|36]]| Schläfli={3,3,5}| CD={{Coxeter–Dynkin diagram|node_1|3|node|3|node|5|node}}| Cell_List=600 ([[W:Tetrahedron|{3,3}]]) [[Image:Tetrahedron.png|20px]]| Face_List=1200 [[W:triangle|{3}]]| Edge_Count=720| Vertex_Count= 120| Petrie_Polygon=[[W:Triacontagon#Petrie polygons|30-gon]]| Coxeter_Group=H₄, [3,3,5], order 14400| Vertex_Figure=[[Image:600-cell verf.svg|80px]]<br>[[W:icosahedron|icosahedron]]| Dual=[[120-cell|120-cell]]| Property_List=[[W:Convex polytope|convex]], [[W:isogonal figure|isogonal]], [[W:isotoxal figure|isotoxal]], [[W:isohedral figure|isohedral]] }} [[File:600-cell net.png|thumb|right|[[W:Net (polyhedron)|Net]]]] In [[geometry]], the '''600-cell''' is the [[W:convex regular 4-polytope|convex regular 4-polytope]] (four-dimensional analogue of a [[W:Platonic solid|Platonic solid]]) with [[W:Schläfli symbol|Schläfli symbol]] {3,3,5}. It is also known as the '''C₆₀₀''', '''hexacosichoron'''<ref>[[W:Norman Johnson (mathematician)|N.W. Johnson]]: ''Geometries and Transformations'', (2018) {{ISBN|978-1-107-10340-5}} Chapter 11: ''Finite Symmetry Groups'', 11.5 ''Spherical Coxeter groups'', p.249</ref> and '''hexacosihedroid'''.<ref>Matila Ghyka, ''The Geometry of Art and Life'' (1977), p.68</ref> It is also called a '''tetraplex''' (abbreviated from "tetrahedral complex") and a '''[[W:polytetrahedron|polytetrahedron]]''', being bounded by tetrahedral [[W:Cell (geometry)|cells]]. The 600-cell's boundary is composed of 600 [[W:Tetrahedron|tetrahedral]] [[W:Cell (mathematics)|cells]] with 20 meeting at each vertex.{{Efn|name=vertex icosahedral pyramid}} Together they form 1200 triangular faces, 720 edges, and 120 vertices. It is the 4-[[W:Four-dimensional space#Dimensional analogy|dimensional analogue]] of the [[W:icosahedron|icosahedron]], since it has five [[W:Tetrahedron|tetrahedra]] meeting at every edge, just as the icosahedron has five [[W:triangle|triangle]]s meeting at every vertex.{{Efn|name=math of dimensional analogy}} Its [[W:dual polytope|dual polytope]] is the [[120-cell|120-cell]]. == Geometry == The 600-cell is the fifth in the sequence of 6 convex regular 4-polytopes (in order of complexity and size at the same radius).{{Efn|name=4-polytopes ordered by size and complexity|group=}} It can be deconstructed into twenty-five overlapping instances of its immediate predecessor the [[24-cell|24-cell]],{{Sfn|Coxeter|1973|loc=§8.51|p=153|ps=; "In fact, the vertices of {3, 3, 5}, each taken 5 times, are the vertices of 25 {3, 4, 3}'s."}} as the 24-cell can be [[24-cell#8-cell|deconstructed]] into three overlapping instances of its predecessor the [[W:Tesseract|tesseract (8-cell)]], and the 8-cell can be [[24-cell#Relationships among interior polytopes|deconstructed]] into two instances of its predecessor the [[16-cell|16-cell]].{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}} 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. The 24-cell's edge length equals its radius, but the 600-cell's edge length is ~0.618 times its radius,{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii), "600-cell" column ₀''R/l'' {{=}} 2𝝓/2}} which is the [[W:golden ratio|golden ratio]]. {{Regular convex 4-polytopes|wiki=W:}} === Coordinates === ==== Unit radius Cartesian coordinates ==== The vertices of a 600-cell of unit radius centered at the origin of 4-space, with edges of length <math>\phi^{-1} \approx 0.618</math> (where <math>\phi = \tfrac12\bigl(1 + \sqrt5~\!\bigr)</math> is the golden ratio), can be given{{Sfn|Coxeter|1973|loc=§8.7 Cartesian coordinates|pp=156-157}} as follows: 8 vertices obtained from :(0, 0, 0, ±1) by permuting coordinates, and 16 vertices of the form: :(±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}, ±{{sfrac|1|2}}) The remaining 96 vertices are obtained by taking [[W:even permutation|even permutation]]s of :(±{{sfrac|φ|2}}, ±{{sfrac|1|2}}, ±{{sfrac|φ⁻¹|2}}, 0) Note that the first 8 are the vertices of a [[16-cell|16-cell]], the second 16 are the vertices of a [[W:tesseract|tesseract]], and those 24 vertices together are the vertices of a [[24-cell|24-cell]]. The remaining 96 vertices are the vertices of a [[W:snub 24-cell|snub 24-cell]], which can be found by partitioning each of the 96 edges of another 24-cell (dual to the first) in the golden ratio in a consistent manner.{{Sfn|Coxeter|1973|loc=§8.4 The snub {3,4,3}|pp=151-153}} When interpreted as [[#Symmetries|quaternions]],{{Efn|name=quaternions}} these are the unit [[W:icosian|icosian]]s. In the 24-cell, there are [[24-cell#Great squares|squares]], [[24-cell#Great hexagons|hexagons]] and [[24-cell#Great triangles|triangles]] that lie on great circles (in central planes through four or six vertices).{{Efn|name=hypercubic chords}} In the 600-cell there are twenty-five overlapping inscribed 24-cells, with each vertex and square shared by five 24-cells, and each hexagon or triangle shared by two 24-cells.{{Efn|In cases where inscribed 4-polytopes of the same kind occupy disjoint sets of vertices (such as the two 16-cells inscribed in the tesseract, or the three 16-cells inscribed in the 24-cell), their sets of vertex chords, central polygons and cells must likewise be disjoint. In the cases where they share vertices (such as the three tesseracts inscribed in the 24-cell, or the 25 24-cells inscribed in the 600-cell), they also share some vertex chords and central polygons.{{Efn|name=disjoint from 8 and intersects 16}}}} In each 24-cell there are three disjoint 16-cells, so in the 600-cell there are 75 overlapping inscribed 16-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}} Each 16-cell constitutes a distinct orthonormal basis for the choice of a [[16-cell#Coordinates|coordinate reference frame]]. The 60 axes and 75 16-cells of the 600-cell constitute a [[W:Configuration (geometry)|geometric configuration]], which in the language of configurations is written as 60₅75₄ to indicate that each axis belongs to 5 16-cells, and each 16-cell contains 4 axes.{{Sfn|Waegell & Aravind|2009|loc=§3.2 The 75 bases of the 600-cell|pp=3-4|ps=; In the 600-cell the configuration's "points" and "lines" are axes ("rays") and 16-cells ("bases"), respectively.}} Each axis is orthogonal to exactly 15 others, and these are just its companions in the 5 16-cells in which it occurs. ==== Hopf spherical coordinates ==== In the 600-cell there are also great circle [[W:pentagon|pentagon]]s and [[W:decagon|decagon]]s (in central planes through ten vertices).{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020}} Only the decagon edges are visible elements of the 600-cell (because they are the edges of the 600-cell). The edges of the other great circle polygons are interior chords of the 600-cell, which are not shown in any of the 600-cell renderings in this article (except where shown as dashed lines).{{Efn|The 600-cell contains 25 distinct 24-cells, bound to each other by pentagonal rings. Each pentagon links five completely disjoint{{Efn|name=completely disjoint}} 24-cells together, the collective vertices of which are the 120 vertices of the 600-cell. Each 24-point 24-cell contains one fifth of all the vertices in the 120-point 600-cell, and is linked to the other 96 vertices (which comprise a [[#Diminished 600-cells|snub 24-cell]]) by the 600-cell's 144 pentagons. Each of the 25 24-cells intersects each of the 144 great pentagons at just one vertex.{{Efn|Each of the 25 24-cells of the 600-cell contains exactly one vertex of each great pentagon.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|p=438}} Six pentagons intersect at each 600-cell vertex, so each 24-cell intersects all 144 great pentagons.|name=distribution of pentagon vertices in 24-cells}} Five 24-cells meet at each 600-cell vertex,{{Efn|name=five 24-cells at each vertex of 600-cell}} so all 25 24-cells are linked by each great pentagon. The 600-cell can be partitioned into five disjoint 24-cells (10 different ways),{{Efn|name=Schoute's ten ways to get five disjoint 24-cells}} and also into 24 disjoint pentagons (inscribed in the 12 Clifford parallel great decagons of one of the 6 [[#Decagons|decagonal fibrations]]) by choosing a pentagon from the same fibration at each 24-cell vertex.|name=24-cells bound by pentagonal fibers}} By symmetry, an equal number of polygons of each kind pass through each vertex; so it is possible to account for all 120 vertices as the intersection of a set of central polygons of only one kind: decagons, hexagons, pentagons, squares, or triangles. For example, the 120 vertices can be seen as the vertices of 15 pairs of [[W:Completely orthogonal|completely orthogonal]]{{Efn|name=Six orthogonal planes of the Cartesian basis}} squares which do not share any vertices, or as 100 ''dual pairs'' of non-orthogonal hexagons between which all axis pairs are orthogonal, or as 144 non-orthogonal pentagons six of which intersect at each vertex. This latter pentagonal symmetry of the 600-cell is captured by the set of [[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]]{{Sfn|Zamboj|2021|pp=10-11|loc=§Hopf coordinates}} (𝜉_''i'', 𝜂, 𝜉_''j''){{Efn|name=Hopf coordinates|The [[W:Rotations in 4-dimensional Euclidean space#Hopf coordinates|Hopf coordinates]] are triples of three angles: : (𝜉_''i'', 𝜂, 𝜉_''j'') that parameterize the [[W:3-sphere#Hopf coordinates|3-sphere]] by numbering points along its great circles. A Hopf coordinate describes a point as a rotation from a polar point (0, 0, 0).{{Efn|name=Hopf coordinate angles|The angles 𝜉_''i'' and 𝜉_''j'' are angles of rotation in the two [[W:completely orthogonal|completely orthogonal]] invariant planes which characterize [[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]]. The angle 𝜂 is the inclination of both these planes from the polar axis, where 𝜂 ranges from 0 to {{sfrac|𝜋|2}}. The (𝜉_''i'', 0, 𝜉_''j'') coordinates describe the great circles which intersect at the north and south pole ("lines of longitude"). The (𝜉_''i'', {{sfrac|𝜋|2}}, 𝜉_''j'') coordinates describe the great circles orthogonal to longitude ("equators"); there is more than one "equator" great circle in a 4-polytope, as the equator of a 3-sphere is a whole 2-sphere of great circles. The other Hopf coordinates (𝜉_''i'', 0 < 𝜂 < {{sfrac|𝜋|2}}, 𝜉_''j'') describe the great circles (''not'' "lines of latitude") which cross an equator but do not pass through the north or south pole.}} Hopf coordinates are a natural alternative to Cartesian coordinates{{Efn|name=Hopf coordinates conversion|The conversion from Hopf coordinates (𝜉_''i'', 𝜂, 𝜉_''j'') to unit-radius Cartesian coordinates (w, x, y, z) is:<br> : w {{=}} cos 𝜉_''i'' sin 𝜂 : x {{=}} cos 𝜉_''j'' cos 𝜂 : y {{=}} sin 𝜉_''j'' cos 𝜂 : z {{=}} sin 𝜉_''i'' sin 𝜂 The Hopf origin pole (0, 0, 0) is Cartesian (0, 1, 0, 0). The conventional "north pole" of Cartesian standard orientation is (0, 0, 1, 0), which is Hopf ({{sfrac|𝜋|2}}, {{sfrac|𝜋|2}}, {{sfrac|𝜋|2}}). Cartesian (1, 0, 0, 0) is Hopf (0, {{sfrac|𝜋|2}}, 0).}} for framing regular convex 4-polytopes, because the group of [[W:Rotations in 4-dimensional Euclidean space|4-dimensional rotations]], denoted SO(4), generates those polytopes.}} given as: : ({<10}{{sfrac|𝜋|5}}, {≤5}{{sfrac|𝜋|10}}, {<10}{{sfrac|𝜋|5}}) where {<10} is the permutation of the ten digits (0 1 2 3 4 5 6 7 8 9) and {≤5} is the permutation of the six digits (0 1 2 3 4 5). The 𝜉_''i'' and 𝜉_''j'' coordinates range over the 10 vertices of great circle decagons; even and odd digits label the vertices of the two great circle pentagons inscribed in each decagon.{{Efn|There are 600 permutations of these coordinates, but there are only 120 vertices in the 600-cell. These are actually the Hopf coordinates of the vertices of the [[120-cell#Cartesian coordinates|120-cell]], which has 600 vertices and can be seen (two different ways) as a compound of 5 disjoint 600-cells.}} === Structure === ==== Polyhedral sections ==== The mutual distances of the vertices, measured in degrees of arc on the circumscribed [[W:hypersphere|hypersphere]], only have the values 36° = {{sfrac|𝜋|5}}, 60° = {{sfrac|𝜋|3}}, 72° = {{sfrac|2𝜋|5}}, 90° = {{sfrac|𝜋|2}}, 108° = {{sfrac|3𝜋|5}}, 120° = {{sfrac|2𝜋|3}}, 144° = {{sfrac|4𝜋|5}}, and 180° = 𝜋. Departing from an arbitrary vertex V one has at 36° and 144° the 12 vertices of an [[W:icosahedron|icosahedron]],{{Efn|name=vertex icosahedral pyramid}} at 60° and 120° the 20 vertices of a [[W:dodecahedron|dodecahedron]], at 72° and 108° the 12 vertices of a larger icosahedron, at 90° the 30 vertices of an [[W:icosidodecahedron|icosidodecahedron]], and finally at 180° the antipodal vertex of V.{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏⁻¹) beginning with a vertex}}{{Sfn|van Oss|1899|ps=; van Oss does not mention the arc distances between vertices of the 600-cell.}}{{Sfn|Buekenhout & Parker|1998}} These can be seen in the H3 [[W:Coxeter plane|Coxeter plane]] projections with overlapping vertices colored.{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}} :[[File:600-cell-polyhedral levels.png|640px]] These polyhedral sections are ''solids'' in the sense that they are 3-dimensional, but of course all of their vertices lie on the surface of the 600-cell (they are hollow, not solid). Each polyhedron lies in Euclidean 4-dimensional space as a parallel cross section through the 600-cell (a hyperplane). In the curved 3-dimensional space of the 600-cell's boundary surface envelope, the polyhedron surrounds the vertex V the way it surrounds its own center. But its own center is in the interior of the 600-cell, not on its surface. V is not actually at the center of the polyhedron, because it is displaced outward from that hyperplane in the fourth dimension, to the surface of the 600-cell. Thus V is the apex of a [[W:Pyramid (geometry)#Polyhedral pyramid|4-pyramid]] based on the polyhedron. {| class=wikitable !colspan=2|Concentric Hulls |- |align=center|[[Image:Hulls of H4only-orthonormal.png|360px]] |The 600-cell is projected to 3D using an orthonormal basis. The vertices are sorted and tallied by their 3D norm. Generating the increasingly transparent hull of each set of tallied norms shows:<br> <br> 1) two points at the origin<br> 2) two icosahedra<br> 3) two dodecahedra<br> 4) two larger icosahedra<br> 5) and a single icosidodecahedron<br> <br> for a total of 120 vertices. This is the view from ''any'' origin vertex. The 600-cell contains 60 distinct sets of these concentric hulls, one centered on each pair of antipodal vertices. |- |} ==== Golden chords ==== [[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{{Efn|[[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.]] The 600-cell geometry is based on the [[24-cell#Hypercubic chords|24-cell]]. The 600-cell rounds out the 24-cell with 2 more great circle polygons (exterior decagon and interior pentagon), adding 4 more chord lengths which alternate with the 24-cell's 4 chord lengths. {{Clear}}|name=hypercubic chords|group=}} with angles of arc. The golden ratio governs{{Efn|name=golden chords|group=}} the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.{{Efn|name=radially golden}}|alt=|400x400px]] {{see also|W:24-cell#Hypercubic chords|label 1=24-cell § Hypercubic chords}} The 120 vertices are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏⁻¹) beginning with a vertex; see column ''a''}} at eight different [[W:Chord (geometry)|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its [[#Geodesics|five great circle polygons]].{{Sfn|Steinbach|1997|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a "fan of chords" diagram like the one here.|p=23|loc=Figure 3}} In ascending order of length, they are: <math>\sqrt{0.382\sim} = \sqrt{2 - \phi} = \phi^{-1} \approx 0.618</math> <math>\sqrt{1}</math> <math>\sqrt{1.382\sim} = \sqrt{3 - \phi} \approx 1.176</math> <math>\sqrt{2}</math> <math>\sqrt{2.618\sim} = \sqrt{1 + \phi} = \phi \approx 1.618</math> <math>\sqrt{3}</math> <math>\sqrt{3.618\sim} = \sqrt{2 + \phi} \approx 1.902</math> <math>\sqrt{4}</math> In the diagram, chord lengths are given as square roots, with a decimal fractional part if necessary, where: <math>\Phi = \phi^{-1} \approx 0.618</math> is the inverse golden ratio, and: <math>\Delta = 1 - \Phi = \Phi^2 \approx 0.382</math> is its square. For example, the 600-cell edge length is: <math>\Phi = \sqrt{0.\Delta} = \sqrt{0.382\sim} \approx 0.618</math> The four [[24-cell#Hypercubic chords|hypercubic chords]] of the 24-cell (<math>\sqrt{1}</math>, <math>\sqrt{2}</math>, <math>\sqrt{3}</math>, <math>\sqrt{4}</math>){{Efn|name=hypercubic chords}} alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of <math>\sqrt{5}</math>.{{Efn|The squares of two of these chord lengths, <math>3 - \phi {{=}} \phi^{-1}\sqrt{5}</math> and <math>2 + \phi {{=}} \phi\sqrt{5}</math>, are [[W:Algebraic conjugate|algebraic conjugate]]s whose product is <math>5</math>.}} The golden chords of the 600-cell exemplify that the [[W:golden ratio|golden ratio]] <math>\phi {{=}} \tfrac{1 + \sqrt{5}}{2} \approx 1.618</math> is a circle ratio related to fifths of <math>\pi</math>. For instance:<br> :<math>\tfrac{\pi}{5} {{=}} \arccos(\tfrac{\phi}{2})</math> is the arc of one 600-cell edge, the <math>\phi^{-1} = \Phi \approx 0.618</math> chord. Reciprocally, in this function discovered by Robert Everest expressing <math>\phi</math> as a function of <math>\pi</math> and the numbers 1, 2, 3 and 5 of the Fibonacci series:<br> : <math>\phi {{=}} 1 - 2 \cos(\tfrac{3\pi}{5})</math> <math>\tfrac{3\pi}{5}</math> is the arc length of the <math>\phi \approx 1.618</math> chord.<ref>{{Cite web|last=Baez|first=John|date=7 March 2017|title=Pi and the Golden Ratio|url=https://johncarlosbaez.wordpress.com/2017/03/07/pi-and-the-golden-ratio/|website=Azimuth|author-link=W:John Carlos Baez|access-date=10 October 2022}}</ref>|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of <math>\sqrt{5}</math>.{{Efn|The 600-cell edges are decagon edges of length <math>\phi^{-1} {{=}} \Phi {{=}} \sqrt{0.\Delta} \approx 0.618</math>, the ''smaller'' golden section of <math>\sqrt{5}</math>; the edges are in the inverse [[W:golden ratio|golden ratio]] <math>\tfrac{1}{\phi} {{=}} \phi^{-1}</math> to the <math>\sqrt{1}</math> hexagon chords (the 24-cell edges). The other fractional-root chords exhibit golden relationships as well. The chord of length <math>\sqrt{3 - \phi} {{=}} \sqrt{1.\Delta}</math> is a pentagon edge. The next fractional-root chord is a decagon diagonal of length <math>\phi {{=}} \sqrt{2.\Phi}</math>, the ''larger'' golden section of <math>\sqrt{5}</math>; it is in the golden ratio{{Efn|name=golden chords|group=}} to the <math>\sqrt{1}</math> chord (and the radius).{{Efn|Notice in the diagram how the <math>\phi</math> chord (the ''larger'' golden section) sums with the adjacent <math>\Phi</math> edge (the ''smaller'' golden section) to <math>\sqrt{5}</math>, as if together they were a <math>\sqrt{5}</math> chord bent to fit inside the <math>\sqrt{4}</math> diameter.}} The last fractional-root chord is the pentagon diagonal of length <math>=\sqrt{2 + \phi} {{=}} \sqrt{3.\Phi}</math>. The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed <math>\sqrt{2 + \phi} / \sqrt{3 - \phi} {{=}} \phi</math>.|name=fractional root chords|group=}} ==== Boundary envelopes ==== [[Image:600-cell.gif|thumb|A 3D projection of a 600-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]. The 3D surface made of 600 tetrahedra is visible.]] The 600-cell ''rounds out'' the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices,{{Efn|name=snub 24-cell|Consider one of the 24-vertex 24-cells inscribed in the 120-vertex 600-cell. The other 96 vertices constitute a [[W:snub 24-cell|snub 24-cell]]. Removing any one 24-cell from the 600-cell produces a snub 24-cell.}} in effect adding twenty-four more overlapping 24-cells inscribed in the 600-cell.{{Efn|The 600-cell contains exactly 25 24-cells, 75 16-cells and 75 8-cells, with each 16-cell and each 8-cell lying in just one 24-cell.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|p=434}}|name=4-polytopes inscribed in the 600-cell}} The new surface thus formed is a tessellation of smaller, more numerous cells{{Efn|Each tetrahedral cell touches, in some manner, 56 other cells. One cell contacts each of the four faces; two cells contact each of the six edges, but not a face; and ten cells contact each of the four vertices, but not a face or edge.|name=tetrahedral cell adjacency}} and faces: 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 [[24-cell#Hypercubic chords|<math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords]]. [[Image:24-cell.gif|thumb|A 3D projection of a [[24-cell|24-cell]] performing a [[24-cell#Simple rotations|simple rotation]]. The 3D surface made of 24 octahedra is visible. It is also present in the 600-cell, but as an invisible interior boundary envelope.]] 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 does not have unit edge-length in a unit-radius coordinate system the way the 24-cell and the tesseract do; unlike those two, the 600-cell is not [[W:Tesseract#Radial equilateral symmetry|radially equilateral]]. Like them it is radially triangular in a special way,{{Efn|All polytopes can be radially triangulated into triangles which meet at their center, each triangle contributing two radii and one edge. There are (at least) three special classes of polytopes which are radially triangular by a special kind of triangle. The ''radially equilateral'' polytopes can be constructed from identical [[W:equilateral triangle|equilateral triangle]]s which all meet at the center.{{Efn|name=radially equilateral|group=}} The ''radially golden'' polytopes can be constructed from identical [[W:Golden triangle (mathematics)|golden triangle]]s which all meet at the center.{{Efn|A [[W:Golden triangle (mathematics)|golden triangle]] is an [[W:Isosceles triangle|isosceles]] [[W:Triangle|triangle]] in which the duplicated side ''a'' is in the [[W:Golden ratio|golden ratio]] to the distinct side ''b'': :<math>\tfrac{a}{b} {{=}} \phi {{=}} \tfrac{1 + \sqrt{5}}{2} \approx 1.618</math> It can be found in a regular [[W:Decagon|decagon]] by connecting any two adjacent vertices to the center, and in the regular [[W:Pentagon|pentagon]] by connecting any two adjacent vertices to the vertex opposite them.<br> The vertex angle is: :<math>\theta = \arccos(\tfrac{\phi}{2}) {{=}} \tfrac{\pi}{5} {{=}} 36^\circ</math> so the base angles are each <math>\tfrac{2\pi}{5} {{=}} 72^\circ</math>. The golden triangle is uniquely identified as the only triangle to have its three angles in 2:2:1 proportions.|name=Golden triangle}} All the [[W:regular polytope|regular polytope]]s are ''radially right'' polytopes which can be constructed, with their various element centers and radii, from identical characteristic [[W:Schläfli orthoscheme|orthoscheme]]s which all meet at the center, subdividing the regular polytope into characteristic [[W:right triangle|right triangle]]s which meet at the center.{{Efn|The [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is the generalization of the [[W:right triangle|right triangle]] to simplex figures of any number of dimensions. Every regular polytope can be radially subdivided into identical [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]]s which meet at its center.{{Efn|name=characteristic orthoscheme}}|name=radially right|group=}}}} but one in which golden triangles rather than equilateral triangles meet at the center.{{Efn|The long radius (center to vertex) of the 600-cell is in the [[W:golden ratio|golden ratio]] to its edge length; thus its radius is <math>\phi</math> if its edge length is 1, and its edge length is <math>\phi^{-1}</math> if its radius is 1.|name=radially golden}} Only a few uniform polytopes have this property, including the four-dimensional 600-cell, the three-dimensional [[W:icosidodecahedron|icosidodecahedron]], and the two-dimensional [[W:Decagon#The golden ratio in decagon|decagon]]. (The icosidodecahedron is the equatorial cross section of the 600-cell, and the decagon is the equatorial cross section of the icosidodecahedron.) '''Radially golden''' polytopes are those which can be constructed, with their radii, from [[W:Golden triangle (mathematics)|golden triangles]].{{Efn|name=Golden triangle}} The boundary envelope of 600 small tetrahedral cells wraps around the twenty-five envelopes of 24 octahedral cells (adding some 4-dimensional space in places between these curved 3-dimensional envelopes). The shape of those interstices must be an [[W:Octahedral pyramid|octahedral 4-pyramid]] of some kind, but in the 600-cell it is [[#Octahedra|not regular]].{{Efn|Beginning with the 16-cell, every regular convex 4-polytope in the unit-radius sequence is inscribed in its successor.{{Sfn|Coxeter|1973|p=305|loc=Table VII: Regular Compounds in Four Dimensions}} Therefore the successor may be constructed by placing [[W:Pyramid (geometry)#Polyhedral pyramid|4-pyramids]] of some kind on the cells of its predecessor. Between the 16-cell and the tesseract, we have 16 right [[5-cell#Irregular 5-cell|tetrahedral pyramids]], with their apexes filling the corners of the tesseract. Between the tesseract and the 24-cell, we have 8 canonical [[W:cubic pyramid|cubic pyramid]]s. But if we place 24 canonical [[W:octahedral pyramid|octahedral pyramid]]s on the 24-cell, we only get another tesseract (of twice the radius and edge length), not the successor 600-cell. Between the 24-cell and the 600-cell there must be 24 smaller, irregular 4-pyramids on a regular octahedral base.|name=truncated irregular octahedral pyramid}} ==== Geodesics ==== The vertex chords of the 600-cell are arranged in [[W:geodesic|geodesic]] [[W:great circle|great circle]] polygons of five kinds: decagons, hexagons, pentagons, squares, and triangles.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=§4 The planes of the 600-cell|pp=437-439}} [[Image:Stereographic polytope 600cell.png|thumb|Cell-centered [[W:stereographic projection|stereographic projection]] of the 600-cell's 72 central decagons onto their great circles. Each great circle is divided into 10 arc-edges at the intersections where 6 great circles cross.]] The <math>\phi^{-1} \approx 0.618</math> edges form 72 flat regular central [[W:decagon|decagon]]s, 6 of which cross at each vertex.{{Efn|name=vertex icosahedral pyramid}} Just as the [[W:icosidodecahedron|icosidodecahedron]] can be partitioned into 6 central decagons (60 edges = 6 × 10), the 600-cell can be partitioned into 72 decagons (720 edges = 72 × 10). The 720 <math>\phi^{-1}</math> edges divide the surface into 1200 triangular faces and 600 tetrahedral cells: a 600-cell. The 720 edges occur in 360 parallel pairs, <math>\sqrt{2 + \phi}</math> apart. As in the decagon and the icosidodecahedron, the edges occur in [[W:Golden triangle (mathematics)|golden triangles]] which meet at the center of the polytope. The 72 great decagons can be divided into 6 sets of 12 non-intersecting [[W:Clifford parallel|Clifford parallel]] geodesics,{{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; various 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, in the 600-cell, visit all 120 vertices just once. For example, each of the 600 tetrahedra participates in 6 great decagons{{Efn|name=tetrahedron linking 6 decagons}} belonging to 6 discrete [[W:Hopf fibration|Hopf fibration]]s, each filling the whole 600-cell. Each [[#Decagons|fibration]] is a bundle of 12 Clifford parallel decagons which form 20 cell-disjoint intertwining rings of 30 tetrahedral cells,{{Efn|name=Boerdijk–Coxeter helix}} each bounded by three of the 12 great decagons.{{Efn|name=Clifford parallel decagons}}|name=Clifford parallels}} such that only one decagonal great circle in each set passes through each vertex, and the 12 decagons in each set reach all 120 vertices.{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} The <math>\sqrt{1}</math> chords form 200 central hexagons (25 sets of 16, with each hexagon in two sets),{{Efn|1=A 24-cell contains 16 hexagons. In the 600-cell, with 25 24-cells, each 24-cell is disjoint from 8 24-cells and intersects each of the other 16 24-cells in six vertices that form a hexagon.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|p=438}} A 600-cell contains 25・16/2 = 200 such hexagons.|name=disjoint from 8 and intersects 16}} 10 of which cross at each vertex{{Efn|The 10 hexagons which cross at each vertex lie along the 20 short radii of the icosahedral vertex figure.{{Efn|name=vertex icosahedral pyramid}}}} (4 from each of five 24-cells that meet at the vertex, with each hexagon in two of those 24-cells).{{Efn|name=five 24-cells at each vertex of 600-cell}} Each set of 16 hexagons consists of the 96 edges and 24 vertices of one of the 25 overlapping inscribed 24-cells. The <math>\sqrt{1}</math> chords join vertices which are two <math>\phi^{-1}</math> edges apart. Each <math>\sqrt{1}</math> chord is the long diameter of a face-bonded pair of tetrahedral cells (a [[W:triangular bipyramid|triangular bipyramid]]), and passes through the center of the shared face. As there are 1200 faces, there are 1200 <math>\sqrt{1}</math> chords, in 600 parallel pairs, <math>\sqrt{3}</math> apart. The hexagonal planes are non-orthogonal (60 degrees apart) but they occur as 100 ''dual pairs'' in which all 3 axes of one hexagon are orthogonal to all 3 axes of its dual.{{Sfn|Waegell & Aravind|2009|loc=§3.4. The 24-cell: points, lines, and Reye's configuration|p=5|ps=; Here Reye's "points" and "lines" are axes and hexagons, respectively. The dual hexagon ''planes'' are not orthogonal to each other, only their dual axis pairs. Dual hexagon pairs do not occur in individual 24-cells, only between 24-cells in the 600-cell.}} The 200 great hexagons can be divided into 10 sets of 20 non-intersecting Clifford parallel geodesics, such that only one hexagonal great circle in each set passes through each vertex, and the 20 hexagons in each set reach all 120 vertices.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} The <math>\sqrt{3 - \phi} \approx 1.176</math> chords form 144 central pentagons, 6 of which cross at each vertex.{{Efn|name=24-cells bound by pentagonal fibers}} The <math>\sqrt{3 - \phi}</math> chords run vertex-to-every-second-vertex in the same planes as the 72 decagons: two pentagons are inscribed in each decagon. They join vertices which are two <math>\phi^{-1}</math> edges apart on a geodesic great circle. The 720 <math>\sqrt{3 - \phi}</math> chords occur in 360 parallel pairs, <math>\phi</math> apart. The <math>\sqrt{2}</math> chords form 450 central squares, 15 of which cross at each vertex (3 from each of the five 24-cells that meet at the vertex). The <math>\sqrt{2}</math> chords join vertices which are three <math>\phi^{-1}</math> edges apart (and two <math>\sqrt{1}</math> chords apart). There are 600 <math>\sqrt{2}</math> chords, in 300 parallel pairs, <math>\sqrt{2}</math> apart. The 450 great squares (225 [[W:Completely orthogonal|completely orthogonal]] pairs) can be divided into 15 sets of 30 non-intersecting Clifford parallel geodesics, such that only one square great circle in each set passes through each vertex, and the 30 squares (15 completely orthogonal pairs) in each set reach all 120 vertices.{{Sfn|Sadoc|2001|p=577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the four-fold screw axis}} The <math>\phi \approx 1.618</math> chords form the legs of 720 central isosceles triangles (72 sets of 10 inscribed in each decagon), 6 of which cross at each vertex. The third edge (base) of each isosceles triangle is of length <math>\sqrt{2 + \phi} \approx 1.902</math>. The <math>\phi</math> chords run vertex-to-every-third-vertex in the same planes as the 72 decagons, joining vertices which are three <math>\phi^{-1}</math> edges apart on a geodesic great circle. There are 720 distinct <math>\phi</math> chords, in 360 parallel pairs, <math>\sqrt{3 - \phi}</math> apart. The <math>\sqrt{3}</math> chords form 400 equilateral central triangles (25 sets of 32, with each triangle in two sets), 10 of which cross at each vertex (4 from each of five [[24-cell#Geodesics|24-cells]], with each triangle in two of the 24-cells). Each set of 32 triangles consists of the 96 <math>\sqrt{3}</math> chords and 24 vertices of one of the 25 overlapping inscribed 24-cells. The <math>\sqrt{3}</math> chords run vertex-to-every-second-vertex in the same planes as the 200 hexagons: two triangles are inscribed in each hexagon. The <math>\sqrt{3}</math> chords join vertices which are four <math>\phi^{-1}</math> edges apart (and two <math>\sqrt{1}</math> chords apart on a geodesic great circle). Each <math>\sqrt{3}</math> chord is the long diameter of two cubic cells in the same 24-cell.{{Efn|The 25 inscribed 24-cells each have 3 inscribed tesseracts, which each have 8 <math>\sqrt{1}</math> cubic cells. The 1200 <math>\sqrt{3}</math> chords are the 4 long diameters of these 600 cubes. The three tesseracts in each 24-cell overlap, and each <math>\sqrt{3}</math> chord is a long diameter of two different cubes, in two different tesseracts, in two different 24-cells. [[24-cell#Relationships among interior polytopes|Each cube belongs to just one tesseract]] in just one 24-cell.|name=600 cubes}} There are 1200 <math>\sqrt{3}</math> chords, in 600 parallel pairs, <math>\sqrt{1}</math> apart. The <math>\sqrt{2 + \phi} \approx 1.902</math> chords (the diagonals of the pentagons) form the legs of 720 central isosceles triangles (144 sets of 5 inscribed in each pentagon), 6 of which cross at each vertex. The third edge (base) of each isosceles triangle is an edge of the pentagon of length <math>\sqrt{3 - \phi} \approx 1.176</math>, so these are [[W:Golden triangle (mathematics)|golden triangles]]. The <math>\sqrt{2 + \phi}</math> chords run vertex-to-every-fourth-vertex in the same planes as the 72 decagons, joining vertices which are four <math>\phi^{-1}</math> edges apart on a geodesic great circle. There are 720 distinct <math>\sqrt{2 + \phi}</math> chords, in 360 parallel pairs, <math>\phi^{-1}</math> apart. The <math>\sqrt{4}</math> chords occur as 60 long diameters (75 sets of 4 orthogonal axes with each set comprising a [[16-cell#Coordinates|16-cell]]), the 120 long radii of the 600-cell. The <math>\sqrt{4}</math> chords join opposite vertices which are five <math>\phi^{-1}</math> edges apart on a geodesic great circle. There are 25 distinct but overlapping sets of 12 diameters, each comprising one of the 25 inscribed 24-cells.{{Efn|name=Schoute's ten ways to get five disjoint 24-cells}} There are 75 distinct but overlapping sets of 4 orthogonal diameters, each comprising one of the 75 inscribed 16-cells. The sum of the squared lengths of all these distinct chords of the 600-cell is 14,400 = 120².{{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}} In this case, <math>(2 - \phi) \cdot 720 + 1 \cdot 1200 + {}\!</math><math>(3 - \phi) \cdot 720 + 2 \cdot 1800 + {}\!</math><math>(1 + \phi)\cdot 720 + 3\cdot 1200 + {}\!</math><math>(2 + \phi) \cdot 720 + 4 \cdot 60</math> is 14,400.}} These are all the central polygons through vertices, but the 600-cell does have one noteworthy great circle that does not pass through any vertices (a 0-gon).{{Efn|Each great decagon central plane is [[W:completely orthogonal|completely orthogonal]] to a great 30-gon{{Efn|A ''[[W:triacontagon|triacontagon]]'' or 30-gon is a thirty-sided polygon. The triacontagon is the largest regular polygon whose interior angle is the sum of the [[W:Interior angle|interior angles]] of smaller polygons: 168° is the sum of the interior angles of the [[W:Equilateral triangle|equilateral triangle]] (60°) and the [[W:Regular pentagon|regular pentagon]] (108°).|name=triacontagon}} central plane which does not intersect any vertices of the 600-cell. The 72 30-gons are each the center axis of a 30-cell [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]],{{Efn|name=Boerdijk–Coxeter helix}} with each segment of the 30-gon passing through a tetrahedron similarly. The 30-gon great circle resides completely in the curved 3-dimensional surface of its 3-sphere;{{Efn|name=0-gon central planes}} its curved segments are not chords. It does not touch any edges or vertices, but it does hit faces. It is the central axis of a spiral skew 30-gram, the [[W:Petrie polygon|Petrie polygon]] of the 600-cell which links all 30 vertices of the 30-cell Boerdijk–Coxeter helix, with three of its edges in each cell.{{Efn|name=Triacontagram}}|name=non-vertex geodesic}} Moreover, 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 600-cell vertices that are helical rather than simply circular; they correspond to isoclinic (diagonal) [[#Rotations|rotations]] rather than simple rotations.{{Efn|name=isoclinic geodesic}} All the geodesic polygons enumerated above lie in central planes of just three kinds, each characterized by a rotation angle: decagon planes ({{sfrac|𝜋|5}} apart), hexagon planes ({{sfrac|𝜋|3}} apart, also in the 25 inscribed 24-cells), and square planes ({{sfrac|𝜋|2}} apart, also in the 75 inscribed 16-cells and the 24-cells). These central planes of the 600-cell can be divided into 4 orthogonal central hyperplanes (3-spaces) each forming an [[W:icosidodecahedron|icosidodecahedron]]. There are 450 great squares 90 degrees apart; 200 great hexagons 60 degrees apart; and 72 great decagons 36 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 are 0 degrees apart in one of the two angles, only one angle is required in 3-space. Great decagons are a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in each angle, and ''may'' be the same angle apart in ''both'' angles.{{Efn|The decagonal planes in the 600-cell occur in equi-isoclinic{{Efn|In 4-space no more than 4 great circles may be Clifford parallel{{Efn|name=Clifford parallels}} and all the same angular distance apart.{{Sfn|Lemmens & Seidel|1973}} Such central planes are mutually ''isoclinic'': each pair of planes is separated by two ''equal'' angles, and an isoclinic [[#Rotations|rotation]] by that angle will bring them together. Where three or four such planes are all separated by the ''same'' angle, they are called ''equi-isoclinic''.|name=equi-isoclinic planes}} groups of 3, everywhere that 3 Clifford parallel decagons 36° ({{sfrac|𝝅|5}}) apart form a 30-cell [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]].{{Efn|name=Boerdijk–Coxeter helix}} Also Clifford parallel to those 3 decagons are 3 equi-isoclinic decagons 72° ({{sfrac|2𝝅|5}}) apart, 3 108° ({{sfrac|3𝝅|5}}) apart, and 3 144° ({{sfrac|4𝝅|5}}) apart, for a total of 12 Clifford parallel [[#Decagons|decagons]] (120 vertices) that comprise a discrete Hopf fibration. Because the great decagons lie in isoclinic planes separated by ''two'' equal angles, their corresponding vertices are separated by a combined vector relative to ''both'' angles. Vectors in 4-space may be combined by [[W:Quaternion#Multiplication of basis elements|quaternionic multiplication]], discovered by [[W:William Rowan Hamilton|Hamilton]].{{Sfn|Mamone, Pileio & Levitt|2010|p=1433|loc=§4.1|ps=; A Cartesian 4-coordinate point (w,x,y,z) is a vector in 4D space from (0,0,0,0). Four-dimensional real space is a vector space: any two vectors can be added or multiplied by a scalar to give another vector. Quaternions extend the vectorial structure of 4D real space by allowing the multiplication of two 4D vectors <small><math>\left(w,x,y,z\right)_1</math></small> and <small><math>\left(w,x,y,z\right)_2</math></small> according to<br> <small><math display=block>\begin{pmatrix} w_2\\ x_2\\ y_2\\ z_2 \end{pmatrix} * \begin{pmatrix} w_1\\ x_1\\ y_1\\ z_1 \end{pmatrix} = \begin{pmatrix} {w_2 w_1 - x_2 x_1 - y_2 y_1 - z_2 z_1}\\ {w_2 x_1 + x_2 w_1 + y_2 z_1 - z_2 y_1}\\ {w_2 y_1 - x_2 z_1 + y_2 w_1 + z_2 x_1}\\ {w_2 z_1 + x_2 y_1 - y_2 x_1 + z_2 w_1} \end{pmatrix} </math></small>}} The corresponding vertices of two great polygons which are 36° ({{sfrac|𝝅|5}}) apart by isoclinic rotation are 60° ({{sfrac|𝝅|3}}) apart in 4-space. The corresponding vertices of two great polygons which are 108° ({{sfrac|3𝝅|5}}) apart by isoclinic rotation are also 60° ({{sfrac|𝝅|3}}) apart in 4-space. The corresponding vertices of two great polygons which are 72° ({{sfrac|2𝝅|5}}) apart by isoclinic rotation are 120° ({{sfrac|2𝝅|3}}) apart in 4-space, and the corresponding vertices of two great polygons which are 144° ({{sfrac|4𝝅|5}}) apart by isoclinic rotation are also 120° ({{sfrac|2𝝅|3}}) apart in 4-space.|name=equi-isoclinic decagons}} Great hexagons may be 60° ({{sfrac|𝝅|3}}) apart in one or ''both'' angles, and may be a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in one or ''both'' angles.{{Efn|The hexagonal planes in the 600-cell occur in equi-isoclinic{{Efn|name=equi-isoclinic planes}} groups of 4, everywhere that 4 Clifford parallel hexagons 60° ({{sfrac|𝝅|3}}) apart form a 24-cell. Also Clifford parallel to those 4 hexagons are 4 equi-isoclinic hexagons 36° ({{sfrac|𝝅|5}}) apart, 4 72° ({{sfrac|2𝝅|5}}) apart, 4 108° ({{sfrac|3𝝅|5}}) apart, and 4 144° ({{sfrac|4𝝅|5}}) apart, for a total of 20 Clifford parallel [[#Hexagons|hexagons]] (120 vertices) that comprise a discrete Hopf fibration.|name=equi-isoclinic hexagons}} Great squares may be 90° ({{sfrac|𝝅|2}}) apart in one or both angles, may be 60° ({{sfrac|𝝅|3}}) apart in one or both angles, and may be a multiple (from 0 to 4) of 36° ({{sfrac|𝝅|5}}) apart in one or both angles.{{Efn|The square planes in the 600-cell occur in equi-isoclinic{{Efn|name=equi-isoclinic planes}} groups of 2, everywhere that 2 Clifford parallel squares 90° ({{sfrac|𝝅|2}}) apart form a 16-cell. Also Clifford parallel to those 2 squares are 4 equi-isoclinic groups of 4, where 3 Clifford parallel 16-cells 60° ({{sfrac|𝝅|3}}) apart form a 24-cell. Also Clifford parallel are 4 equi-isoclinic groups of 3: 3 36° ({{sfrac|𝝅|5}}) apart, 3 72° ({{sfrac|2𝝅|5}}) apart, 3 108° ({{sfrac|3𝝅|5}}) apart, and 3 144° ({{sfrac|4𝝅|5}}) apart, for a total of 30 Clifford parallel [[#Squares|squares]] (120 vertices) that comprise a discrete Hopf fibration.|name=equi-isoclinic squares}} Planes which are separated by two equal angles are called ''[[24-cell#Clifford parallel polytopes|isoclinic]]''.{{Efn|name=equi-isoclinic planes}} Planes which are isoclinic have [[W:Clifford parallel|Clifford parallel]] great circles.{{Efn|name=Clifford parallels}} A great hexagon and a great decagon ''may'' be isoclinic, but more often they are separated by a {{sfrac|𝝅|3}} (60°) angle ''and'' a multiple (from 1 to 4) of {{sfrac|𝝅|5}} (36°) angle.|name=two angles between central planes}} Each great square plane is [[W:Completely orthogonal|completely orthogonal]] to another great square plane.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Each great hexagon plane is completely orthogonal to a plane which intersects only two vertices (one <math>\sqrt{4}</math> long diameter): a great [[W:digon|digon]] plane.{{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=digon planes}} Each great decagon plane is completely orthogonal to a plane which intersects ''no'' vertices: a great 0-gon plane.{{Efn|The 600-cell has 72 great 30-gons: 6 sets of 12 Clifford parallel 30-gon central planes, each completely orthogonal to a decagon central plane. Unlike the great circles of the unit-radius 600-cell that pass through its vertices, this 30-gon is not actually a great circle of the unit-radius 3-sphere. Because it passes through face centers rather than vertices, it has a shorter radius and lies on a smaller 3-sphere. Of course, there is also a unit-radius great circle in this central plane completely orthogonal to a decagon central plane, but as a great circle polygon it is a 0-gon, not a 30-gon, because it intersects ''none'' of the points of the 600-cell. In the 600-cell, the great circle polygon completely orthogonal to each great decagon is a 0-gon. |name=0-gon central planes}} ==== Fibrations of great circle polygons ==== Each set of similar great circle polygons (squares or hexagons or decagons) can be divided into bundles of non-intersecting Clifford parallel great circles (of 30 squares or 20 hexagons or 12 decagons).{{Efn|name=Clifford parallels}} Each [[W:fiber bundle|fiber bundle]] of Clifford parallel great circles{{Efn|name=equi-isoclinic planes}} is a discrete [[W:Hopf fibration|Hopf fibration]] which fills the 600-cell, visiting all 120 vertices just once.{{Sfn|Sadoc|2001|pp=575-578|loc=§2 Geometry of the {3,3,5}-polytope in S₃|ps=; Sadoc studied all the Hopf fibrations of the 600-cell into sets of {4}, {6} or {10} great circle fibers on different screw axes, gave their Hopf maps, and fully illustrated the characteristic decagonal cell rings.}} Each discrete Hopf fibration has its 3-dimensional ''base'' which is a distinct polyhedron that acts as a ''map'' or scale model of the fibration.{{Efn|name=Hopf fibration base}} The great circle polygons in each bundle spiral around each other, delineating helical rings of face-bonded cells which nest into each other, pass through each other without intersecting in any cells and exactly fill the 600-cell with their disjoint cell sets. The different fiber bundles with their cell rings each fill the same space (the 600-cell) but their fibers run Clifford parallel in different "directions"; great circle polygons in different fibrations are not Clifford parallel.{{Sfn|Tyrrell & Semple|1971|loc=§4. Isoclinic planes in Euclidean space E₄|pp=6-7}} ===== Decagons ===== [[File:Regular_star_figure_6(5,2).svg|thumb|200px|[[W:Triacontagon#Triacontagram|Triacontagram {30/12}=6{5/2}]] is the [[Schläfli double six|Schläfli double six]] configuration 30₂12₅ characteristic of the H₄ polytopes. The 30 vertex circumference is the skew Petrie polygon.{{Efn|name=Petrie polygons of the 120-cell}} The interior angle between adjacent edges is 36°, also the isoclinic angle between adjacent Clifford parallel decagon planes.{{Efn|name=two angles between central planes}}]] The fibrations of the 600-cell include 6 fibrations of its 72 great decagons: 6 fiber bundles of 12 great decagons.{{Efn|name=Clifford parallel decagons}} The 12 Clifford parallel decagons in each bundle are completely disjoint. Adjacent parallel decagons are spanned by edges of other great decagons. Each fiber bundle{{Efn|name=equi-isoclinic decagons}} delineates [[#Boerdijk–Coxeter helix rings|20 helical rings]] of 30 tetrahedral cells each,{{Efn|name=Boerdijk–Coxeter helix}} with five rings nesting together around each decagon.{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} The Hopf map of this fibration is the [[W:icosahedron|icosahedron]], where each of 12 vertices lifts to a great decagon, and each of 20 triangular faces lifts to a 30-cell ring.{{Efn|name=Hopf fibration base}} Each tetrahedral cell occupies only one of the 20 cell rings in each of the 6 fibrations. The tetrahedral cell contributes each of its 6 edges to a decagon in a different fibration, but contributes that edge to five distinct cell rings in the fibration.{{Efn|name=tetrahedron linking 6 decagons}} The 12 great circles and [[#Boerdijk–Coxeter helix rings|30-cell ring]]s of the 600-cell's 6 characteristic [[W:Hopf fibration|Hopf fibration]]s make the 600-cell a [[W:Configuration (geometry)|geometric configuration]] of 30 "points" and 12 "lines" written as 30₂12₅. It is called the [[Schläfli double six|Schläfli double six]] configuration after [[W:Ludwig Schläfli|Ludwig Schläfli]],{{Sfn|Schläfli|1858|ps=; this paper of Schläfli's describing the [[Schläfli double six|double six configuration]] was one of the only fragments of his discovery of the [[W:Regular polytopes (book)|regular polytopes]] in higher dimensions to be published during his lifetime.{{Sfn|Coxeter|1973|p=211|loc=§11.x Historical remarks|ps=; "The finite group [3^{2, 2, 1}] is isomorphic with the group of incidence-preserving permutations of the 27 lines on the general cubic surface. (For the earliest description of these lines, see Schlafli 2.)".}}}} the Swiss mathematician who discovered the 600-cell and the complete set of regular polytopes in ''n'' dimensions.{{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."}} ===== Hexagons ===== The [[24-cell#Cell rings|fibrations of the 24-cell]] include 4 fibrations of its 16 great hexagons: 4 fiber bundles of 4 great hexagons. The 4 Clifford parallel hexagons in each bundle are completely disjoint. Adjacent parallel hexagons are spanned by edges of other great hexagons. Each fiber bundle delineates 4 helical rings of 6 octahedral cells each, with three rings nesting together around each hexagon. Each octahedral cell occupies only one cell ring in each of the 4 fibrations. The octahedral cell contributes 3 of its 12 edges to 3 different Clifford parallel hexagons in each fibration, but contributes each edge to three distinct cell rings in the fibration. The 600-cell contains 25 24-cells, and can be seen (10 different ways) as a compound of 5 disjoint 24-cells.{{Efn|name=24-cells bound by pentagonal fibers}} It has 10 fibrations of its 200 great hexagons: 10 fiber bundles of 20 great hexagons. The 20 Clifford parallel hexagons in each bundle are completely disjoint. Adjacent parallel hexagons are spanned by edges of great decagons.{{Efn|name=equi-isoclinic hexagons}} Each fiber bundle delineates 20 helical rings of 6 octahedral cells each, with three rings nesting together around each hexagon. The Hopf map of this fibration is the [[W:dodecahedron|dodecahedron]], where the 20 vertices each lift to a bundle of great hexagons.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} Each octahedral cell occupies only one of the 20 6-octahedron rings in each of the 10 fibrations. The 20 6-octahedron rings belong to 5 disjoint 24-cells of 4 6-octahedron rings each; each hexagonal fibration of the 600-cell consists of 5 disjoint 24-cells. ===== Squares ===== The [[16-cell#Helical construction|fibrations of the 16-cell]] include 3 fibrations of its 6 great squares: 3 fiber bundles of 2 great squares. The 2 Clifford parallel squares in each bundle are completely disjoint. Adjacent parallel squares are spanned by edges of other great squares. Each fiber bundle delineates 2 helical rings of 8 tetrahedral cells each. Each tetrahedral cell occupies only one cell ring in each of the 3 fibrations. The tetrahedral cell contributes each of its 6 edges to a different square (contributing two opposite non-intersecting edges to each of the 3 fibrations), but contributes each edge to both of the two distinct cell rings in the fibration. The 600-cell contains 75 16-cells, and can be seen (10 different ways) as a compound of 15 disjoint 16-cells. It has 15 fibrations of its 450 great squares: 15 fiber bundles of 30 great squares. The 30 Clifford parallel squares in each bundle are completely disjoint. Adjacent parallel squares are spanned by edges of great decagons.{{Efn|name=equi-isoclinic squares}} Each fiber bundle delineates 30 cell-disjoint helical rings of 8 tetrahedral cells each.{{Efn|These are the <math>\sqrt{2}</math> tetrahedral cells of the 75 inscribed 16-cells, ''not'' the <math>\phi^{-1}</math> tetrahedral cells of the 600-cell.|name=two different tetrahelixes}} The Hopf map of this fibration is the [[W:icosidodecahedron|icosidodecahedron]], where the 30 vertices each lift to a bundle of great squares.{{Sfn|Sadoc|2001|p=577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the four-fold screw axis}} Each tetrahedral cell occupies only one of the 30 8-tetrahedron rings in each of the 15 fibrations. ===== Clifford parallel cell rings ===== The densely packed helical cell rings{{Sfn|Coxeter|1970|ps=, 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]].{{Efn|name=orthoscheme ring}} He found that cell rings follow [[W:Petrie polygon|Petrie polygon]]s 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 the regular 4-polytopes with tetrahedral cells (5-cell, 16-cell, 600-cell) have twisted cell rings, and the others (whose cells have opposing faces) do not.{{Efn|name=directly congruent versus twisted cell rings}} Separately, he categorized cell rings by whether they form their honeycombs in hyperbolic or Euclidean space, the latter being those found in the 4-polytopes which can tile 4-space by translation to form Euclidean honeycombs (16-cell, 8-cell, 24-cell).}}{{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 decompositions composed of meridian and equatorial cell rings with illustrations.}}{{Sfn|Sadoc|2001|pp=575-578|loc=§2 Geometry of the {3,3,5}-polytope in S₃|ps=; Sadoc studied all the Hopf fibrations of the 600-cell into sets of {4}, {6} or {10} great circle fibers on different screw axes, gave their Hopf maps, and fully illustrated the characteristic decagonal cell rings.}} of fibrations are cell-disjoint, but they share vertices, edges and faces. Each fibration of the 600-cell can be seen as a dense packing of cell rings with the corresponding faces of adjacent cell rings face-bonded to each other.{{Efn|name=fibrations are distinguished only by rotations}} The same fibration can also be seen as a minimal ''sparse'' arrangement of fewer ''completely disjoint'' cell rings that do not touch at all.{{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}} The fibrations of great decagons can be seen (five different ways) as 4 completely disjoint 30-cell rings with spaces separating them, rather than as 20 face-bonded cell rings, by leaving out all but one cell ring of the five that meet at each decagon.{{Sfn|Sadoc|2001|loc=§2.6 The {3, 3, 5} polytope: a set of four helices|p=578}} The five different ways you can do this are equivalent, in that all five correspond to the same discrete fibration (in the same sense that the 6 decagonal fibrations are equivalent, in that all 6 cover the same 600-cell). The 4 cell rings still constitute the complete fibration: they include all 12 Clifford parallel decagons, which visit all 120 vertices.{{Efn|The only way to partition the 120 vertices of the 600-cell into 4 completely disjoint 30-vertex, 30-cell rings{{Efn|name=Boerdijk–Coxeter helix}} is by partitioning each of 15 completely disjoint 16-cells similarly into 4 symmetric parts: 4 antipodal vertex pairs lying on the 4 orthogonal axes of the 16-cell. The 600-cell contains 75 distinct 16-cells which can be partitioned into sets of 15 completely disjoint 16-cells. In any set of 4 completely disjoint 30-cell rings, there is a set of 15 completely disjoint 16-cells, with one axis of each 16-cell in each 30-cell ring.|name=fifteen 16-cells partitioned among four 30-cell rings}} This subset of 4 of 20 cell rings is dimensionally analogous{{Efn|One might ask whether dimensional analogy "always works", or if it is perhaps "just guesswork" that might sometimes be incapable of producing a correct dimensionally analogous figure, especially when reasoning from a lower to a higher dimension. Apparently dimensional analogy in both directions has firm mathematical foundations. Dechant{{Sfn|Dechant|2021|loc=§1. Introduction}} derived the 4D symmetry groups from their 3D symmetry group counterparts by induction, demonstrating that there is nothing in 4D symmetry that is not already inherent in 3D symmetry. He showed that neither 4D symmetry nor 3D symmetry is more fundamental than the other, as either can be derived from the other. This is true whether dimensional analogies are computed using Coxeter group theory, or Clifford geometric algebra. These two rather different kinds of mathematics contribute complementary geometric insights. Another profound example of dimensional analogy mathematics is the [[W:Hopf fibration|Hopf fibration]], a mapping between points on the 2-sphere and disjoint (Clifford parallel) great circles on the 3-sphere.|name=math of dimensional analogy}} to the subset of 12 of 72 decagons, in that both are sets of completely disjoint [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]] which visit all 120 vertices.{{Efn|Unlike their bounding decagons, the 20 cell rings themselves are ''not'' all Clifford parallel to each other, because only completely disjoint polytopes are Clifford parallel.{{Efn|name=completely disjoint}} The 20 cell rings have 5 different subsets of 4 Clifford parallel cell rings. Each cell ring is bounded by 3 Clifford parallel great decagons, so each subset of 4 Clifford parallel cell rings is bounded by a total of 12 Clifford parallel great decagons (a discrete Hopf fibration). In fact each of the 5 different subsets of 4 cell rings is bounded by the ''same'' 12 Clifford parallel great decagons (the same Hopf fibration); there are 5 different ways to see the same 12 decagons as a set of 4 cell rings (and equivalently, just one way to see them as a single set of 20 cell rings).}} The subset of 4 of 20 cell rings is one of 5 fibrations ''within'' the fibration of 12 of 72 decagons: a fibration of a fibration. All the fibrations have this two level structure with ''subfibrations''. The fibrations of the 24-cell's great hexagons can be seen (three different ways) as 2 completely disjoint 6-cell rings with spaces separating them, rather than as 4 face-bonded cell rings, by leaving out all but one cell ring of the three that meet at each hexagon. Therefore each of the 10 fibrations of the 600-cell's great hexagons can be seen as 2 completely disjoint octahedral cell rings. The fibrations of the 16-cell's great squares can be seen (two different ways) as a single 8-tetrahedral-cell ring with an adjacent cell-ring-sized empty space, rather than as 2 face-bonded cell rings, by leaving out one of the two cell rings that meet at each square. Therefore each of the 15 fibrations of the 600-cell's great squares can be seen as a single tetrahedral cell ring.{{Efn|name=two different tetrahelixes}} The sparse constructions of the 600-cell's fibrations correspond to lower-symmetry decompositions of the 600-cell, 24-cell or [[16-cell#Helical construction|16-cell]] with cells of different colors to distinguish the cell rings from the spaces between them.{{Efn|Note that the differently colored helices of cells are different cell rings (or ring-shaped holes) in the same fibration, ''not'' the different fibrations of the 4-polytope. Each fibration is the entire 4-polytope.}} The particular lower-symmetry form of the 600-cell corresponding to the sparse construction of the great decagon fibrations is dimensionally analogous{{Efn|name=math of dimensional analogy}} to the [[W:Icosahedron#Pyritohedral symmetry|snub tetrahedron]] form of the icosahedron (which is the ''base''{{Efn|Each [[W:Hopf fibration|Hopf fibration]] of the 3-sphere into Clifford parallel great circle fibers has a map (called its ''base'') which is an ordinary [[W:2-sphere#Dimensionality|2-sphere]].{{Sfn|Zamboj|2021}} On this map each great circle fiber appears as a single point. The base of a great decagon fibration of the 600-cell is the [[W:icosahedron|icosahedron]], in which each vertex represents one of the 12 great decagons.{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} To a toplogist the base is not necessarily any part of the thing it maps: the base icosahedron is not expected to be a cell or interior feature of the 600-cell, it is merely the dimensionally analogous sphere,{{Efn|name=math of dimensional analogy}} useful for reasoning about the fibration. But in fact the 600-cell does have [[#Icosahedra|icosahedra]] in it: 120 icosahedral [[W:Vertex figure|vertex figure]]s,{{Efn|name=vertex icosahedral pyramid}} any of which can be seen as its base: a 3-dimensional 1:10 scale model of the whole 4-dimensional 600-cell. Each 3-dimensional vertex icosahedron is ''lifted'' to the 4-dimensional 600-cell by a 720 degree [[24-cell#Isoclinic rotations|isoclinic rotation]],{{Efn|name=isoclinic geodesic}} which takes each of its 4 disjoint triangular faces in a circuit around one of 4 disjoint 30-vertex [[#Boerdijk–Coxeter helix rings|rings of 30 tetrahedral cells]] (each [[W:Braid|braid]]ed of 3 Clifford parallel great decagons), and so visits all 120 vertices of the 600-cell, ''generating'' the 600-cell. Since the 12 decagonal great circles (of the 4 rings) are Clifford parallel [[#Decagons|decagons of the same fibration]], we can see geometrically how the icosahedron works as a map of a Hopf fibration of the entire 600-cell, and how the Hopf fibration's characteristic isoclinic rotation generates the 600-cell, since the Hopf fibration is an expression of an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic symmetry]].{{Efn|Sadoc studied twisted 3-dimensional molecules, and imagined them embedded in 4-dimensional space, as the the Hopf fibrations of regular 4-polytopes. He found that these molecules would close-pack perfectly in 4-space without exhibiting any torsion, although their packing in 3-space was imperfect, "frustrated" by their torsion. <blockquote>The frustration, which arises when the molecular orientation is transported along the two [circular] 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]]....{{Efn|name=Petrie polygon of a honeycomb}} 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=dense fabric of pole-circles}} 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>|name=Sadoc frustration}}|name=Hopf fibration base}} of these fibrations on the 2-sphere). Each of the 20 [[#Boerdijk–Coxeter helix rings|Boerdijk-Coxeter cell rings]]{{Efn|name=Boerdijk–Coxeter helix}} is ''lifted'' from a corresponding ''face'' of the icosahedron.{{Efn|The 4 {{Background color|red}} faces of the [[W:Icosahedron#Pyritohedral symmetry|snub tetrahedron]] correspond to the 4 completely disjoint cell rings of the sparse construction of the fibration (its ''subfibration''). The red faces are centered on the vertices of an inscribed tetrahedron, and lie in the center of the larger faces of an inscribing tetrahedron.}} === Constructions === The 600-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, the 120-cell, and the polygons {7} and above.{{Sfn|Coxeter|1973|loc=Table VI (iii): 𝐈𝐈 = {3,3,5}|p=303}} Consequently, there are numerous ways to construct or deconstruct the 600-cell, but none of them are trivial. The construction of the 600-cell from its regular predecessor the 24-cell can be difficult to visualize. ==== Gosset's construction ==== [[W:Thorold Gosset|Thorold Gosset]] discovered the [[W:Semiregular polytope|semiregular 4-polytopes]], including the [[W:Snub 24-cell|snub 24-cell]] with 96 vertices, which falls between the 24-cell and the 600-cell in the sequence of convex 4-polytopes of increasing size and complexity in the same radius. Gosset's construction of the 600-cell from the 24-cell is in two steps, using the snub 24-cell as an intermediate form. In the first, more complex step (described [[W:Snub 24-cell#Constructions|elsewhere]]) the snub 24-cell is constructed by a special snub truncation of a 24-cell at the [[W:Golden ratio|golden sections]] of its edges.{{Sfn|Coxeter|1973|loc=§8.4 The snub {3,4,3}|pp=151-153}} In the second step the 600-cell is constructed in a straightforward manner by adding 4-pyramids (vertices) to facets of the snub 24-cell.{{Sfn|Coxeter|1973|loc=§8.5 Gosset's construction for {3,3,5}|p=153}} The snub 24-cell is a diminished 600-cell from which 24 vertices (and the cluster of 20 tetrahedral cells around each) have been truncated,{{Efn|name=snub 24-cell}} leaving a "flat" icosahedral cell in place of each removed icosahedral pyramid.{{Efn|name=vertex icosahedral pyramid}} The snub 24-cell thus has 24 icosahedral cells and the remaining 120 tetrahedral cells. The second step of Gosset's construction of the 600-cell is simply the reverse of this diminishing: an icosahedral pyramid of 20 tetrahedral cells is placed on each icosahedral cell. Constructing the unit-radius 600-cell from its precursor the unit-radius 24-cell by Gosset's method actually requires ''three'' steps. The 24-cell precursor to the snub-24 cell is ''not'' of the same radius: it is larger, since the snub-24 cell is its truncation. Starting with the unit-radius 24-cell, the first step is to reciprocate it around its [[W:Midsphere|midsphere]] to construct its outer [[W:Dual polyhedra#Canonical duals|canonical dual]]: a larger 24-cell, since the 24-cell is self-dual. That larger 24-cell can then be snub truncated into a unit-radius snub 24-cell. ==== Cell clusters ==== Since it is so indirect, Gosset's construction may not help us very much to directly visualize how the 600 tetrahedral cells fit together into a curved 3-dimensional [[#Boundary envelopes|surface envelope]], or how they lie on the underlying surface envelope of the 24-cell's octahedral cells. For that it is helpful to build up the 600-cell directly from clusters of tetrahedral cells.{{Efn|name=tetrahedral cell adjacency}} Most of us have difficulty [[#Visualization|visualizing]] the 600-cell ''from the outside'' in 4-space, or recognizing an [[#3D projections|outside view]] of the 600-cell due to our total lack of sensory experience in 4-dimensional spaces,{{Sfn|Borovik|2006|ps=; "The environment which directed the evolution of our brain never provided our ancestors with four-dimensional experiences.... [Nevertheless] we humans are blessed with a remarkable piece of mathematical software for image processing hardwired into our brains. Coxeter made full use of it, and expected the reader to use it.... Visualization is one of the most powerful interiorization techniques. It anchors mathematical concepts and ideas into one of the most powerful parts of our brain, the visual processing module. Coxeter Theory [of polytopes generated by] finite reflection groups allow[s] an approach to their study based on a systematic reduction of complex geometric configurations to much simpler two- and three-dimensional special cases."}} but we should be able to visualize the surface envelope of 600 cells ''from the inside'' because that volume is a 3-dimensional honeycomb{{Sfn|Coxeter|1970|loc=§9. The 120-cell and the 600-cell|p=19}} that we could actually "walk around in" and explore.{{Sfn|Miyazaki|1990|ps=; Miyazaki showed that the surface envelope of the 600-cell can be realized architecturally in our ordinary 3-dimensional space as physical buildings (geodesic domes).}} In these exercises of building the 600-cell up from cell clusters, we are entirely within a 3-dimensional space, albeit a strangely small, [[W:Elliptic geometry#Hyperspherical model|closed curved space]], in which we can go a mere ten edge lengths away in a straight line in any direction and return to our starting point. ===== Icosahedra ===== [[File:Uniform polyhedron-43-h01.svg|thumb|A regular icosahedron colored in [[W:Regular icosahedron#Symmetries|snub octahedron]] symmetry.{{Efn|Because the octahedron can be [[W:Snub (geometry)|snub truncated]] yielding an icosahedron,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7}} another name for the icosahedron is [[W:Regular icosahedron#Symmetries|snub octahedron]]. This term refers specifically to a [[W:Icosahedron#Pyritohedral symmetry|lower symmetry]] arrangement of the icosahedron's faces (with 8 faces of one color and 12 of another).|name=snub octahedron}} Icosahedra in the 600-cell are face bonded to each other at the yellow faces, and to clusters of 5 tetrahedral cells at the blue faces. The apex of the [[W:Icosahedral pyramid|icosahedral pyramid]] (not visible) is a 13th 600-cell vertex inside the icosahedron (but above its hyperplane).|alt=|200x200px]] [[File:5-cell net.png|thumb|A cluster of 5 tetrahedral cells: four cells face-bonded around a fifth cell (not visible). The four cells lie in different hyperplanes.|alt=|200x200px]] The [[W:Vertex figure|vertex figure]] of the 600-cell is the [[W:Icosahedron|icosahedron]].{{Efn|In the curved 3-dimensional space of the 600-cell's boundary surface, at each vertex one finds the twelve nearest other vertices surrounding the vertex the way an icosahedron's vertices surround its center. Twelve 600-cell edges converge at the icosahedron's center, where they appear to form six straight lines which cross there. However, the center is actually displaced in the 4th dimension (radially outward from the center of the 600-cell), out of the hyperplane defined by the icosahedron's vertices. Thus the vertex icosahedron is actually a canonical [[W:Icosahedral pyramid|icosahedral pyramid]],{{Efn|name=120 overlapping icosahedral pyramids}} composed of 20 regular tetrahedra on a regular icosahedron base, and the vertex is its apex.{{Efn|name=radially equilateral icosahedral pyramid}}|name=vertex icosahedral pyramid|group=}} Twenty tetrahedral cells meet at each vertex, forming an [[W:Icosahedral pyramid|icosahedral pyramid]] whose apex is the vertex, surrounded by its base icosahedron. It is remarkable that twenty regular tetrahedra fit inside a regular icosahedral pyramid in 4-space. In 3-space, twenty triangular pyramids fit inside a regular icosahedron around its center but they are ''not'' regular tetrahedra, because the regular icosahedron's radius is not the same as its edge length.{{Efn|In Euclidean 3-space, the icosahedron is not [[W:Cuboctahedron#Radial equilateral symmetry|radially equilateral like the cuboctahedron]]. The icosahedron's radii are shorter than its edge length. But in the [[W:3-sphere|spherical 3-space]] of the 600-cell's surface the center of a regular icosahedron is lifted orthogonally out of its 3-space hyperplane: remarkably, just far enough to make its radii the same length as its edges. As a figure in Euclidean 4-space, this radially equilateral spherical icosahedron is an [[W:Icosahedral pyramid|icosahedral pyramid]]. In 4-space the 12 edges radiating from its apex are not actually its radii: the apex of the icosahedral pyramid is not its center, just one of its vertices. But in curved 3-space the 12 edges radiating symmetrically from the apex ''are'' radii, so the icosahedron is radially equilateral ''in that spherical space''. In Euclidean 4-space there are only two radially equilateral figures: 24 edges radiating symmetrically from a central point make the [[24-cell#Tetrahedral constructions|radially equilateral 24-cell]], and a symmetrical subset of 16 of those edges make the [[W:tesseract#Radial equilateral symmetry|radially equilateral tesseract]].|name=radially equilateral icosahedral pyramid}} The 600-cell has a [[W:Dihedral angle|dihedral angle]] of {{nowrap|{{sfrac|𝜋|3}} + arccos(−{{sfrac|1|4}}) ≈ 164.4775°}}.{{Sfn|Coxeter|1973|p=293|ps=; 164°29'}} An entire 600-cell can be assembled from 24 such icosahedral pyramids (bonded face-to-face at 8 of the 20 faces of the icosahedron, colored yellow in the illustration), plus 24 clusters of 5 tetrahedral cells (four cells face-bonded around one) which fill the voids remaining between the icosahedra. Each icosahedron is face-bonded to each adjacent cluster of 5 cells by two blue faces that share an edge (which is also one of the six edges of the central tetrahedron of the five). Six clusters of 5 cells surround each icosahedron, and six icosahedra surround each cluster of 5 cells. Five tetrahedral cells surround each icosahedron edge: two from inside the icosahedral pyramid, and three from outside it.{{Efn|An icosahedron edge between two blue faces is surrounded by two blue-faced icosahedral pyramid cells and 3 cells from an adjacent cluster of 5 cells (one of which is the central tetrahedron of the five)}} The apexes of the 24 icosahedral pyramids are the vertices of a 24-cell inscribed in the 600-cell. The other 96 vertices (the vertices of the icosahedra) are the vertices of an inscribed [[W:Snub 24-cell|snub 24-cell]], which has exactly the same [[W:Snub 24-cell#Structure|structure]] of icosahedra and tetrahedra described here, except that the icosahedra are not 4-pyramids filled by tetrahedral cells; they are only "flat" 3-dimensional icosahedral cells, because the central apical vertex is missing. The 24-cell edges joining icosahedral pyramid apex vertices run through the centers of the yellow faces. Coloring the icosahedra with 8 yellow and 12 blue faces can be done in 5 distinct ways.{{Efn|The pentagonal pyramids around each vertex of the "[[W:Regular icosahedron#Symmetries|snub octahedron]]" icosahedron all look the same, with two yellow and three blue faces. Each pentagon has five distinct rotational orientations. Rotating any pentagonal pyramid rotates all of them, so the five rotational positions are the only five different ways to arrange the colors.}} Thus each icosahedral pyramid's apex vertex is a vertex of 5 distinct 24-cells,{{Efn|Five 24-cells meet at each icosahedral pyramid apex{{Efn|name=vertex icosahedral pyramid}} of the 600-cell. Each 24-cell shares not just one vertex but 6 vertices (one of its four hexagonal central planes) with each of the other four 24-cells.{{Efn|name=disjoint from 8 and intersects 16}}|name=five 24-cells at each vertex of 600-cell}} and the 120 vertices comprise 25 (not 5) 24-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}} The icosahedra are face-bonded into geodesic "straight lines" by their opposite yellow faces, bent in the fourth dimension into a ring of 6 icosahedral pyramids. Their apexes are the vertices of a [[24-cell#Great hexagons|great circle hexagon]]. This hexagonal geodesic traverses a ring of 12 tetrahedral cells, alternately bonded face-to-face and vertex-to-vertex. The long diameter of each face-bonded pair of tetrahedra (each [[W:Triangular bipyramid|triangular bipyramid]]) is a hexagon edge (a 24-cell edge). There are 4 rings of 6 icosahedral pyramids intersecting at each apex-vertex, just as there are 4 cell-disjoint interlocking [[24-cell#Cell rings|rings of 6 octahedra]] in the 24-cell (a [[#Hexagons|hexagonal fibration]]).{{Efn|There is a vertex icosahedron{{Efn|name=vertex icosahedral pyramid}} inside each 24-cell octahedral central section (not inside a <math>\sqrt{1}</math>octahedral cell, but in the larger <math>\sqrt{2}</math> octahedron that lies in a central hyperplane), and a larger icosahedron inside each 24-cell cuboctahedron. The two different-sized icosahedra are the second and fourth [[#Polyhedral sections|sections of the 600-cell (beginning with a vertex)]]. The octahedron and the cuboctahedron are the central sections of the 24-cell (beginning with a vertex and beginning with a cell, respectively).{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections}} The cuboctahedron, large icosahedron, octahedron, and small icosahedron nest like [[W:Russian dolls|Russian dolls]] and are related by a helical contraction.{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids}} The contraction begins with the square faces of the cuboctahedron folding inward along their diagonals to form pairs of triangles.{{Efn|Notice that the contraction is chiral, since there are two choices of diagonal on which to begin folding the square faces.}} The 12 vertices of the cuboctahedron move toward each other to the points where they form a regular icosahedron (the large icosahedron); they move slightly closer together until they form a [[W:Jessen's icosahedron|Jessen's icosahedron]]; they continue to spiral toward each other until they merge into the 6 vertices of the octahedron;{{Sfn|Itoh & Nara|2021|loc=§4. From the 24-cell onto an octahedron|ps=; "This article addresses the 24-cell and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the [[W:Jitterbug transformation|Jitterbug]] by [[W:Buckminster Fuller|Buckminster Fuller]]."}} and they continue moving along the same helical paths, separating again into the 12 vertices of the snub octahedron (the small icosahedron).{{Efn|name=snub octahedron}} The geometry of this sequence of transformations{{Efn|These transformations are not among the orthogonal transformations of the Coxeter groups generated by reflections.{{Efn|name=transformations}} They are transformations of the [[W:Tetrahedral symmetry#Pyritohedral symmetry|pyritohedral 3D symmetry group]], the unique polyhedral point group that is neither a rotation group nor a reflection group.{{Sfn|Koca et. al.|2016|loc=4. Pyritohedral Group and Related Polyhedra|p=145|ps=; see Table 1.}}}} in [[W:3-sphere|S³]] is similar to the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]] and the [[W:Tensegrity#Tensegrity icosahedra|tensegrity icosahedron]] in [[W:Three-dimensional space|R³]]. The twisting, expansive-contractive transformations between these polyhedra were named [[Kinematics of the cuboctahedron#Jitterbug transformations|Jitterbug transformations]] by [[W:Buckminster Fuller|Buckminster Fuller]].<ref>{{cite journal | last = Verheyen | first = H. F. | doi = 10.1016/0898-1221(89)90160-0 | issue = 1–3 | journal = [[W:Computers and Mathematics with Applications|Computers and Mathematics with Applications]] | mr = 0994201 | pages = 203–250 | title = The complete set of Jitterbug transformers and the analysis of their motion | volume = 17 | year = 1989| doi-access = free }}</ref>}} The tetrahedral cells are face-bonded into [[W:Boerdijk-Coxeter helix|triple helices]], bent in the fourth dimension into [[#Boerdijk–Coxeter helix rings|rings of 30 tetrahedral cells]].{{Efn|Since tetrahedra{{Efn|name=tetrahedron linking 6 decagons}} do not have opposing faces, the only way they can be stacked face-to-face in a straight line is in the form of a twisted chain called a [[W:Boerdijk-Coxeter helix|Boerdijk-Coxeter helix]]. This is a Clifford parallel{{Efn|name=Clifford parallels}} triple helix as shown in the [[#Boerdijk–Coxeter helix rings|illustration]]. In the 600-cell we find them bent in the fourth dimension into geodesic rings. Each ring has 30 cells and touches 30 vertices. The cells are each face-bonded to two adjacent cells, but one of the six edges of each tetrahedron belongs only to that cell, and these 30 edges form 3 Clifford parallel great decagons which spiral around each other.{{Efn|name=Clifford parallel decagons}} 5 30-cell rings meet at and spiral around each decagon (as 5 tetrahedra meet at each edge). A bundle of 20 such cell-disjoint rings fills the entire 600-cell, thus constituting a discrete [[W:Hopf fibration|Hopf fibration]]. There are 6 distinct such Hopf fibrations, covering the same space but running in different "directions".|name=Boerdijk–Coxeter helix}} The three helices are geodesic "straight lines" of 10 edges: [[#Hopf spherical coordinates|great circle decagons]] which run Clifford parallel{{Efn|name=Clifford parallels}} to each other. Each tetrahedron, having six edges, participates in six different decagons{{Efn|The six great decagons which pass by each tetrahedral cell along its edges do not all intersect with each other, because the 6 edges of the tetrahedron do not all share a vertex. Each decagon intersects four of the others (at 60 degrees), but just misses one of the others as they run past each other (at 90 degrees) along the opposite and perpendicular [[W:Skew lines|skew edges]] of the tetrahedron. Each tetrahedron links three pairs of decagons which do ''not'' intersect at a vertex of the tetrahedron. However, none of the six decagons are Clifford parallel;{{Efn|name=Clifford parallels}} each belongs to a different [[W:Hopf fibration|Hopf fiber bundle]] of 12. Only one of the tetrahedron's six edges may be part of a helix in any one [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple helix ring]].{{Efn|name=Boerdijk–Coxeter helix}} Incidentally, this footnote is one of a tetrahedron of four footnotes about Clifford parallel decagons{{Efn|name=Clifford parallel decagons}} that all reference each other.|name=tetrahedron linking 6 decagons}} and thereby in all 6 of the [[#Decagons|decagonal fibrations of the 600-cell]]. The partitioning of the 600-cell into clusters of 20 cells and clusters of 5 cells is artificial, since all the cells are the same. One can begin by picking out an icosahedral pyramid cluster centered at any arbitrarily chosen vertex, so there are 120 overlapping icosahedra in the 600-cell.{{Efn|The 120-point 600-cell has 120 overlapping icosahedral pyramids.{{Efn|name=vertex icosahedral pyramid}}|name=120 overlapping icosahedral pyramids}} Their 120 apexes are each a vertex of five 24-vertex 24-cells, so there are 5*120/24 = 25 overlapping 24-cells.{{Efn|name=24-cells bound by pentagonal fibers}} ===== Octahedra ===== There is another useful way to partition the 600-cell surface, into 24 clusters of 25 tetrahedral cells, which reveals more structure{{Sfn|Coxeter|1973|p=299|loc=Table V: (iv) Simplified sections of {3,3,5} ... beginning with a cell}} and a direct construction of the 600-cell from its predecessor the 24-cell. Begin with any one of the clusters of 5 cells (above), and consider its central cell to be the center object of a new larger cluster of tetrahedral cells. The central cell is the first section of the 600-cell beginning with a cell. By surrounding it with more tetrahedral cells, we can reach the deeper sections beginning with a cell. First, note that a cluster of 5 cells consists of 4 overlapping pairs of face-bonded tetrahedra ([[W:Triangular dipyramid|triangular dipyramid]]s) whose long diameter is a 24-cell edge (a hexagon edge) of length <math>\sqrt{1}</math> Six more triangular dipyramids fit into the concavities on the surface of the cluster of 5,{{Efn|These 12 cells are edge-bonded to the central cell, face-bonded to the exterior faces of the cluster of 5, and face-bonded to each other in pairs. They are blue-faced cells in the 6 different icosahedral pyramids surrounding the cluster of 5.}} so the exterior chords connecting its 4 apical vertices are also 24-cell edges of length <math>\sqrt{1}</math>. They form a tetrahedron of edge length <math>\sqrt{1}</math>, which is the second section of the 600-cell beginning with a cell.{{Efn|The <math>\sqrt{1}</math> tetrahedron has a volume of 9 <math>\phi^{-1}</math> tetrahedral cells. In the curved 3-dimensional volume of the 600 cells, it encloses the cluster of 5 cells, which do not entirely fill it. The 6 dipyramids (12 cells) which fit into the concavities of the cluster of 5 cells overfill it: only one third of each dipyramid lies within the <math>\sqrt{1}</math> tetrahedron. The dipyramids contribute one-third of each of 12 cells to it, a volume equivalent to 4 cells.|name=}} There are 600 of these <math>\sqrt{1}</math> tetrahedral sections in the 600-cell. With the six triangular dipyamids fit into the concavities, there are 12 new cells and 6 new vertices in addition to the 5 cells and 8 vertices of the original cluster. The 6 new vertices form the third section of the 600-cell beginning with a cell, an octahedron of edge length <math>\sqrt{1}</math>, obviously the cell of a 24-cell.{{Efn|The 600-cell also contains 600 ''octahedra''. The first section of the 600-cell beginning with a cell is tetrahedral, and the third section is octahedral. These internal octahedra are not ''cells'' of the 600-cell because they are not volumetrically disjoint, but they are each a cell of one of the 25 internal 24-cells. The 600-cell also contains 600 cubes, each a cell of one of its 75 internal 8-cell tesseracts.{{Efn|name=600 cubes}}|name=600 octahedra}} As partially filled so far (by 17 tetrahedral cells), this <math>\sqrt{1}</math> octahedron has concave faces into which a short triangular pyramid fits; it has the same volume as a regular tetrahedral cell but an irregular tetrahedral shape.{{Efn|Each <math>\sqrt{1}</math> edge of the octahedral cell is the long diameter of another tetrahedral dipyramid (two more face-bonded tetrahedral cells). In the 24-cell, three octahedral cells surround each edge, so one third of the dipyramid lies inside each octahedron, split between two adjacent concave faces. Each concave face is filled by one-sixth of each of the three dipyramids that surround its three edges, so it has the same volume as one tetrahedral cell.}} Each octahedron surrounds 1 + 4 + 12 + 8 = 25 tetrahedral cells: 17 regular tetrahedral cells plus 8 volumetrically equivalent tetrahedral cells each consisting of 6 one-sixth fragments from 6 different regular tetrahedral cells that each span three adjacent octahedral cells.{{Efn|A <math>\sqrt{1}</math> octahedral cell (of any 24-cell inscribed in the 600-cell) has six vertices which all lie in the same hyperplane: they bound an octahedral section (a flat three-dimensional slice) of the 600-cell. The same <math>\sqrt{1}</math> octahedron filled by 25 tetrahedral cells has a total of 14 vertices lying in three parallel three-dimensional sections of the 600-cell: the 6-point <math>\sqrt{1}</math> octahedral section, a 4-point <math>\sqrt{1}</math> tetrahedral section, and a 4-point <math>\phi^{-1}</math> tetrahedral section. In the curved three-dimensional space of the 600-cell's surface, the <math>\sqrt{1}</math> octahedron surrounds the <math>\sqrt{1}</math> tetrahedron which surrounds the <math>\phi^{-1}</math> tetrahedron, as three concentric hulls. This 14-vertex 4-polytope is a 4-pyramid with a regular octahedron base: not a canonical [[W:Octahedral pyramid|octahedral pyramid]] with one apex (which has only 7 vertices) but an irregular truncated octahedral pyramid. Because its base is a regular octahedron which is a 24-cell octahedral cell, this 4-pyramid ''lies on'' the surface of the 24-cell.}} Thus the unit-radius 600-cell may be constructed directly from its predecessor,{{Efn||name=truncated irregular octahedral pyramid}} the unit-radius 24-cell, by placing on each of its octahedral facets a truncated{{Efn|The apex of a canonical <math>\sqrt{1}</math> [[W:Octahedral pyramid|octahedral pyramid]] has been truncated into a regular tetrahedral cell with shorter <math>\phi^{-1}</math> edges, replacing the apex with four vertices. The truncation has also created another four vertices (arranged as a <math>\sqrt{1}</math> tetrahedron in a hyperplane between the octahedral base and the apex tetrahedral cell), and linked these eight new vertices with <math>\phi^{-1}</math> edges. The truncated pyramid thus has eight 'apex' vertices above the hyperplane of its octahedral base, rather than just one apex: 14 vertices in all. The original pyramid had flat sides: the five geodesic routes from any base vertex to the opposite base vertex ran along two <math>\sqrt{1}</math> edges (and just one of those routes ran through the single apex). The truncated pyramid has rounded sides: five geodesic routes from any base vertex to the opposite base vertex run along three <math>\phi^{-1}</math> edges (and pass through two 'apexes').}} irregular octahedral pyramid of 14 vertices{{Efn|The uniform 4-polytopes which this 14-vertex, 25-cell irregular 4-polytope most closely resembles may be the 10-vertex, 10-cell [[W:Rectified 5-cell|rectified 5-cell]] and its dual (it has characteristics of both).}} constructed (in the above manner) from 25 regular tetrahedral cells of edge length <math>\phi^{-1}</math>. ===== Union of two tori ===== There is yet another useful way to partition the 600-cell surface into clusters of tetrahedral cells, which reveals more structure{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5}|ps=; "Let us now proceed to a toroidal decomposition of the {3, 3, 5} polytope."}} and the [[#Decagons|decagonal fibrations]] of the 600-cell. An entire 600-cell can be assembled around two rings of 5 icosahedral pyramids, bonded vertex-to-vertex into two geodesic "straight lines". [[File:100 tets.jpg|thumb|100 tetrahedra in a 10×10 array forming a [[W:Clifford torus|Clifford torus]] boundary in the 600 cell.{{Efn|name=why 100}} Its opposite edges are identified, forming a [[W:Duocylinder|duocylinder]].]] The [[120-cell|120-cell]] can be decomposed into [[120-cell#Intertwining rings|two disjoint tori]]. Since it is the dual of the 600-cell, this same dual tori structure exists in the 600-cell, although it is somewhat more complex. The 10-cell geodesic path in the 120-cell corresponds to the 10-vertex decagon path in the 600-cell.{{Sfn|Coxeter|1970|loc=§9. The 120-cell and the 600-cell|pp=19-23}} Start by assembling five tetrahedra around a common edge. This structure looks somewhat like an angular "flying saucer". Stack ten of these, vertex to vertex, "pancake" style. Fill in the annular ring between each pair of "flying saucers" with 10 tetrahedra to form an icosahedron. You can view this as five vertex-stacked [[W:Icosahedral pyramids|icosahedral pyramids]], with the five extra annular ring gaps also filled in.{{Efn|The annular ring gaps between icosahedra are filled by a ring of 10 face-bonded tetrahedra that all meet at the vertex where the two icosahedra meet. This 10-cell ring is shaped like a [[W:Pentagonal antiprism|pentagonal antiprism]] which has been hollowed out like a bowl on both its top and bottom sides, so that it has zero thickness at its center. This center vertex, like all the other vertices of the 600-cell, is itself the apex of an icosahedral pyramid where 20 tetrahedra meet.{{Efn|name=120 overlapping icosahedral pyramids}} Therefore the annular ring of 10 tetrahedra is itself an equatorial ring of an icosahedral pyramid, containing 10 of the 20 cells of its icosahedral pyramid.|name=annular ring}} The surface is the same as that of ten stacked [[W:pentagonal antiprism|pentagonal antiprism]]s: a triangular-faced column with a pentagonal cross-section.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5}, Fig. 2. A five fold symmetry column|ps=; in caption (sic) dodecagons should be decagons.}} Bent into a columnar ring this is a torus consisting of 150 cells, ten edges long, with 100 exposed triangular faces,{{Efn|The 100-face surface of the triangular-faced 150-cell column could be scissors-cut lengthwise along a 10 edge path and peeled and laid flat as a 10×10 parallelogram of triangles.|name=triangles 10×10}} 150 exposed edges, and 50 exposed vertices. Stack another tetrahedron on each exposed face. This will give you a somewhat bumpy torus of 250 cells with 50 raised vertices, 50 valley vertices, and 100 valley edges.{{Efn|Because the 100-face surface of the 150-cell torus is alternately convex and concave, 100 tetrahedra stack on it in face-bonded pairs, as 50 [[W:Triangular bipyramid|triangular bipyramid]]s which share one raised vertex and bury one formerly exposed valley edge. The triangular bipyramids are vertex-bonded to each other in 5 parallel lines of 5 bipyramids (10 tetrahedra) each, which run straight up and down the outside surface of the 150-cell column.}} The valleys are 10 edge long closed paths and correspond to other instances of the 10-vertex decagon path mentioned above (great circle decagons). These decagons spiral around the center core decagon,{{Efn|5 decagons spiral clockwise and 5 spiral counterclockwise, intersecting each other at the 50 valley vertices.}} but mathematically they are all equivalent (they all lie in central planes). Build a second identical torus of 250 cells that interlinks with the first. This accounts for 500 cells. These two tori mate together with the valley vertices touching the raised vertices, leaving 100 tetrahedral voids that are filled with the remaining 100 tetrahedra that mate at the valley edges. This latter set of 100 tetrahedra are on the exact boundary of the [[W:Duocylinder|duocylinder]] and form a [[W:Clifford torus|Clifford torus]].{{Efn|A [[W:Clifford torus|Clifford torus]] is the [[W:Hopf fibration|Hopf fiber bundle]] of a distinct [[W:SO(4)#Isoclinic rotations|isoclinic rotation]] of a rigid [[W:3-sphere|3-sphere]], involving all of its points. The [[W:SO(4)#Visualization of 4D rotations|torus embedded in 4-space]], like the double rotation, is the [[W:Cartesian product|Cartesian product]] of two [[W:Completely orthogonal|completely orthogonal]] [[W:Great circle|great circle]]s. It is a filled [[W:Doughnut|doughnut]] not a ring doughnut; there is no hole in the 3-sphere except the [[W:4-ball (mathematics)|4-ball]] it encloses. A regular 4-polytope has a distinct number of characteristic Clifford tori, because it has a distinct number of characteristic rotational symmetries. Each forms a discrete fibration that reaches all the discrete points once each, in an isoclinic rotation with a distinct set of pairs of completely orthogonal invariant planes.|name=Clifford torus}} They can be "unrolled" into a square 10×10 array. Incidentally this structure forms one tetrahedral layer in the [[W:Tetrahedral-octahedral honeycomb|tetrahedral-octahedral honeycomb]]. There are exactly 50 "egg crate" recesses and peaks on both sides that mate with the 250 cell tori.{{Efn|How can a bumpy "egg crate" square of 100 tetrahedra lie on the smooth surface of the Clifford torus?{{Efn|name=Clifford torus}} How can a flat 10x10 square represent the 120-vertex 600-cell (where are the other 20 vertices)? In the isoclinic rotation of the 600-cell in [[#Decagons|great decagon invariant planes]], the Clifford torus is a smooth [[W:Clifford torus|Euclidean 2-surface]] which intersects the mid-edges of exactly 100 tetrahedral cells. Edges are what tetrahedra have 6 of. The mid-edges are not vertices of the 600-cell, but they are all 600 vertices of its equal-radius dual polytope, the 120-cell. The 120-cell has 5 disjoint 600-cells inscribed in it, two different ways. This distinct smooth Clifford torus (this rotation) is a discrete fibration of the 120-cell in 60 decagon invariant planes, and a discrete fibration of the 600-cell in 12 decagon invariant planes.|name=why 100}} In this case into each recess, instead of an octahedron as in the honeycomb, fits a [[W:Triangular bipyramid|triangular bipyramid]] composed of two tetrahedra. This decomposition of the 600-cell has [[W:Coxeter notation|symmetry]] [10,2⁺,10], order 400, the same symmetry as the [[W:Grand antiprism|grand antiprism]].{{Sfn|Dechant|2021|pp=20-22|loc=§7. The Grand Antiprism and H₂ × H₂}} The grand antiprism is just the 600-cell with the two above 150-cell tori removed, leaving only the single middle layer of 300 tetrahedra, dimensionally analogous{{Efn|name=math of dimensional analogy}} to the 10-face belt of an icosahedron with the 5 top and 5 bottom faces removed (a [[W:Pentagonal antiprism|pentagonal antiprism]]).{{Efn|The same 10-face belt of an icosahedral pyramid is an annular ring of 10 tetrahedra around the apex.{{Efn|name=annular ring}}}} The two 150-cell tori each contain 6 Clifford parallel great decagons (five around one), and the two tori are Clifford parallel to each other, so together they constitute a complete [[#Clifford parallel cell rings|fibration of 12 decagons]] that reaches all 120 vertices, despite filling only half the 600-cell with cells. ===== Boerdijk–Coxeter helix rings ===== The 600-cell can also be partitioned into 20 cell-disjoint intertwining rings of 30 cells,{{Sfn|Sadoc|2001|pp=577-578|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries}} each ten edges long, forming a discrete [[W:Hopf fibration|Hopf fibration]] which fills the entire 600-cell.{{Sfn|Zamboj|2021|pp=6-12|loc=§2 Mathematical background}} Each ring of 30 face-bonded tetrahedra is a cylindrical [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]] bent into a ring in the fourth dimension. {| class="wikitable" width="600" |[[File:600-cell tet ring.png|200px]]<br>A single 30-tetrahedron [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]] ring within the 600-cell, seen in stereographic projection.{{Efn|name=Boerdijk–Coxeter helix}} |[[File:600-cell Coxeter helix-ring.png|200px]]<br>A 30-tetrahedron ring can be seen along the perimeter of this 30-gonal orthogonal projection of the 600-cell.{{Efn|name=non-vertex geodesic}} |[[File:Regular_star_polygon_30-11.svg|200px]]<br>The 30-cell ring as a {30/11} polygram of 30 edges wound into a helix that twists around its axis 11 times. This projection along the axis of the 30-cell cylinder shows the 30 vertices 12° apart around the cylinder's circular cross section, with the edges connecting every 11th vertex on the circle.{{Efn|The 30 vertices and 30 edges of the 30-cell ring lie on a [[W:Skew polygon|skew]] {30/11} [[W:Star polygon|star polygon]] with a [[W:Winding number|winding number]] of 11 called a [[W:Triacontagon#Triacontagram|triacontagram₁₁]], a continuous tight corkscrew [[W:Helix|helix]] bent into a loop of 30 edges (the {{Background color|magenta|magenta}} edges in the [[#Boerdijk–Coxeter helix rings|triple helix illustration]]), which [[W:Density (polytope)#Polygons|winds]] 11 times around itself in the course of a single revolution around the 600-cell, accompanied by a single 360 degree twist of the 30-cell ring.{{Sfn|Sadoc|2001|pp=577-578|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries}} The same 30-cell ring can also be [[W:Density (polytope)|characterized]] as the [[W:Petrie polygon|Petrie polygon]] of the 600-cell.{{Efn|name=Petrie polygon in 30-cell ring}}|name=Triacontagram}} |- |colspan=3|[[File:Coxeter_helix_edges.png|625px]]<br>The 30-vertex, 30-tetrahedron [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]] ring, cut and laid out flat in 3-dimensional space. Three {{Background color|cyan|cyan}} Clifford parallel great decagons bound the ring.{{Efn|name=Clifford parallel decagons}} They are bridged by a skew 30-gram helix of 30 {{Background color|magenta|magenta}} edges linking all 30 vertices: the [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] of the 600-cell.{{Efn|name=Petrie polygon in 30-cell ring}} The 15 {{Background color|orange|orange}} edges and 15 {{Background color|yellow|yellow}} edges form separate 15-gram helices, the edge-paths of ''isoclines''. |} The 30-cell ring is the 3-dimensional space occupied by the 30 vertices of three {{Background color|cyan|cyan}} Clifford parallel great decagons that lie adjacent to each other, 36° = {{sfrac|𝜋|5}} = one 600-cell edge length apart at all their vertex pairs.{{Efn|name=triple-helix of three central decagonal planes}} The 30 {{Background color|magenta|magenta}} edges joining these vertex pairs form a helical [[W:Triacontagon#Triacontagram|triacontagram]], a skew 30-gram spiral of 30 edge-bonded triangular faces, that is the [[W:Petrie polygon#The Petrie polygon of regular polychora (4-polytopes)|Petrie polygon]] of the 600-cell.{{Efn|The 600-cell's [[W:Petrie polygon|Petrie polygon]] is a skew [[W:Skew polygon#Regular skew polygons in four dimensions|triacontagon {30}]]. It can be [[#Decagons|seen in orthogonal projection as the circumference]] of a [[W:Triacontagon#Triacontagram|triacontagram {30/3}=3{10}]] helix which zig-zags 60° left and right, bridging the space between the 3 Clifford parallel great decagons of the 30-cell ring. In the completely orthogonal plane it projects to the regular [[W:Triacontagon#Triacontagram|triacontagram {30/11}]].{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii)|ps=; ''600-cell h₁ h₂''.}}|name=Petrie polygon in 30-cell ring}} The dual of the 30-cell ring (the skew 30-gon made by connecting its cell centers) is the [[W:Skew polygon#Regular skew polygons in four dimensions|Petrie polygon]] of the [[120-cell|120-cell]], the 600-cell's [[W:Dual polytope|dual polytope]].{{Efn|The [[W:Skew polygon#Regular skew polygons in four dimensions|regular skew 30-gon]] is the [[W:Petrie polygon|Petrie polygon]] of the 600-cell and its dual the [[120-cell|120-cell]]. The Petrie polygons of the 120-cell occur in the 600-cell as duals of the 30-cell [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter helix rings]]: connecting their 30 cell centers together produces the Petrie polygons of the dual 120-cell, as noticed by Rolfdieter Frank (circa 2001). Thus he discovered that the vertex set of the 120-cell partitions into 20 non-intersecting Petrie polygons. This set of 20 disjoint Clifford parallel skew polygons is a discrete [[W:Hopf fibration|Hopf fibration]] of the 120-cell (just as their 20 dual 30-cell rings are a discrete [[#Decagons|fibration]] of the 600-cell).|name=Petrie polygons of the 120-cell}} The central axis of the 30-cell ring is a great 30-gon geodesic that passes through the center of 30 faces, but does not intersect any vertices.{{Efn|name=non-vertex geodesic}} The 15 {{Background color|orange|orange}} edges and 15 {{Background color|yellow|yellow}} edges form separate 15-gram helices. Each orange or yellow edge crosses between two {{Background color|cyan|cyan}} great decagons. Successive orange or yellow edges of these 15-gram helices do not lie on the same great circle; they lie in different central planes inclined at 36° = {{sfrac|𝝅|5}} to each other.{{Efn|name=two angles between central planes}} Each 15-gram helix is noteworthy as the edge-path of an [[#Rotations on polygram isoclines|isocline]], the [[W:Geodesic|geodesic]] path of an isoclinic [[#Rotations|rotation]].{{Efn|name=isoclinic geodesic}} The isocline is a circular curve which intersects every ''second'' vertex of the 15-gram, missing the vertex in between. A single isocline runs twice around each orange (or yellow) 15-gram through every other vertex, hitting half the vertices on the first loop and the other half of them on the second loop. The two connected loops forms a single [[W:Möbius loop|Möbius loop]], a skew {15/2} [[W:Pentadecagram|pentadecagram]]. The pentadecagram is not shown in these illustrations (but [[#Decagons and pentadecagrams|see below]]), because its edges are invisible chords between vertices which are two orange (or two yellow) edges apart, and no chords are shown in these illustrations. Although the 30 vertices of the 30-cell ring do not lie in one great 30-gon central plane,{{Efn|The 30 vertices of the [[#Boerdijk–Coxeter helix rings|Boerdijk–Coxeter triple-helix ring]] lie in 3 decagonal central planes which intersect only at one point (the center of the 600-cell), even though they are not completely orthogonal or orthogonal at all: they are {{sfrac|{{pi}}|5}} apart.{{Efn|name=two angles between central planes}} Their decagonal great circles are Clifford parallel: one 600-cell edge-length apart at every point.{{Efn|name=Clifford parallels}} They are ordinary 2-dimensional great circles, ''not'' helices, but they are [[W:link (knot theory)|linked]] Clifford parallel circles.|name=triple-helix of three central decagonal planes}} these invisible [[#Decagons and pentadecagrams|pentadecagram isoclines]] are true geodesic circles of a special kind, that wind through all four dimensions rather than lying in a 2-dimensional plane as an ordinary geodesic great circle does.{{Efn|name=4-dimensional great circles}} Five of these 30-cell [[W:Helix|helices]] nest together and spiral around each of the 10-vertex decagon paths, forming the 150-cell torus described in the [[#Union of two tori|grand antiprism decomposition]] above.{{Sfn|Dechant|2021|pp=20-22|loc=§7. The Grand Antiprism and H₂ × H₂}} Thus ''every'' great decagon is the center core decagon of a 150-cell torus.{{Efn|The 20 30-cell rings are [[W:Chiral|chiral]] objects; they either spiral clockwise (right) or counterclockwise (left). The 150-cell torus (formed by five cell-disjoint 30-cell rings of the same chirality surrounding a great decagon) is not itself a chiral object, since it can be decomposed into either five parallel left-handed rings or five parallel right-handed rings. Unlike the 20-cell rings, the 150-cell tori are directly congruent with no [[W:Torsion of a curve|torsion]], like the octahedral [[24-cell#6-cell rings|6-cell rings of the 24-cell]]. Each great decagon has five left-handed 30-cell rings surrounding it, and also five right-handed 30-cell rings surrounding it; but left-handed and right-handed 30-cell rings are not cell-disjoint and belong to different distinct rotations: the left and right rotations of the same fibration. In either distinct isoclinic rotation (left or right), the vertices of the 600-cell move along the axial [[#Decagons and pentadecagrams|15-gram isoclines]] of 20 left 30-cell rings or 20 right 30-cell rings. Thus the great decagons, the 30-cell rings, and the 150-cell tori all occur as sets of Clifford parallel interlinked circles,{{Efn|name=Clifford parallels}} although the exact way they nest together, avoid intersecting each other, and pass through each other to form a [[W:Hopf link|Hopf link]] is not identical for these three different kinds of [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]], in part because the linked pairs are variously of no inherent chirality (the decagons), the same chirality (the 30-cell rings), or no net torsion and both left and right interior organization (the 150-cell tori) but tracing the same chirality of interior organization in any distinct left or right rotation.|name=chirality of cell rings}} The 600-cell may be decomposed into 20 30-cell rings, or into two 150-cell tori and 10 30-cell rings, but not into four 150-cell tori of this kind.{{Efn|A point on the icosahedron Hopf map{{Efn|name=Hopf fibration base}} of the 600-cell's decagonal fibration lifts to a great decagon; a triangular face lifts to a 30-cell ring; and a pentagonal pyramid of 5 faces lifts to a 150-cell torus.{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5}, Fig. 2. A five fold symmetry column|ps=; in caption (sic) dodecagons should be decagons.}} In the [[#Union of two tori|grand antiprism decomposition]], two completely disjoint 150-cell tori are lifted from antipodal pentagons, leaving an equatorial ring of 10 icosahedron faces between them: a Petrie decagon of 10 triangles, which lift to 10 30-cell rings. The two completely disjoint 150-cell tori contain 12 disjoint (Clifford parallel) decagons and all 120 vertices, so they comprise a complete Hopf fibration; there is no room for more 150-cell tori of this kind. To get a decomposition of the 600-cell into four 150-cell tori of this kind, the icosahedral map would have to be decomposed into four pentagons, centered at the vertices of an inscribed tetrahedron, and the icosahedron cannot be decomposed that way.}} The 600-cell ''can'' be decomposed into four 150-cell tori of a different kind.{{Efn|Sadoc describes the decomposition of the 600-cell into four tori.{{Sfn|Sadoc|2001|loc=§2.6 The {3, 3, 5} polytope: a set of four helices|p=578}} It is the same [[#Decagons|fibration of 12 great decagons and 20 30-cell rings]], seen as a [[#Clifford parallel cell rings|fibration of four completely disjoint 30-cell rings]]{{Efn|name=completely disjoint}} with spaces between them, which still encompasses all 12 decagons and all 120 vertices. If we look closely at the spaces between the four disjoint 30-cell rings, we ''can'' discern four 150-cell rings of 5 30-cell rings each. But these 150-cell rings do not have 5 30-cell rings around a common decagon axis, and 6 decagons each. Their axis is a 30-cell ring, not a decagon, and they contain only 3 decagons each. To construct them, on each of the four completely disjoint 30-cell rings, face-bond three more 30-cell rings to the exterior faces, making four stellated ("bumpy") rings containing four 30-cell rings (120 cells) each. Collectively they contain 16 of the 20 30-cell rings: there are still four 30-cell ring "holes" left to fill in the 600-cell. To do that, fill some of the surface concavities of each 120-tetrahedron ring by wrapping a fifth 30-cell ring around its circumference, completely orthogonal to the axial 30-cell ring you started with. The result is four 150-cell tori, of 5 30-cell rings each, each having two completely orthogonal 30-cell ring axes, either of which can be seen as either an axis or a circumference: it is both. On the icosahedron Hopf map,{{Efn|name=Hopf fibration base}} the four 30-cell rings lift from a star of four icosahedron faces (three faces edge-bonded around one). The fifth 30-cell ring lifts from a fifth face edge-bonded to the star, a sort of "extra flap" like the sixth square flap of the [[W:Cube#Orthogonal projections|net of a cube]] before you fold it up into a cube. It does not matter which of the six possible adjacent faces you choose as the flap, but the choice determines the choice for all four 150-cell rings. There are six choices because there are six decagonal fibrations; this is when you fix which fibration you are taking. Thus ''every'' 30-cell ring is the center core of a 150-cell ring.}} ==== Radial golden triangles ==== The 600-cell can be constructed radially from 720 [[W:Golden triangle (mathematics)|golden triangle]]s of edge lengths <math>1, 1, \phi^{-1}</math> which meet at the center of the 4-polytope, each contributing two <math>\sqrt{1}</math> radii and a <math>\phi^{-1}</math> edge. They form 1200 triangular pyramids with their apexes at the center: irregular tetrahedra with equilateral <math>\phi^{-1}</math> bases (the faces of the 600-cell). These form 600 tetrahedral pyramids with their apexes at the center: irregular 5-cells with regular <math>\phi^{-1}</math> tetrahedron bases (the cells of the 600-cell). ==== Characteristic orthoscheme ==== {| class="wikitable floatright" !colspan=6|Characteristics of the 600-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "600-cell"}} |- !align=right| !align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}} !colspan=2 align=center|arc !colspan=2|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}} |- !align=right|𝒍 |align=center|<small><math>phi{-1} \approx 0.618</math></small> |align=center|<small>36°</small> |align=center|<small><math>\tfrac{\pi}{5}</math></small> |align=center|<small>164°29′</small> |align=center|<small><math>\pi-2\psi</math></small> |- | | | | | |- !align=right|𝟀 |align=center|<small><math>\sqrt{\tfrac{2}{3\phi^2}} \approx 0.505</math></small> |align=center|<small>22°15′20″</small> |align=center|<small><math>\tfrac{\pi}{3} - \eta</math></small> |align=center|<small>60°</small> |align=center|<small><math>\tfrac{\pi}{3}</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}{2\phi^2}} \approx 0.437</math></small> |align=center|<small>18°</small> |align=center|<small><math>\tfrac{\pi}{10}</math></small> |align=center|<small>36°</small> |align=center|<small><math>\tfrac{\pi}{5}</math></small> |- !align=right|𝟁 |align=center|<small><math>\sqrt{\tfrac{1}{6\phi^2}} \approx 0.252</math></small> |align=center|<small>17°44′40″</small> |align=center|<small><math>\eta - \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{3}{4\phi^2}} \approx 0.535</math></small> |align=center|<small>22°15′20″</small> |align=center|<small><math>\tfrac{\pi}{3} - \eta</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\phi^2}} \approx 0.309</math></small> |align=center|<small>18°</small> |align=center|<small><math>\tfrac{\pi}{10}</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}{12\phi^2}} \approx 0.178</math></small> |align=center|<small>17°44′40″</small> |align=center|<small><math>\eta - \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>\tfrac12\sqrt{2 + \phi} \approx 0.951</math></small> |align=center| |align=center| |align=center| |align=center| |- !align=right|<small><math>_2R^4/l</math></small> |align=center|<small><math>\sqrt{\tfrac{\phi^2}{3}} \approx 0.934</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{\phi^4}{8}} \approx 0.926</math></small> |align=center| |align=center| |align=center| |align=center| |- | | | | | |- !align=right|<small><math>\eta</math></small> |align=center| |align=center|<small>37°44′40″</small> |align=center|<small><math>\tfrac{\text{arc sec }4}{2}</math></small> |align=center| |align=center| |} Every regular 4-polytope has its characteristic 4-orthoscheme, an [[5-cell#Irregular 5-cells|irregular 5-cell]].{{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}} The '''characteristic 5-cell of the regular 600-cell''' is represented by the [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] {{Coxeter–Dynkin diagram|node|3|node|3|node|5|node}}, which can be read as a list of the dihedral angles between its mirror facets. It is an irregular [[W:Pyramid (mathematics)#Polyhedral pyramid|tetrahedral pyramid]] based on the [[W:Tetrahedron#Orthoschemes|characteristic tetrahedron of the regular tetrahedron]]. The regular 600-cell is subdivided by its symmetry hyperplanes into 14400 instances of its characteristic 5-cell that all meet at its center.{{Efn|‟The Petrie polygons of the Platonic solid <small><math>\{p, q\}</math></small> correspond to equatorial polygons of the truncation <small><math>\{\tfrac{p}{q}\}</math></small> and to ''equators'' of the simplicially subdivided spherical tessellation <small><math>\{p, q\}</math></small>. This "[[W:Schläfli orthoscheme#Characteristic simplex of the general regular polytope|simplicial subdivision]]" is the arrangement of <small><math>g = g_{p, q}</math></small> right-angled spherical triangles into which the sphere is decomposed by the planes of symmetry of the solid. The great circles that lie in these planes were formerly called "lines of symmetry", but perhaps a more vivid name is ''reflecting circles''. The analogous simplicial subdivision of the spherical honeycomb <small><math>\{p, q, r\}</math></small> consists of the <small><math>g = g_{p, q, r}</math></small> tetrahedra '''0123''' into which a hypersphere (in Euclidean 4-space) is decomposed by the hyperplanes of symmetry of the polytope <small><math>\{p, q, r\}</math></small>. The great spheres which lie in these hyperplanes are naturally called ''reflecting spheres''. Since the orthoscheme has no obtuse angles, it entirely contains the arc that measures the absolutely shortest distance 𝝅/''h'' [between the] 2''h'' tetrahedra [that] are strung like beads on a necklace, or like a "rotating ring of tetrahedra" ... whose opposite edges are generators of a helicoid. The two opposite edges of each tetrahedron are related by a screw-displacement.{{Efn|name=transformations}} Hence the total number of spheres is 2''h''.”{{Sfn|Coxeter|1973|pp=227−233|loc=§12.7 A necklace of tetrahedral beads}}|name=orthoscheme ring}} 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 600-cell).{{Efn|The four edges of each 4-orthoscheme which meet at the center of a 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 600-cell has unit radius and edge length <small><math>\ell = \phi^{-1} \approx 0.618</math></small>, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{2}{3\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{2\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6\phi^2}}</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{3}{4\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{12\phi^2}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the regular tetrahedron), plus <small><math>1</math></small>, <small><math>\tfrac12\sqrt{2 + \phi}</math></small>, <small><math>\sqrt{\tfrac{\phi^2}{3}}</math></small>, <small><math>\sqrt{\tfrac{\phi^4}{8}}</math></small> (edges which are the characteristic radii of the 600-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{2\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{6\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{1}{4\phi^2}}</math></small>, <small><math>\sqrt{\tfrac{\phi^4}{8}}</math></small>, first from a 600-cell vertex to a 600-cell edge center, then turning 90° to a 600-cell face center, then turning 90° to a 600-cell tetrahedral cell center, then turning 90° to the 600-cell center. ==== Reflections ==== The 600-cell can be 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}}{{Sfn|Dechant|2017|pp=410-419|loc=§6. The Coxeter Plane; see p. 416, Table 1. Summary of the factorisations of the Coxeter versors of the 4D root systems|ps=; "Coxeter (reflection) groups in the Clifford framework ... afford a uniquely simple prescription for reflections. Via the Cartan-Dieudonné theorem, performing two reflections successively generates a rotation, which in Clifford algebra is described by a spinor that is simply the geometric product of the two vectors generating the reflections."}} so for example an ''n''-dimensional reflection is an (''n''+1)-dimensional half-turn.{{Sfn|Coxeter|1973|loc=§12-34|p=220}} A full isoclinic revolution of the 600-cell in decagonal invariant planes takes ''each'' of the 120 vertices through 15 vertices and back to itself, on a skew pentadecagram₂ geodesic [[#Decagons and pentadecagrams|isocline]] of circumference 5𝝅 that winds around the 3-sphere, as each great decagon rotates (like a wheel) and also tilts sideways (like a coin flipping) with the completely orthogonal plane.{{Efn|name=one true 5𝝅 circle}} Any set of four orthogonal pairs of antipodal vertices (the 8 vertices of one of the 75 inscribed 16-cells){{Efn|name=fifteen 16-cells partitioned among four 30-cell rings}} performing such an orbit visits 15 * 8 = 120 distinct vertices and [[24-cell#Clifford parallel polytopes|generates the 600-cell]] sequentially in one full isoclinic rotation, just as any single characteristic 5-cell reflecting itself in its own mirror walls generates the 120 vertices simultaneously by reflection.{{Efn|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). 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> 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). [If we assume the [[W:Galilean relativity|Galilean principle of relativity]], every displacement in 4-space can be viewed as either of those, because we can view any QT as a Q² in a linearly moving (translating) reference frame. Therefore any transformation from one inertial reference frame to another is expressable as a Q². By the same principle, we can view any QT or Q² as an isoclinic (equi-angled) Q² by appropriate choice of reference frame.{{Efn|[[W:Arthur Cayley|Cayley]] showed that any rotation in 4-space can be decomposed into two isoclinic rotations,{{Efn|name=double rotation}} 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}} That is to say, Coxeter's relation is a mathematical statement of the principle of relativity, on group-theoretic grounds.{{Efn|Notice that Coxeter's relation correctly captures the limits to relativity, in that we can only exchange the translation (T) for ''one'' of the two rotations (Q). An observer in any inertial reference frame can always measure the presence, direction and velocity of ''one'' rotation up to uncertainty, and can always also distinguish the direction and velocity of his own proper time arrow.}}] Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} ==== Weyl orbits ==== Another construction method uses [[#Symmetries|quaternions]] and the [[W:Icosahedral symmetry|icosahedral symmetry]] of [[W:Weyl group|Weyl group]] orbits <math>O(\Lambda)=W(H_4)=I</math> of order 120.{{Sfn|Koca, Al-Ajmi, & Ozdes Koca|2011|loc=6. Dual of the snub 24-cell|pp=986-988}} The following are the orbits of weights of D4 under the Weyl group W(D4): : O(0100) : T = {±1,±e1,±e2,±e3,(±1±e1±e2±e3)/2} : O(1000) : V1 : O(0010) : V2 : O(0001) : V3 [[File:120Cell-SimpleRoots-Quaternion-Tp.png|600px]] With quaternions <math>(p,q)</math> where <math>\bar p</math> is the conjugate of <math>p</math> and <math>[p,q]:r\rightarrow r'=prq</math> and <math>[p,q]^*:r\rightarrow r''=p\bar rq</math>, then the [[W:Coxeter group|Coxeter group]] <math>W(H_4)=\lbrace[p,\bar p] \oplus [p,\bar p]^*\rbrace </math> is the symmetry group of the 600-cell and the [[120-cell|120-cell]] of order 14400. Given <math>p \in T</math> such that <math>\bar p=\pm p^4, \bar p^2=\pm p^3, \bar p^3=\pm p^2, \bar p^4=\pm p</math> and <math>p^\dagger</math> as an exchange of <math>-1/\varphi \leftrightarrow \varphi</math> within <math>p</math>, we can construct: * the [[W:Snub 24-cell|snub 24-cell]] <math>S=\sum_{i=1}^4\oplus p^i T</math> * the 600-cell <math>I=T+S=\sum_{i=0}^4\oplus p^i T</math> * the [[120-cell|120-cell]] <math>J=\sum_{i,j=0}^4\oplus p^i\bar p^{\dagger j}T'</math> === 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)]], the [[W:Orthogonal group|group]] of rotations{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes|pp=1438-1439|ps=; the 600-cell has 14,400 symmetry operations (rotations and reflections) as enumerated in Table 2, symmetry group 𝛨₄.{{Efn|name=distinct rotations}}}} about a fixed point in 4-dimensional Euclidean space.{{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 [[W:Completely orthogonal|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|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}} 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), 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, similar but not identical to two simple rotations through the ''same'' angle.{{Efn|A point under isoclinic rotation traverses the diagonal{{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]].{{Efn|name=isoclinic geodesic}} 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=isoclinic rotation to non-adjacent vertices}} For example, when the unit-radius 600-cell rotates isoclinically 36 degrees in a decagon invariant plane and 36 degrees in its completely orthogonal invariant plane,{{Efn|name=non-vertex geodesic}} each vertex is displaced to another vertex <math>\sqrt{1}</math> (60°) distant, moving <math>\sqrt{1/4} {{=}} 1/2</math> (half the <math>\sqrt{1}</math> overall displacement) in four orthogonal directions.|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.{{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 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-space analogues{{Efn|name=math of dimensional analogy}} of 2-dimensional great circles in 3-space (great 1-spheres).|name=4-dimensional great circles}} They are true circles,{{Efn|name=one true 5𝝅 circle}} and even form [[W:Hopf fibration|fibrations]] like ordinary 2-dimensional great circles. These '''[[#Rotations on polygram isoclines|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 [[W:Chiral|chiral]] pairs as [[W:Villarceau circle|Villarceau circle]]s on the [[W:Clifford torus|Clifford torus]],{{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 5𝝅 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}} 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 of a 4-polytope's [[W:Skew polygon#Regular skew polygons in four dimensions|skew]] '''Clifford polygon'''.{{Efn|name=Clifford polygon}}|name=isoclinic geodesic}}|name=identical rotations}} The 600-cell is generated by [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotations]]{{Efn|name=isoclinic geodesic}} of the 24-cell by 36° = {{sfrac|𝜋|5}} (the arc of one 600-cell edge length).{{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{{Efn|name=isoclinic 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:William Kingdon Clifford|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 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 rotates sideways.{{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}} ''All'' central polygons (of every kind) rotate by the same angle (though not all do so invariantly), and are also displaced sideways by the same angle to a Clifford parallel polygon (of the same kind).|name=Clifford displacement}} ==== Twenty-five 24-cells ==== There are 25 inscribed 24-cells in the 600-cell.{{sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020}}{{Efn|The 600-cell has 7200 distinct rotational displacements, each with its invariant rotation plane. The 7200 distinct central planes can be grouped into sets of Clifford parallel invariant rotation planes of 25 distinct ''isoclinic'' rotations, and are usually given as those sets.{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2}}|name=distinct rotations}} Therefore there are also 25 inscribed snub 24-cells, 75 inscribed tesseracts and 75 inscribed 16-cells.{{Efn|name=4-polytopes inscribed in the 600-cell}} The 8-vertex 16-cell has 4 long diameters inclined at 90° = {{sfrac|𝜋|2}} to each other, often taken as the 4 orthogonal axes or [[16-cell#Coordinates|basis]] of the coordinate system.{{Efn|name=Six orthogonal planes of the Cartesian basis}} The 24-vertex 24-cell has 12 long diameters inclined at 60° = {{sfrac|𝜋|3}} to each other: 3 disjoint sets of 4 orthogonal axes, each set comprising the diameters of one of 3 inscribed 16-cells, isoclinically rotated by {{sfrac|𝜋|3}} with respect to each other.{{Efn|The three 16-cells in the 24-cell are rotated by 60° ({{sfrac|𝜋|3}}) isoclinically with respect to each other. Because an isoclinic rotation is a rotation in two completely orthogonal planes at the same time, this means their corresponding vertices are 120° ({{sfrac|2𝜋|3}}) apart. In a unit-radius 4-polytope, vertices 120° apart are joined by a <math>\sqrt{3}</math> chord.|name=120° apart}} The 120-vertex 600-cell has 60 long diameters: ''not just'' 5 disjoint sets of 12 diameters, each comprising one of 5 inscribed 24-cells (as we might suspect by analogy), but 25 distinct but overlapping sets of 12 diameters, each comprising one of 25 inscribed 24-cells.{{Sfn|Waegell & Aravind|2009|loc=§3. The 600-cell|pp=2-5}} There ''are'' 5 disjoint 24-cells in the 600-cell, but not ''just'' 5: there are 10 different ways to partition the 600-cell into 5 disjoint 24-cells.{{Efn|name=Schoute's ten ways to get five disjoint 24-cells|[[W:Pieter Hendrik Schoute|Schoute]] was the first to state (a century ago) that there are exactly ten ways to partition the 120 vertices of the 600-cell into five disjoint 24-cells. The 25 24-cells can be placed in a 5 x 5 array such that each row and each column of the array partitions the 120 vertices of the 600-cell into five disjoint 24-cells. The rows and columns of the array are the only ten such partitions of the 600-cell.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|p=434}}}} Like the 16-cells and 8-cells inscribed in the 24-cell, the 25 24-cells inscribed in the 600-cell are mutually [[24-cell#Clifford parallel polytopes|isoclinic polytopes]]. The rotational distance between inscribed 24-cells is always {{sfrac|𝜋|5}} in each invariant plane of rotation.{{Efn|There is a single invariant plane in each simple rotation, and a completely orthogonal fixed plane. There are an infinite number of pairs of [[W:Completely orthogonal|completely orthogonal]] invariant planes in each isoclinic rotation, all rotating through the same angle;{{Efn|name=dense fabric of pole-circles}} nonetheless, not all [[#Geodesics|central planes]] are [[24-cell#Isoclinic rotations|invariant planes of rotation]]. The invariant planes of an isoclinic rotation constitute a [[#Fibrations of great circle polygons|fibration]] of the entire 4-polytope.{{Sfn|Kim|Rote|2016|loc=§8.2 Equivalence of an Invariant Family and a Hopf Bundle|pp=13-14}} In every isoclinic rotation of the 600-cell taking vertices to vertices either 12 Clifford parallel great [[#Decagons|decagons]], ''or'' 20 Clifford parallel great [[#Hexagons|hexagons]] ''or'' 30 Clifford parallel great [[#Squares|squares]] are invariant planes of rotation.|name=isoclinic invariant planes}} Five 24-cells are disjoint because they are Clifford parallel: their corresponding vertices are {{sfrac|𝜋|5}} apart on two non-intersecting Clifford parallel{{Efn|name=Clifford parallels}} decagonal great circles (as well as {{sfrac|𝜋|5}} apart on the same decagonal great circle).{{Efn|Two Clifford parallel{{Efn|name=Clifford parallels}} great decagons don't intersect, but their corresponding vertices are linked by one edge of another decagon. The two parallel decagons and the ten linking edges form a double helix ring. Three decagons can also be parallel (decagons come in parallel [[W:Hopf fibration|fiber bundles]] of 12) and three of them may form a triple helix ring. If the ring is cut and laid out flat in 3-space, it is a [[W:Boerdijk–Coxeter helix|Boerdijk–Coxeter helix]]{{Efn|name=Boerdijk–Coxeter helix}} 30 tetrahedra{{Efn|name=tetrahedron linking 6 decagons}} long. The three Clifford parallel decagons can be seen as the {{Background color|cyan}} edges in the [[#Boerdijk–Coxeter helix rings|triple helix illustration]]. Each {{Background color|magenta}} edge is one edge of another decagon linking two parallel decagons.|name=Clifford parallel decagons}} An isoclinic rotation of decagonal planes by {{sfrac|𝜋|5}} takes each 24-cell to a disjoint 24-cell (just as an [[24-cell#Clifford parallel polytopes|isoclinic rotation of hexagonal planes]] by {{sfrac|𝜋|3}} takes each 16-cell to a disjoint 16-cell).{{Efn|name=isoclinic geodesic displaces every central polytope}} Each isoclinic rotation occurs in two chiral forms: there are 4 disjoint 24-cells to the ''left'' of each 24-cell, and another 4 disjoint 24-cells to its ''right''.{{Efn|A ''disjoint'' 24-cell reached by an isoclinic rotation is not any of the four adjacent 24-cells; the double rotation{{Efn|name=identical rotations}} takes it past (not through) the adjacent 24-cell it rotates toward,{{Efn|Five 24-cells meet at each vertex of the 600-cell,{{Efn|name=five 24-cells at each vertex of 600-cell}} so there are four different directions in which the vertices can move to rotate the 24-cell (or all the 24-cells at once in an [[24-cell#Isoclinic rotations|isoclinic rotation]]{{Efn|name=isoclinic geodesic displaces every central polytope}}) directly toward an adjacent 24-cell.|name=four directions toward another 24-cell}} and left or right to a more distant 24-cell from which it is completely disjoint.{{Efn|name=completely disjoint}} The four directions reach 8 different 24-cells{{Efn|name=disjoint from 8 and intersects 16}} because in an isoclinic rotation each vertex moves in a spiral along two completely orthogonal great circles at once. Four paths are right-hand [[W:Screw thread#Handedness|threaded]] (like most screws and bolts), moving along the circles in the "same" directions, and four are 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}}|name=rotations to 8 disjoint 24-cells}} The left and right rotations reach different 24-cells; therefore each 24-cell belongs to two different sets of five disjoint 24-cells. All [[24-cell#Clifford parallel polytopes|Clifford parallel polytopes]] are isoclinic, but not all isoclinic polytopes are Clifford parallels (completely disjoint objects).{{Efn|All isoclinic ''polygons'' are Clifford parallels (completely disjoint).{{Efn||name=completely disjoint}} Polyhedra (3-polytopes) and polychora (4-polytopes) may be isoclinic and ''not'' disjoint, if all of their corresponding central polygons are either Clifford parallel, or cocellular (in the same hyperplane) or coincident (the same object, shared). For example, the 24-cell, 600-cell and 120-cell contain pairs of inscribed tesseracts (8-cells) which are isoclinically rotated by {{sfrac|𝜋|3}} with respect to each other, yet are not disjoint: they share a [[16-cell#Octahedral dipyramid|16-cell]] (8 vertices, 6 great squares and 4 octahedral central hyperplanes), and some corresponding pairs of their great squares are cocellular (intersecting) rather than Clifford parallel (disjoint).|name=isoclinic and not disjoint}} Each 24-cell is isoclinic ''and'' Clifford parallel to 8 others, and isoclinic but ''not'' Clifford parallel to 16 others.{{Efn|name=disjoint from 8 and intersects 16}} With each of the 16 it shares 6 vertices: a hexagonal central plane.{{Efn|name=five 24-cells at each vertex of 600-cell}} Non-disjoint 24-cells are related by a [[W:Rotations in 4-dimensional Euclidean space#Simple rotations|simple rotation]] by {{sfrac|𝜋|5}} in an invariant plane intersecting only two vertices of the 600-cell,{{Efn|name=digon planes}} a rotation in which the completely orthogonal [[24-cell#Simple rotations|fixed plane]] is their common hexagonal central plane. They are also related by an [[W:Rotations in 4-dimensional Euclidean space#Isoclinic rotations|isoclinic rotation]] in which both planes rotate by {{sfrac|𝜋|5}}.{{Efn|In the 600-cell, there is a [[24-cell#Simple rotations|simple rotation]] which will take any vertex ''directly'' to any other vertex, also moving most or all of the other vertices but leaving 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 decagon, a great hexagon, a great square or a great [[W:Digon|digon]],{{Efn|name=digon planes}} and the completely orthogonal fixed plane intersects 0 vertices (a 30-gon),{{Efn|name=non-vertex geodesic}} 2 vertices (a digon), 4 vertices (a square) or 6 vertices (a hexagon) respectively. Two ''non-disjoint'' 24-cells are related by a [[24-cell#Simple rotations|simple rotation]] through {{sfrac|𝜋|5}} of the digon central plane completely orthogonal to their common hexagonal central plane. In this simple rotation, the hexagon does not move. The two ''non-disjoint'' 24-cells are also related by an isoclinic rotation in which the shared hexagonal plane ''does'' move.{{Efn|name=rotations to 16 non-disjoint 24-cells}}|name=direct simple rotations}} There are two kinds of {{sfrac|𝜋|5}} isoclinic rotations which take each 24-cell to another 24-cell.{{Efn|Any isoclinic rotation by {{sfrac|𝜋|5}} in decagonal invariant planes{{Efn|Any isoclinic rotation in a decagonal invariant plane is an isoclinic rotation in 24 invariant planes: 12 Clifford parallel decagonal planes,{{Efn|name=isoclinic invariant planes}} and the 12 Clifford parallel 30-gon planes completely orthogonal to each of those decagonal planes.{{Efn|name=non-vertex geodesic}} As the invariant planes rotate in two completely orthogonal directions at once,{{Efn|name=helical geodesic}} all points in the planes move with them (stay in their planes and rotate with them), describing helical isoclines{{Efn|name=isoclinic geodesic}} through 4-space. Note however that in a ''discrete'' decagonal fibration of the 600-cell (where 120 vertices are the only points considered), the 12 30-gon planes contain ''no'' points.}} takes ''every'' [[#Geodesics|central polygon]], [[#Clifford parallel cell rings|geodesic cell ring]] or inscribed 4-polytope{{Efn|name=4-polytopes inscribed in the 600-cell}} in the 600-cell to a [[24-cell#Clifford parallel polytopes|Clifford parallel polytope]] {{sfrac|𝜋|5}} away.|name=isoclinic geodesic displaces every central polytope}} ''Disjoint'' 24-cells are related by a {{sfrac|𝜋|5}} isoclinic rotation of an entire [[#Decagons|fibration of 12 Clifford parallel ''decagonal'' invariant planes]]. (There are 6 such sets of fibers, and a right or a left isoclinic rotation possible with each set, so there are 12 such distinct rotations.){{Efn|name=rotations to 8 disjoint 24-cells}} ''Non-disjoint'' 24-cells are related by a {{sfrac|𝜋|5}} isoclinic rotation of an entire [[#Hexagons|fibration of 20 Clifford parallel ''hexagonal'' invariant planes]].{{Efn|Notice the apparent incongruity of rotating ''hexagons'' by {{sfrac|𝜋|5}}, since only their opposite vertices are an integral multiple of {{sfrac|𝜋|5}} apart. However, [[#Icosahedra|recall]] that 600-cell vertices which are one hexagon edge apart are exactly two decagon edges and two tetrahedral cells (one triangular dipyramid) apart. The hexagons have their own [[#Hexagons|10 discrete fibrations]] and [[#Clifford parallel cell rings|cell rings]], not Clifford parallel to the [[#Decagons|decagonal fibrations]] but also by fives{{Efn|name=24-cells bound by pentagonal fibers}} in that five 24-cells meet at each vertex, each pair sharing a hexagon.{{Efn|name=five 24-cells at each vertex of 600-cell}} Each hexagon rotates ''non-invariantly'' by {{sfrac|𝜋|5}} in a [[#Hexagons and hexagrams|hexagonal isoclinic rotation]] between ''non-disjoint'' 24-cells.{{Efn|name=rotations to 16 non-disjoint 24-cells}} Conversely, in all [[#Decagons and pentadecagrams|{{sfrac|𝜋|5}} isoclinic rotations in ''decagonal'' invariant planes]], all the vertices travel along isoclines{{Efn|name=isoclinic geodesic}} which follow the edges of ''hexagons''.|name=apparent incongruity}} (There are 10 such sets of fibers, so there are 20 such distinct rotations.){{Efn|At each vertex, a 600-cell has four adjacent (non-disjoint){{Efn||name=completely disjoint}} 24-cells that can each be reached by a simple rotation in that direction.{{Efn|name=four directions toward another 24-cell}} Each 24-cell has 4 great hexagons crossing at each of its vertices, one of which it shares with each of the adjacent 24-cells; in a simple rotation that hexagonal plane remains fixed (its vertices do not move) as the 600-cell rotates ''around'' the common hexagonal plane. The 24-cell has 16 great hexagons altogether, so it is adjacent (non-disjoint) to 16 other 24-cells.{{Efn|name=disjoint from 8 and intersects 16}} In addition to being reachable by a simple rotation, each of the 16 can also be reached by an isoclinic rotation in which the shared hexagonal plane is ''not'' fixed: it rotates (non-invariantly) through {{sfrac|𝜋|5}}. The double rotation reaches an adjacent 24-cell ''directly'' as if indirectly by two successive simple rotations:{{Efn|name=double rotation}} first to one of the ''other'' adjacent 24-cells, and then to the destination 24-cell (adjacent to both of them).|name=rotations to 16 non-disjoint 24-cells}} On the other hand, each of the 10 sets of five ''disjoint'' 24-cells is Clifford parallel because its corresponding great ''hexagons'' are Clifford parallel. (24-cells do not have great decagons.) The 16 great hexagons in each 24-cell can be divided into 4 sets of 4 non-intersecting Clifford parallel [[24-cell#Geodesics|geodesics]], each set of which covers all 24 vertices of the 24-cell. The 200 great hexagons in the 600-cell can be divided into 10 sets of 20 non-intersecting Clifford parallel [[#Geodesics|geodesics]], each set of which covers all 120 vertices and constitutes a discrete [[#Hexagons|hexagonal fibration]]. Each of the 10 sets of 20 disjoint hexagons can be divided into five sets of 4 disjoint hexagons, each set of 4 covering a disjoint 24-cell. Similarly, the corresponding great ''squares'' of disjoint 24-cells are Clifford parallel. ==== Rotations on polygram isoclines ==== The regular convex 4-polytopes each have their characteristic kind of right (and left) [[W:Isoclinic rotation|isoclinic rotation]], corresponding to their characteristic kind of discrete [[W:Hopf fibration|Hopf fibration]] of great circles.{{Efn|The poles of the invariant axis of a rotating 2-sphere are dimensionally analogous to the pair of invariant planes of a rotating 3-sphere. The poles of the rotating 2-sphere are dimensionally analogous to linked great circles on the 3-sphere. By dimensional analogy, each 1D point in 3D lifts to a 2D line in 4D, in this case a circle.{{Efn|name=Hopf fibration base}} The two antipodal rotation poles lift to a pair of circular Hopf fibers which are not merely Clifford parallel and interlinked,{{Efn|name=Clifford parallels}} but also [[W:Completely orthogonal|completely orthogonal]]. ''The invariant great circles of the 4D rotation are its poles.'' In the case of an isoclinic rotation, there is not merely one such pair of 2D poles (completely orthogonal Hopf great circle fibers), there are many such pairs: a finite number of circle-pairs if the 3-sphere fibration is discrete (e.g. a regular polytope with a finite number of vertices), or else an infinite number of orthogonal circle-pairs, entirely filling the 3-sphere. Every point in the curved 3-space of the 3-sphere lies on ''one'' such circle (never on two, since the completely orthogonal circles, like all the Clifford parallel Hopf great circle fibers, do not intersect). Where a 2D rotation has one pole, and a 3D rotation of a 2-sphere has 2 poles, ''an isoclinic 4D rotation of a 3-sphere has nothing but poles'', an infinite number of them. In a discrete 4-polytope, all the Clifford parallel invariant great polygons of the rotation are poles, and they fill the 4-polytope, passing through every vertex just once. ''In one full revolution of such a rotation, every point in the space loops exactly once through its pole-circle.''{{Efn|Consider the statement: ''In one full revolution of an isoclinic rotation, every point in the space loops exactly once through its great circle Hopf fiber.'' It can be found in the literature, expressed in the mathematical language of the Hopf fibration,{{Sfn|Kim|Rote|2016|loc= 8 The Construction of Hopf Fibrations|pp=12-16|ps=; see Theorem 13.}} but as a plain language statement of Euclidean geometry, how exactly should we visualize it? It paints a clear picture of all the great circles of a Hopf fibration rotating as rigid wheels, in parallel. That is a correct visualization, except for the fact that points moving under isoclinic rotation traverse an invariant great circle only in the sense that they stay on that circle as the whole circle itself is tilting sideways, rotating in parallel with the completely orthogonal great circle.{{Efn|name=isoclinic geodesic}} With respect to the stationary reference frame, the points move diagonally on a helical isocline, they do not move on a planar great circle.{{Efn|name=helical geodesic}} Each helical isocline is itself a kind of circle, but it is not a planar great circle of the [[W:Hopf fibration|Hopf fibration]]: it is a special kind of geodesic circle whose circumference is greater than 2𝝅''r'', and it is not pictured explicitly at all by the plain statement we are trying to visualize.{{Efn|name=isocline circumference.}} We cannot easily visualize this statement about the Hopf great circles in a stationary reference frame. The statement does ''not'' simply mean that in an isoclinic rotation every point on a stationary Hopf great circle loops through its stationary great circle. Rather, it means that every point on every Hopf great circle loops through its great circle ''as every great circle itself is moving orthogonally'', flipping like a coin in the plane completely orthogonal to its own plane (at any instant, because of course the completely orthogonal plane is moving too). This simultaneous ''twisting'' rotation in two completely orthogonal planes is a double rotation; if the angle of rotation in the two completely orthogonal planes is exactly the same, it is isoclinic. An isoclinic rotation takes each rigid planar Hopf great circle to the stationary position of another Hopf great circle, while simultaneously each Hopf great circle also rotates like a wheel. This fibration of doubly rotating rigid wheels is undoubtably hard to visualize. In any graphical animation (whether actually rendered or merely imagined) it will be difficult to track the motions of the different rotating wheels, because Clifford parallel circles are not parallel in the ordinary sense, and every great circle is moving in a different direction at any one instant. There is one more way in which this simple statement belies the full complexity of the isoclinic motion. While it is true that every point loops through its Hopf great circle exactly once ''in a full isoclinic revolution, every vertex moves more than 360 degrees,'' as measured in the stationary reference frame. In any distinct isoclinic rotation, all the vertices move the same angular distance in the stationary reference frame in one full revolution, but each distinct pair of left-right isoclinic rotations corresponds to a unique Hopf fibration,{{Sfn|Kim|Rote|2016|loc=§8.2 Equivalence of an Invariant Family and a Hopf Bundle|pp=13-14}} and the characteristic distance moved is different for each kind of Hopf fibration. For example, in the [[24-cell#Isoclinic rotations|isoclinic rotation of a great hexagon fibration of the 24-cell]], each vertex moves 720 degrees in the stationary reference frame (2 times the distance it moves within its moving Hopf great circle);{{Efn|name=4𝝅 rotation}} but in the [[#Decagons and pentadecagrams|isoclinic rotation of a great decagon fibration of the 600-cell]], each vertex moves 900 degrees in the stationary reference frame (2.5 times its great circle distance).}} The circles are arranged with a surprising symmetry, so that ''each pole-circle links with every other pole-circle'', like a maximally dense fabric of 4D [[W:Chain mail|chain mail]] in which all the circles are linked through each other, but no two circles ever intersect.|name=dense fabric of pole-circles}} For example, the 600-cell can be fibrated six different ways into a set of Clifford parallel [[#Decagons|great decagons]], so the 600-cell has six distinct right (and left) isoclinic rotations in which those great decagon planes are [[24-cell#Isoclinic rotations|invariant planes of rotation]]. We say these isoclinic rotations are ''characteristic'' of the 600-cell because the 600-cell's edges lie in their invariant planes. These rotations only emerge in the 600-cell, although they are also found in larger regular polytopes (the 120-cell) which contain inscribed instances of the 600-cell. Just as the [[#Geodesics|geodesic]] ''polygons'' (decagons or hexagons or squares) in the 600-cell's central planes form [[#Fibrations of great circle polygons|fiber bundles of Clifford parallel ''great circles'']], the corresponding geodesic [[W:Skew polygon|skew]] ''[[W:Polygram (geometry)|polygrams]]'' (which trace the paths on the [[W:Clifford torus|Clifford torus]] of vertices under isoclinic rotation){{Sfn|Perez-Gracia & Thomas|2017|loc=§2. Isoclinic rotations|pp=2−3}} form [[W:Fiber bundle|fiber bundle]]s of Clifford parallel ''isoclines'': helical circles which wind through all four dimensions.{{Efn|name=isoclinic geodesic}} Since isoclinic rotations are [[W:Chiral|chiral]], occurring in left-handed and right-handed forms, each polygon fiber bundle has corresponding left and right polygram fiber bundles.{{Sfn|Kim|Rote|2016|p=12-16|loc=§8 The Construction of Hopf Fibrations; see §8.3}} All the fiber bundles are aspects of the same discrete [[W:Hopf fibration|Hopf fibration]], because the fibration is the various expressions of the same distinct left-right pair of isoclinic rotations. Cell rings are another expression of the Hopf fibration. Each discrete fibration has a set of cell-disjoint cell rings that tesselates the 4-polytope.{{Efn|The choice of a partitioning of a regular 4-polytope into cell rings is arbitrary, because all of its cells are identical. No particular fibration is distinguished, ''unless'' the 4-polytope is rotating. In isoclinic rotations, one set of cell rings (one fibration) is distinguished as the unique container of that distinct left-right pair of rotations and its isoclines.|name=fibrations are distinguished only by rotations}} The isoclines in each chiral bundle spiral around each other: they are axial geodesics of the rings of face-bonded cells. The [[#Clifford parallel cell rings|Clifford parallel cell rings]] of the fibration nest into each other, pass through each other without intersecting in any cells, and exactly fill the 600-cell with their disjoint cell sets. Isoclinic rotations rotate a rigid object's vertices along parallel paths, each vertex circling within two orthogonal moving great circles, the way a [[W:Loom|loom]] weaves a piece of fabric from two orthogonal sets of parallel fibers. A bundle of Clifford parallel great circle polygons and a corresponding bundle of Clifford parallel skew polygram isoclines are the [[W:Warp and woof|warp and woof]] of the same distinct left or right isoclinic rotation, which takes Clifford parallel great circle polygons to each other, flipping them like coins and rotating them through a Clifford parallel set of central planes. Meanwhile, because the polygons are also rotating individually like wheels, vertices are displaced along helical Clifford parallel isoclines (the chords of which form the skew polygram), through vertices which lie in successive Clifford parallel polygons.{{Efn|name=helical geodesic}} In the 600-cell, each family of isoclinic skew polygrams (moving vertex paths in the decagon {10}, hexagon {6}, or square {4} great polygon rotations) can be divided into bundles of non-intersecting Clifford parallel polygram isoclines.{{Sfn|Perez-Gracia & Thomas|2017|loc=§1. Introduction|ps=; "This article [will] derive a spectral decomposition of isoclinic rotations and explicit formulas in matrix and Clifford algebra for the computation of Cayley's [isoclinic] factorization."{{Efn|name=double rotation}}}} The isocline bundles occur in pairs of ''left'' and ''right'' chirality; the isoclines in each rotation act as [[W:Chiral|chiral]] objects, as does each distinct isoclinic rotation itself.{{Efn|The fibration's [[#Clifford parallel cell rings|Clifford parallel cell rings]] may or may not be [[W:Chiral|chiral]] objects, depending upon whether the 4-polytope's cells have opposing faces or not. The characteristic cell rings of the 16-cell and 600-cell (with tetrahedral cells) are chiral: they twist either clockwise or counterclockwise. Isoclines acting with either left or right chirality (not both) run through cell rings of this kind, though each fibration contains both left and right cell rings.{{Efn|Each isocline has no inherent chirality but can act as either a left or right isocline; it is shared by a distinct left rotation and a distinct right rotation of different fibrations.|name=isoclines have no inherent chirality}} The characteristic cell rings of the 8-cell tesseract, 24-cell and 120-cell (with cubical, octahedral, and dodecahedral cells respectively) are directly congruent, not chiral: there is only one kind of characteristic cell ring in each of these 4-polytopes, and it is not twisted (it has no [[W:Torsion of a curve|torsion]]). Pairs of left-handed and right-handed isoclines run through cell rings of this kind. The left and right isoclines are enantiomorphously congruent (mirror images) of each other. Note that all these 4-polytopes (except the 16-cell) contain fibrations of their inscribed [[#Geometry|predecessors]]' characteristic cell rings in addition to their own characteristic fibrations, so the 600-cell contains both chiral and directly congruent cell rings.|name=directly congruent versus twisted cell rings}} Each fibration contains an equal number of left and right isoclines, in two disjoint bundles, which trace the paths of the 600-cell's vertices during the fibration's left or right isoclinic rotation respectively. Each left or right fiber bundle of isoclines ''by itself'' constitutes a discrete Hopf fibration which fills the entire 600-cell, visiting all 120 vertices just once. It is a ''different bundle of fibers'' than the bundle of Clifford parallel polygon great circles, but the two fiber bundles describe the ''same discrete fibration'' because they enumerate those 120 vertices together in the same distinct right (or left) isoclinic rotation, by their intersection as a fabric of cross-woven parallel fibers. Each isoclinic rotation involves pairs of completely orthogonal invariant central planes of rotation, which both rotate through the same angle. There are two ways they can do this: by both rotating in the "same" direction, or by rotating 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). The right polygram and right isoclinic rotation conventionally correspond to invariant pairs rotating in the same direction; the left polygram and left isoclinic rotation correspond to pairs rotating in opposite directions.{{Sfn|Perez-Gracia & Thomas|2017|loc=§5. A useful mapping|pp=12−13}} Left and right isoclines are different paths that go to different places. In addition, each distinct isoclinic rotation (left or right) can be performed in a positive or negative direction along the circular parallel fibers. A fiber bundle of Clifford parallel isoclines is the set of helical vertex circles described by a distinct left or right isoclinic rotation. Each moving vertex travels along an isocline contained within a (moving) cell ring. While the left and right isoclinic rotations each double-rotate the same set of Clifford parallel invariant [[24-cell#Planes of rotation|planes of rotation]], they step through different sets of great circle polygons because left and right isoclinic rotations hit alternate vertices of the great circle {2p} polygon (where p is a prime ≤ 5).{{Efn|name={2p} isoclinic rotations}} The left and right rotation share the same Hopf bundle of {2p} polygon fibers, which is ''both'' a left and right bundle, but they have different bundles of {p} polygons{{Sfn|Kim|Rote|2016|p=14|loc=§8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle}} because the discrete fibers are opposing left and right {p} polygons inscribed in the {2p} polygon.{{Efn|Each discrete fibration of a regular convex 4-polytope is characterized by a unique left-right pair of isoclinic rotations and a unique bundle of great circle {2p} polygons (0 ≤ p ≤ 5) in the invariant planes of that pair of rotations. Each distinct rotation has a unique bundle of left (or right) {p} polygons inscribed in the {2p} polygons, and a unique bundle of skew {2p} polygrams which are its discrete left (or right) isoclines. The {p} polygons weave the {2p} polygrams into a bundle, and vice versa.}} A [[24-cell#Simple rotations|simple rotation]] is direct and local, taking some vertices to adjacent vertices along great circles, and some central planes to other central planes within the same hyperplane. (The 600-cell has four orthogonal [[#Polyhedral sections|central hyperplanes]], each of which is an icosidodecahedron.) In a simple rotation, there is just a single pair of completely orthogonal invariant central planes of rotation; it does not constitute a fibration. An [[24-cell#Isoclinic rotations|isoclinic rotation]] is diagonal and global, taking ''all'' the vertices to ''non-adjacent'' vertices{{Efn|Isoclinic rotations take each vertex to a non-adjacent vertex. In the characteristic isoclinic rotations of the 5-cell, 16-cell, 24-cell and 600-cell, the non-adjacent vertex is the opposite vertex of a neighboring cell. In the 8-cell it is three zig-zag edge-lengths away in the same cell: the opposite vertex of a cube. In the 120-cell it is four zig-zag edges away in the same cell: the opposite vertex of a dodecahedron. |name=isoclinic rotation to non-adjacent vertices}} along diagonal isoclines, and ''all'' the central plane polygons to Clifford parallel polygons (of the same kind). A left-right pair of isoclinic rotations constitutes a discrete fibration. All the Clifford parallel central planes of the fibration are invariant planes of rotation, separated by ''two'' equal angles and lying in different hyperplanes.{{Efn|name=two angles between central planes}} The diagonal isocline{{Efn|name=isoclinic 4-dimensional diagonal}} is a shorter route between the non-adjacent vertices than the multiple simple routes between them available along edges: it is the ''shortest route'' on the 3-sphere, the [[W:Geodesic|geodesic]]. ==== Decagons and pentadecagrams ==== The [[#Fibrations of great circle polygons|fibrations of the 600-cell]] include 6 [[#Decagons|fibrations of its 72 great decagons]]: 6 fiber bundles of 12 great decagons,{{Efn|name=Clifford parallel decagons}} each delineating [[#Boerdijk–Coxeter helix rings|20 chiral cell rings]] of 30 tetrahedral cells each,{{Efn|name=Boerdijk–Coxeter helix}} with three great decagons bounding each cell ring, and five cell rings nesting together around each decagon. The 12 Clifford parallel decagons in each bundle are completely disjoint. Adjacent parallel decagons are spanned by edges of other great decagons.{{Efn|name=equi-isoclinic decagons}} Each fibration corresponds to a distinct left (and right) isoclinic rotation of the 600-cell in 12 great decagon invariant planes on 5𝝅 isoclines. The bundle of 12 Clifford parallel decagon fibers is divided into a bundle of 12 left pentagon fibers and a bundle of 12 right pentagon fibers, with each left-right pair of pentagons inscribed in a decagon.{{Sfn|Kim|Rote|2016|loc=§8.3 Properties of the Hopf Fibration|pp=14-16}} 12 great polygons comprise a fiber bundle covering all 120 vertices in a discrete [[W:Hopf fibration|Hopf fibration]]. There are 20 cell-disjoint 30-cell rings in the fibration, but only 4 completely disjoint 30-cell rings.{{Efn|name=completely disjoint}} The 600-cell has six such discrete [[#Decagons|decagonal fibrations]], and each is the domain (container) of a unique left-right pair of isoclinic rotations (left and right fiber bundles of 12 great pentagons).{{Efn|There are six congruent decagonal fibrations of the 600-cell. Choosing one decagonal fibration means choosing a bundle of 12 directly congruent Clifford parallel decagonal great circles, and a cell-disjoint set of 20 directly congruent 30-cell rings which tesselate the 600-cell. The fibration and its great circles are not chiral, but it has distinct left and right expressions in a left-right pair of isoclinic rotations. In the right (left) rotation the vertices move along a right (left) Hopf fiber bundle of Clifford parallel isoclines and intersect a right (left) Hopf fiber bundle of Clifford parallel great pentagons. The 30-cell rings are the only chiral objects, other than the ''bundles'' of isoclines or pentagons.{{Sfn|Kim|Rote|2016|p=14|loc=§8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle}} A right (left) pentagon bundle contains 12 great pentagons, inscribed in the 12 Clifford parallel great [[#Decagons|decagons]]. A right (left) isocline bundle contains 20 Clifford parallel pentadecagrams, one in each 30-cell ring.|name=decagonal fibration of chiral bundles}} Each great decagon belongs to just one fibration,{{Sfn|Kim|Rote|2016|p=14|loc=§8.3 Corollary 9. Every great circle belongs to a unique right [(and left)] Hopf bundle}} but each 30-cell ring belongs to 5 of the six fibrations (and is completely disjoint from 1 other fibration). The 600-cell contains 72 great decagons, divided among six fibrations, each of which is a set of 20 cell-disjoint 30-cell rings (4 completely disjoint 30-cell rings), but the 600-cell has only 20 distinct 30-cell rings altogether. Each 30-cell ring contains 3 of the 12 Clifford parallel decagons in each of 5 fibrations, and 30 of the 120 vertices. In these ''decagonal'' isoclinic rotations, vertices travel along isoclines which follow the edges of ''hexagons'',{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} advancing a pythagorean distance of one hexagon edge in each double 36°×36° rotational unit.{{Efn||name=apparent incongruity}} In an isoclinic rotation, each successive hexagon edge travelled lies in a different great hexagon, so the isocline describes a skew polygram, not a polygon. In a 60°×60° isoclinic rotation (as in the [[24-cell#Isoclinic rotations|24-cell's characteristic hexagonal rotation]], and [[#Hexagons and hexagrams|below in the ''hexagonal'' rotations of the 600-cell]]) this polygram is a <s>[[W:Hexagram|hexagram]]</s>: the isoclinic rotation follows a 6-edge circular path, just as a simple hexagonal rotation does, although it takes ''two'' revolutions to enumerate all the vertices in it, because the isocline is a double loop through every other vertex, and its chords are <math>\sqrt{3}</math> chords of the hexagon instead of <math>\sqrt{1}</math> hexagon edges.{{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. 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 <math>\sqrt{3}</math> longer, its circumference is 4𝝅 instead of 2𝝅,{{Efn|name=4𝝅 rotation}} it circles through four dimensions instead of two,{{Efn|name=Villarceau circles}} and it acts in two chiral forms (left and right) although all such circles of the same circumference are directly or enantiomorphously congruent.{{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.|name=one true 4𝝅 circle}} But in the 600-cell's 36°×36° characteristic ''decagonal'' rotation, successive great hexagons are closer together and more numerous, and the isocline polygram formed by their 15 hexagon ''edges'' is a pentadecagram (15-gram).{{Efn|name=one true 5𝝅 circle}} It is not only not the same period as the hexagon or the simple decagonal rotation, it is not even an integer multiple of the period of the hexagon, or the decagon, or either's simple rotation. Only the compound {30/4}=2{15/2} triacontagram (30-gram), which is two 15-grams rotating in parallel (a black and a white), is a multiple of them all, and so constitutes the rotational unit of the decagonal isoclinic rotation.{{Efn|The analogous relationships among three kinds of {2p} isoclinic rotations, in [[#Fibrations of great circle polygons|Clifford parallel bundles of {4}, {6} or {10} great polygon invariant planes]] respectively, are at the heart of the complex nested relationship among the [[#Geometry|regular convex 4-polytopes]].{{Efn|name=4-polytopes ordered by size and complexity}} In the <math>\sqrt{1}</math> [[#Hexagons and hexagrams|hexagon {6} rotations characteristic of the 24-cell]], the [[#Rotations on polygram isoclines|isocline chords (polygram edges)]] are simply <math>\sqrt{3}</math> chords of the great hexagon, so the [[24-cell#Simple rotations|simple {6} hexagon rotation]] and the [[24-cell#Isoclinic rotations|isoclinic <s>{6/2} hexagram</s> rotation]] both rotate circles of 6 vertices. The <s>hexagram</s> isocline, a special kind of great circle, has a circumference of 4𝝅 compared to the hexagon 2𝝅 great circle.{{Efn|name=one true 4𝝅 circle}} The invariant central plane completely orthogonal to each {6} great hexagon is a {2} great digon,{{Efn|name=digon planes}} so an [[#Hexagons and hexagrams|isoclinic {6} rotation of <s>hexagrams</s>]] is also a {2} rotation of ''axes''.{{Efn|name=direct simple rotations}} In the <math>\sqrt{2}</math> [[#Squares and octagrams|square {4} rotations characteristic of the 16-cell]], the isocline polygram is an [[16-cell#Helical construction|octagram]], and the isocline's chords are its <math>\sqrt{2}</math> edges and its <math>\sqrt{4}</math> diameters, so the isocline is a circle of circumference 4𝝅. In an isoclinic rotation, the eight vertices of the {8/3} octagram change places, each making one complete revolution through 720° as the isocline [[W:Winding number|winds]] ''three'' times around the 3-sphere. The invariant central plane completely orthogonal to each {4} great square is another {4} great square <math>\sqrt{4}</math> distant, so a ''right'' {4} rotation of squares is also a ''left'' {4} rotation of squares. The 16-cell's [[W:Dural polytope|dual polytope]] the [[W:8-cell|8-cell tesseract]] inherits the same simple {4} and isoclinic {8/3} rotations, but its characteristic isoclinic rotation takes place in completely orthogonal invariant planes which contain a {4} great ''rectangle'' or a {2} great digon (from its successor the 24-cell). In the 8-cell this is a rotation of <math>\sqrt{1}</math> × <math>\sqrt{3}</math> great rectangles, and also a rotation of <math>\sqrt{4}</math> axes, but it is the same isoclinic rotation as the 24-cell's characteristic rotation of {6} great hexagons (in which the great rectangles are inscribed), as a consequence of the unique circumstance that [[24-cell#Geometry|the 8-cell and 24-cell have the same edge length]]. In the <math>\phi^{-1}</math> [[#Decagons|decagon {10} rotations characteristic of the 600-cell]], the isocline ''chords'' are <math>\sqrt{1}</math> hexagon ''edges'', the isocline polygram is a pentadecagram, and the isocline has a circumference of 5𝝅.{{Efn|name=one true 5𝝅 circle}} The [[#Decagons and pentadecagrams|isoclinic {15/2} pentadecagram rotation]] rotates a circle of {15} vertices in the same time as the simple decagon rotation of {10} vertices. The invariant central plane completely orthogonal to each {10} great decagon is a {0} great 0-gon,{{Efn|name=0-gon central planes}} so a {10} rotation of decagons is also a {0} rotation of planes containing no vertices. The 600-cell's dual polytope the [[120-cell#Chords|120-cell inherits]] the same simple {10} and isoclinic {15/2} rotations, but its characteristic isoclinic rotation takes place in completely orthogonal invariant planes which contain {2} great [[W:Digon|digon]]s (from its successor the 5-cell).{{Efn|120 regular 5-cells are inscribed in the 120-cell. The [[5-cell#Geodesics and rotations|5-cell has digon central planes]], no two of which are orthogonal. It has 10 digon central planes, where each vertex pair is an edge, not an axis. The 5-cell is self-dual, so by reciprocation the 120-cell can be inscribed in a regular 5-cell of larger radius. Therefore the finite sequence of 6 regular 4-polytopes{{Efn|name=4-polytopes ordered by size and complexity}} nested like [[W:Russian dolls|Russian dolls]] can also be seen as an infinite sequence.|name=infinite inscribed sequence}} This is a rotation of [[120-cell#Chords|irregular great hexagons]] {6} of two alternating edge lengths (analogous to the tesseract's great rectangles), where the two different-length edges are three 120-cell edges and three [[5-cell#Boerdijk–Coxeter helix|5-cell edges]].|name={2p} isoclinic rotations}} In the 30-cell ring, the non-adjacent vertices linked by isoclinic rotations are two edge-lengths apart, with three other vertices of the ring lying between them.{{Efn|In the 30-cell ring, each isocline runs from a vertex to a non-adjacent vertex in the third shell of vertices surrounding it. Three other vertices between these two vertices can be seen in the 30-cell ring, two adjacent in the first [[#Polyhedral sections|surrounding shell]], and one in the second surrounding shell.}} The two non-adjacent vertices are linked by a <math>\sqrt{1}</math> chord of the isocline which is a great hexagon edge (a 24-cell edge). The <math>\sqrt{1}</math> chords of the 30-cell ring (without the <math>\phi^{-1}</math> 600-cell edges) form a skew [[W:Triacontagram|triacontagram]]_{{30/4}=2{15/2}} which contains 2 disjoint {15/2} Möbius double loops, a left-right pair of [[W:Pentadecagram|pentadecagram]]₂ isoclines. Each left (or right) bundle of 12 pentagon fibers is crossed by a left (or right) bundle of 8 Clifford parallel pentadecagram fibers. Each distinct 30-cell ring has 2 double-loop pentadecagram isoclines running through its even or odd (black or white) vertices respectively.{{Efn|Isoclinic rotations{{Efn|name=isoclinic geodesic}} partition the 600 cells (and the 120 vertices) of the 600-cell into two disjoint subsets of 300 cells (and 60 vertices), even and odd (or black and white), which shift places among themselves on black or white isoclines, in a manner dimensionally analogous{{Efn|name=math of dimensional analogy}} to the way the [[W:Bishop (chess)|bishops]]' diagonal moves restrict them to the white or the black squares of the [[W:Chessboard|chessboard]].{{Efn|Left and right isoclinic rotations partition the 600 cells (and 120 vertices) into black and white in the same way.{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}} 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.{{Sfn|Coxeter|1973|p=156|loc=|ps=: "...the chess-board has an n-dimensional analogue."}} ''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]], '''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. (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.) Things which have [[W:Chirality|chirality]] come in '''''right or left''''' enantiomorphous forms: '''isoclinic rotations''' and '''chiral objects''' which include '''[[#Characteristic orthoscheme|characteristic orthoscheme]]s''', '''pairs of Clifford parallel great polygon planes''',{{Sfn|Kim|Rote|2016|p=8|loc=Left and Right Pairs of Isoclinic Planes}} '''[[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 [[#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 [[24-cell#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 part of left or right chiral objects when they are vertex paths in a left or right rotation, although they have no inherent chirality themselves.{{Efn|name=isoclines have no inherent chirality}} 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}} The black and white subsets are also divided among black and white invariant great circle polygons of the isoclinic rotation. In a discrete rotation (as of a 4-polytope with a finite number of vertices) the black and white subsets correspond to sets of inscribed great polygons {p} in invariant great circle polygons {2p}. For example, in the 600-cell a black and a white great pentagon {5} are inscribed in an invariant great decagon {10} of the characteristic decagonal isoclinic rotation. Importantly, a black and white pair of polygons {p} of the same distinct isoclinic rotation are never inscribed in the same {2p} polygon; there is always a black and a white {p} polygon inscribed in each invariant {2p} polygon, but they belong to distinct isoclinic rotations: the left and right rotation of the same fibraton, which share the same set of invariant planes. Black (white) isoclines intersect only black (white) great {p} polygons, so each vertex is either black or white.|name=black and white}} The pentadecagram helices have no inherent chirality, but each acts as either a left or right isocline in any distinct isoclinic rotation.{{Efn|name=isoclines have no inherent chirality}} The 2 pentadecagram fibers belong to the left and right fiber bundles of 5 different fibrations. At each vertex, there are six great decagons and six pentadecagram isoclines (six black or six white) that cross at the vertex.{{Efn|Each axis of the 600-cell touches a left isocline of each fibration at one end and a right isocline of the fibration at the other end. Each 30-cell ring's axial isocline passes through only one of the two antipodal vertices of each of the 30 (of 60) 600-cell axes that the isocline's 30-vertex, 30-cell ring touches (at only one end).}} Eight pentadecagram isoclines (four black and four white) comprise a unique right (or left) fiber bundle of isoclines covering all 120 vertices in the distinct right (or left) isoclinic rotation. Each fibration has a unique left and right isoclinic rotation, and corresponding unique left and right fiber bundles of 12 pentagons and 8 pentadecagram isoclines. There are only 20 distinct black isoclines and 20 distinct white isoclines in the 600-cell. Each distinct isocline belongs to 5 fiber bundles. {| class="wikitable" width="450" !colspan=4|Three sets of 30-cell ring chords from the same [[W:Orthogonal projection|orthogonal projection]] viewpoint |- ![[W:Pentadecagon#Pentadecagram|Pentadecagram {15/2}]] ![[W:Triacontagon#Triacontagram|Triacontagram {30/4}=2{15/2}]] ![[W:Triacontagon#Triacontagram|Triacontagram {30/6}=6{5}]] |- |colspan=2 align=center|All edges are [[W:Pentadecagram|pentadecagram]] isocline chords of length <math>\sqrt{1}</math>, which are also [[24-cell#Great hexagons|great hexagon]] edges of 24-cells inscribed in the 600-cell. |colspan=1 align=center|Only [[#Golden chords|great pentagon edges]] of length <math>\sqrt{3 - \phi} \approx 1.176</math>. |- |[[File:Regular_star_polygon_15-2.svg|200px]] |[[File:Regular_star_figure_2(15,2).svg|200px]] |[[File:Regular_star_figure_6(5,1).svg|200px]] |- |valign=top|A single black (or white) isocline is a Möbius double loop skew pentadecagram {15/2} of circumference 5𝝅.{{Efn|name=one true 5𝝅 circle}} The <math>\sqrt{1}</math> chords are 24-cell edges (hexagon edges) from different inscribed 24-cells. These chords are invisible (not shown) in the [[#Boerdijk–Coxeter helix rings|30-cell ring illustration]], where they join opposite vertices of two face-bonded tetrahedral cells that are two orange edges apart or two yellow edges apart. |valign=top|The 30-cell ring as a skew compound of two disjoint pentadecagram {15/2} isoclines (a black-white pair, shown here as orange-yellow).{{Efn|name=black and white}} The <math>\sqrt{1}</math> chords of the isoclines link every 4th vertex of the 30-cell ring in a straight chord under two orange edges or two yellow edges. The doubly-curved isocline is the geodesic (shortest path) between those vertices; they are also two edges apart by three different angled paths along the edges of the face-bonded tetrahedra. |valign=top|Each pentadecagram isocline (at left) intersects all six great pentagons (above) in two or three vertices. The pentagons lie on flat 2𝝅 great circles in the decagon invariant planes of rotation. The pentadecagrams are ''not'' flat: they are helical 5𝝅 isocline circles whose 15 chords lie in successive great ''hexagon'' planes inclined at 𝝅/5 = 36° to each other. The isocline circle is said to be twisting either left or right with the rotation, but all such pentadecagrams are directly or enantiomorphously congruent. |- |colspan=3|No 600-cell edges appear in these illustrations, only [[#Hopf spherical coordinates|invisible interior chords of the 600-cell]]. In this article, they should all properly be drawn as dashed lines. |} Two 15-gram double-loop isoclines are axial to each 30-cell ring. The 30-cell rings are chiral; each fibration contains 10 right (clockwise-spiraling) rings and 10 left (counterclockwise spiraling) rings. The right and left isoclines in each 3-cell ring are enantiomorphously congruent (mirror images).{{Efn|The chord-path of an isocline 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 [[#Boerdijk–Coxeter helix 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 two 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,{{Efn|An isoclinic rotation by 36° is two simple rotations by 36° at the same time.{{Efn|The composition of two simple 36° rotations in a pair of completely orthogonal invariant planes is a 36° isoclinic rotation in ''twelve'' pairs of completely orthogonal invariant planes.{{Efn|name=double rotation}} Thus the isoclinic rotation is the compound of twelve simple rotations, and all 120 vertices rotate in invariant decagon planes, versus just 10 vertices in a simple rotation.}} It moves all the vertices 60° at the same time, in various different directions. Fifteen successive diagonal rotational increments, of 36°×36° each, move each vertex 900° through 15 vertices on a Möbius double loop of circumference 5𝝅 called an ''isocline'', winding around the 600-cell and back to its point of origin, in one-and-one-half the time (15 rotational increments) that it would take a simple rotation to take the vertex once around the 600-cell on an ordinary {10} great circle (in 10 rotational increments).{{Efn|name=double threaded}} The helical double loop 5𝝅 isocline is just a special kind of ''single'' full circle, of 1.5 the period (15 chords instead of 10) as the simple great circle. The isocline is ''one'' true circle, as perfectly round and geodesic as the simple great circle, even through its chords are φ longer, its circumference is 5𝝅 instead of 2𝝅, it circles through four dimensions instead of two, and it acts in two chiral forms (left and right) even though all such circles of the same circumference are directly or enantiomorphously congruent.{{Efn|name=isocline circumference.}} 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=isoclinic geodesic}}|name=one true 5𝝅 circle}} with no net [[W:Torsion of a curve|torsion]].{{Efn|name=Sadoc frustration}} An isocline acts as a left (or right) isocline when traversed by a left (or right) rotation (of different fibrations).|name=Clifford polygon}} Each acts as a left (or right) isocline a left (or right) rotation, but has no inherent chirality.{{Efn|name=isoclines have no inherent chirality}} The fibration's 20 left and 20 right 15-grams altogether contain 120 disjoint open pentagrams (60 left and 60 right), the open ends of which are adjacent 600-cell vertices (one <math>\phi^{-1}</math> edge-length apart). The 30 chords joining the isocline's 30 vertices are <math>\sqrt{1}</math> hexagon edges (24-cell edges), connecting 600-cell vertices which are ''two'' 600-cell <math>\phi^{-1}</math> edges apart on a decagon great circle. {{Efn|Because the 600-cell's [[#Decagons and pentadecagrams|helical pentadecagram₂ geodesic]] is bent into a twisted ring in the fourth dimension like a [[W:Möbius strip|Möbius strip]], its [[W:Screw thread|screw thread]] doubles back across itself after each revolution, without ever reversing its direction of rotation (left or right). The 30-vertex isoclinic path follows a Möbius double loop, forming a single continuous 15-vertex loop traversed in two revolutions. The Möbius helix is a geodesic "straight line" or ''[[#Decagons and pentadecagrams|isocline]]''. The isocline connects the vertices of a lower frequency (longer wavelength) skew polygram than the Petrie polygon. The Petrie triacontagon has <math>\phi^{-1}</math> edges; the isoclinic pentadecagram₂ has <math>\sqrt{1}</math> edges which join vertices which are two <math\phi^{-1}</math> edges apart. Each <math>\sqrt{1}</math> edge belongs to a different [[#Hexagons|great hexagon]], and successive <math>\sqrt{1}</math> edges belong to different 24-cells, as the isoclinic rotation takes hexagons to Clifford parallel hexagons and passes through successive Clifford parallel 24-cells.|name=double threaded}} These isocline chords are both hexa''gon'' edges and penta''gram'' edges. The 20 Clifford parallel isoclines (30-cell ring axes) of each left (or right) isocline bundle do not intersect each other. Either distinct decagonal isoclinic rotation (left or right) rotates all 120 vertices (and all 600 cells), but pentadecagram isoclines and pentagons are connected such that vertices alternate as 60 black and 60 white vertices (and 300 black and 300 white cells), like the black and white squares of the [[W:Chessboard|chessboard]].{{Efn|name=isoclinic chessboard}} In the course of the rotation, the vertices on a left (or right) isocline rotate within the same 15-vertex black (or white) isocline, and the cells rotate within the same black (or white) 30-cell ring. ==== Hexagons and <s>hexagrams</s> ==== [[File:Regular_star_figure_2(10,3).svg|thumb|[[W:Icosagon#Related polygons|Icosagram {20/6}{{=}}2{10/3}]] contains 2 disjoint {10/3} decagrams (red and orange) which connect vertices 3 apart on the {10} and 6 apart on the {20}. In the 600-cell the edges are great pentagon edges spanning 72°.]]The [[#Fibrations of great circle polygons|fibrations of the 600-cell]] include 10 [[#Hexagons|fibrations of its 200 great hexagons]]: 10 fiber bundles of 20 great hexagons. The 20 Clifford parallel hexagons in each bundle are completely disjoint. Adjacent parallel hexagons are spanned by edges of great decagons.{{Efn|name=equi-isoclinic hexagons}} Each fibration corresponds to a distinct left (and right) isoclinic rotation of the 600-cell in 20 great hexagon invariant planes on 4𝝅 isoclines. Each fiber bundle delineates 20 disjoint directly congruent [[24-cell#6-cell rings|cell rings of 6 octahedral cells]] each, with three cell rings nesting together around each hexagon. The bundle of 20 Clifford parallel hexagon fibers is divided into a bundle of 20 black <math>\sqrt{3}</math> [[24-cell#Great triangles|great triangle]] fibers and a bundle of 20 white great triangle fibers, with a black and a white triangle inscribed in each hexagon and 6 black and 6 white triangles in each 6-octahedron ring. The black or white triangles are joined by three intersecting black or white isoclines, each of which is a special kind of helical great circle{{Efn|name=one true 4𝝅 circle}} through the corresponding vertices in 10 Clifford parallel black (or white) great triangles. The 10 <math>\sqrt{3 - \phi}</math> chords of each isocline form a skew [[W:Decagon#decagram|decagram {10/3}]], 10 great pentagon edges joined end-to-end in a helical loop, [[W:Winding number|winding]] 3 times around the 600-cell through all four dimensions rather than lying flat in a central plane. Each pair of black and white isoclines (intersecting antipodal great hexagon vertices) forms a compound 20-gon [[W:Icosagon#Related polygons|icosagram {20/6}{{=}}2{10/3}]]. Notice the relation between the [[24-cell#Helical dodecagrams and their isoclines|24-cell's characteristic rotation in great hexagon invariant planes]] (on <s>hexagram</s> isoclines), and the 600-cell's own version of the rotation of great hexagon planes (on decagram isoclines). They are exactly the same isoclinic rotation: they have the same isocline. They have different numbers of the same isocline because the 600-cell contains multiple 24-cells, and the 600-cell's <math>\sqrt{3 - \phi}</math> isocline chord is shorter than the 24-cell's <math>\sqrt{3}</math> isocline chord because the isocline intersects more vertices in the 600-cell (10) than it does in the 24-cell (6), but both Clifford polygrams have a 4𝝅 circumference.{{Efn|The 24-cell rotates hexagons on <s>[[24-cell#Helical dodecagrams and their isoclines|hexagrams]]</s>, while the 600-cell rotates hexagons on decagrams, but these are discrete instances of the same kind of isoclinic rotation in hexagon invariant planes. In particular, their directly or enantiomorphously congruent isoclines are all a geodesic circle of circumference 4𝝅.{{Efn|All 3-sphere isoclines{{Efn|name=isoclinic geodesic}} 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 2𝝅''r''; simple rotations take place on these isoclines. Double rotations have isoclines of more than 2𝝅''r'' circumference, because their circle does not close in a single revolution. The ''characteristic rotation'' of a regular 4-polytope is the isoclinic (equi-angled) double 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''r'' circumference. The 600-cell edge-rotates on isoclines of 5𝝅''r'' circumference.|name=isocline circumference.}}|name=4𝝅 rotation}} They have different isocline polygrams only because the isocline curve intersects more vertices in the 600-cell than it does in the 24-cell.{{Efn|The 600-cell's helical {20/6}{{=}}2{10/3} [[W:20-gon|icosagram]] is a compound of the 24-cell's <s>helical {6/2} hexagram</s>, which is inscribed within it just as the 24-cell is inscribed in the 600-cell.}} ==== Squares and octagrams ==== [[File:Regular_star_polygon_24-5.svg|thumb|The Clifford polygon of the 600-cell's isoclinic rotation in great square invariant planes is a skew regular [[W:24-gon#Related polygons|{24/5} 24-gram]], with <math>\phi</math> edges that connect vertices 5 apart on the 24-vertex circumference, which is a unique 24-cell (<math>\sqrt{1}</math> edges not shown).]]The [[#Fibrations of great circle polygons|fibrations of the 600-cell]] include 15 [[#Squares|fibrations of its 450 great squares]]: 15 fiber bundles of 30 great squares. The 30 Clifford parallel squares in each bundle are completely disjoint. Adjacent parallel squares are spanned by edges of great decagons.{{Efn|name=equi-isoclinic squares}} Each fibration corresponds to a distinct left (and right) isoclinic rotation of the 600-cell in 30 great square invariant planes (15 completely orthogonal pairs) on 4𝝅 isoclines. Each fiber bundle delineates 30 chiral [[16-cell#Helical construction|cell rings of 8 tetrahedral cells]] each,{{Efn|name=two different tetrahelixes}} with a left and right cell ring nesting together to fill each of the 15 disjoint 16-cells inscribed in the 600-cell. Axial to each 8-tetrahedron ring is a special kind of helical great circle, an isocline.{{Efn|name=isoclinic geodesic}} In a left (or right) isoclinic rotation of the 600-cell in great square invariant planes, all the vertices circulate on one of 15 Clifford parallel isoclines. The 30 Clifford parallel squares in each bundle are joined by four Clifford parallel 24-gram isoclines (one through each vertex), each of which intersects one vertex in 24 of the 30 squares, and all 24 vertices of just one of the 600-cell's 25 24-cells. Each isocline is a 24-gram circuit intersecting all 25 24-cells, 24 of them just once and one of them 24 times. The 24 vertices in each 24-gram isocline comprise a unique 24-cell; there are 25 such distinct isoclines in the 600-cell. Each isocline is a skew {24/5} 24-gram, 24 <math>\phi</math> chords joined end-to-end in a helical loop, winding 5 times around one 24-cell through all four dimensions rather than lying flat in a central plane. Adjacent vertices of the 24-cell are one <math>\sqrt{1}</math> chord apart, and 5 <math>\phi</math> chords apart on its isocline. A left (or right) isoclinic rotation through 720° takes each 24-cell to and through every other 24-cell. Notice the relations between the [[16-cell#Helical construction|16-cell's rotation in just 2 completely orthogonal great square planes]], the [[24-cell#Helical octagrams and their isoclines|24-cell's rotation in 6 Clifford parallel great squares]], and this rotation of the 600-cell in 30 Clifford parallel great squares. These three rotations are the same rotation, taking place on exactly the same kind of isocline circles, which happen to intersect more vertices in the 600-cell (24) than they do in the 16-cell (8).{{Efn|The 16-cell rotates squares on [[16-cell#Helical construction|{8/3} octagrams]], the 24-cell rotates squares on [[24-cell#Helical octagrams and their isoclines|{24/9}=3{8/3} octagrams]], and the 600 rotates squares on {24/5} 24-grams, but these are discrete instances of the same kind of isoclinic rotation in great square invariant planes. In particular, their directly or enantiomorphously congruent isoclines are all exactly the same geodesic circle of circumference 4𝝅. They have different isocline polygrams only because the isocline curve intersects more vertices in the 600-cell than it does in the 24-cell or the 16-cell. The 600-cell's helical {24/5} 24-gram is a compound of the 24-cell's helical {24/9} octagram, which is inscribed within the 600-cell just as the 16-cell's helical {8/3} octagram is inscribed within the 24-cell.}} In the 16-cell's rotation the distance between vertices on an isocline curve is the <math>\sqrt{4}</math> diameter. In the 600-cell vertices are closer together, and its <math>\phi</math> chord is the distance between adjacent vertices on the same isocline, but all these isoclines have a 4𝝅 circumference.{{Efn|name=isocline circumference.}} === As a configuration === This [[W:Regular 4-polytope#As configurations|configuration matrix]]{{Sfn|Coxeter|1973|p=12|loc=§1.8. Configurations}} represents the 600-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 600-cell. The non-diagonal numbers say how many of the column's element occur in or at the row's element. <math>\begin{bmatrix}\begin{matrix}120 & 12 & 30 & 20 \\ 2 & 720 & 5 & 5 \\ 3 & 3 & 1200 & 2 \\ 4 & 6 & 4 & 600 \end{matrix}\end{bmatrix}</math> Here is the configuration expanded with ''k''-face elements and ''k''-figures. The diagonal element counts are the ratio of the full [[W:Coxeter group|Coxeter group]] order, 14400, divided by the order of the subgroup with mirror removal. {| class=wikitable !H₄||{{Coxeter–Dynkin diagram|node_1|3|node|3|node|5|node}} ! [[W:k-face|''k''-face]]||f_''k''||f₀ || f₁||f₂||f₃||[[W:Vertex figure|''k''-fig]] !Notes |- align=right |H₃ || {{Coxeter–Dynkin diagram|node_x|2|node|3|node|5|node}} ||( ) !f₀ || 120 || 12 || 30 || 20 ||[[W:icosahedron|{3,5}]] || H₄/H₃ = 14400/120 = 120 |- align=right |A₁H₂ ||{{Coxeter–Dynkin diagram|node_1|2|node_x|2|node|5|node}} ||{ } !f₁ || 2 || 720 || 5 || 5 || [[W:pentagon|{5}]] || H₄/H₂A₁ = 14400/10/2 = 720 |- align=right |A₂A₁ ||{{Coxeter–Dynkin diagram|node_1|3|node|2|node_x|2|node}} ||[[W:equilateral triangle|{3}]] !f₂ || 3 || 3 || 1200 || 2 || { } || H₄/A₂A₁ = 14400/6/2 = 1200 |- align=right |A₃ ||{{Coxeter–Dynkin diagram|node_1|3|node|3|node|2|node_x}} ||[[W:tetrahedron|{3,3}]] !f₃ || 4 || 6 || 4 || 600|| ( ) || H₄/A₃ = 14400/24 = 600 |} == Symmetries == The [[W:Icosian|icosian]]s are a specific set of Hamiltonian [[W:Quaternion|quaternion]]s with the same symmetry as the 600-cell.{{Sfn|van Ittersum|2020|loc=§4.3|pp=80-95}} The icosians lie in the ''golden field'', (''a'' + ''b''{{radic|5}}) + (''c'' + ''d''{{radic|5}})'''i''' + (''e'' + ''f''{{radic|5}})'''j''' + (''g'' + ''h''{{radic|5}})'''k''', where the eight variables are [[W:Rational number|rational number]]s.{{Sfn|Steinbach|1997|p=24}} The finite sums of the 120 [[W:Icosian#Unit icosians|unit icosians]] are called the [[W:Icosian#Icosian ring|icosian ring]]. When interpreted as quaternions,{{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 120 vertices of the 600-cell form a [[W:group (mathematics)|group]] under quaternionic multiplication. This group is often called the [[W:Binary icosahedral group|binary icosahedral group]] and denoted by ''2I'' as it is the double cover of the ordinary [[W:Icosahedral group|icosahedral group]] ''I''.{{Sfn|Stillwell|2001|loc=The Poincaré Homology Sphere|pp=22-23}} It occurs twice in the rotational symmetry group ''RSG'' of the 600-cell as an [[W:Invariant subgroup|invariant subgroup]], namely as the subgroup ''2I_L'' of quaternion left-multiplications and as the subgroup ''2I_R'' of quaternion right-multiplications. Each rotational symmetry of the 600-cell is generated by specific elements of ''2I_L'' and ''2I_R''; the pair of opposite elements generate the same element of ''RSG''. The [[W:Center of a group|centre]] of ''RSG'' consists of the non-rotation ''Id'' and the central inversion ''−Id''. We have the isomorphism ''RSG ≅ (2I_L × 2I_R) / {Id, -Id}''. The order of ''RSG'' equals {{sfrac|120 × 120|2}} = 7200. The [[W:Quaternion algebra|quaternion algebra]] as a tool for the treatment of 3D and 4D rotations, and as a road to the full understanding of the theory of [[W:Rotations in 4-dimensional Euclidean space|rotations in 4-dimensional Euclidean space]], is described by Mebius.{{Sfn|Mebius|1994|p=1|loc="''[[W:Quaternion algebra|Quaternion algebra]]'' is the tool ''par excellence'' for the treatment of three- and four- dimensional (3D and 4D) rotations. Obviously only 3D and by implication 2D rotations have an everyday practical meaning, but the [[W:Rotations in 4-dimensional Euclidean space|theory of 4D rotations]] turns out to offer the easiest road to the representation of 3D rotations by quaternions."}} The binary icosahedral group is [[W:Isomorphic|isomorphic]] to [[W:special linear group|SL(2,5)]]. The full [[W:Symmetry group|symmetry group]] of the 600-cell is the [[W:H4 (mathematics)|Coxeter group H₄]].{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=§2 The Labeling of H₄}} This is a [[W:Group (mathematics)|group]] of order 14400. It consists of 7200 [[W:Rotation (mathematics)|rotations]] and 7200 rotation-reflections. The rotations form an [[W:Invariant subgroup|invariant subgroup]] of the full symmetry group. The rotational symmetry group was first described by S.L. van Oss.{{Sfn|van Oss|1899||pp=1-18}} The H₄ group and its Clifford algebra construction from 3-dimensional symmetry groups by induction is described by Dechant.{{Sfn|Dechant|2021|loc=Abstract|ps=; "[E]very 3D root system allows the construction of a corresponding 4D root system via an 'induction theorem'. In this paper, we look at the icosahedral case of H3 → H4 in detail and perform the calculations explicitly. Clifford algebra is used to perform group theoretic calculations based on the versor theorem and the Cartan-Dieudonné theorem ... shed[ding] light on geometric aspects of the H4 root system (the 600-cell) as well as other related polytopes and their symmetries ... including the construction of the Coxeter plane, which is used for visualising the complementary pairs of invariant polytopes.... This approach therefore constitutes a more systematic and general way of performing calculations concerning groups, in particular reflection groups and root systems, in a Clifford algebraic framework."}} == Visualization == The symmetries of the 3-D surface of the 600-cell are somewhat difficult to visualize due to both the large number of tetrahedral cells,{{Efn||name=tetrahedral cell adjacency}} and the fact that the tetrahedron has no opposing faces or vertices.{{Efn|name=directly congruent versus twisted cell rings}} One can start by realizing the 600-cell is the dual of the 120-cell. One may also notice that the 600-cell also contains the vertices of a dodecahedron,{{Sfn|Coxeter|1973|loc=Table VI (iii): 𝐈𝐈 = {3,3,5}|p=303}} which with some effort can be seen in most of the below perspective projections. === 2D projections === The H3 [[W:Decagon|decagon]]al projection shows the plane of the [[W:van Oss polygon|van Oss polygon]]. {| class="wikitable" width=600 |+ [[W:Orthographic projection|Orthographic projection]]s by [[W:Coxeter plane|Coxeter plane]]s{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}} |- align=center !H₄ ! - !F₄ |- align=center |[[File:600-cell graph H4.svg|200px]]<br>[30]<br>(Red=1) |[[File:600-cell t0 p20.svg|200px]]<br>[20]<br>(Red=1) |[[File:600-cell t0 F4.svg|200px]]<br>[12]<br>(Red=1) |- align=center !H₃ !A₂ / B₃ / D₄ !A₃ / B₂ |- align=center |[[File:600-cell t0 H3.svg|200px]]<br>[10]<br>(Red=1,orange=5,yellow=10) |[[File:600-cell t0 A2.svg|200px]]<br>[6]<br>(Red=1,orange=3,yellow=6) |[[File:600-cell t0.svg|200px]]<br>[4]<br>(Red=1,orange=2,yellow=4) |} === 3D projections === A three-dimensional model of the 600-cell, in the collection of the [[W:Institut Henri Poincaré|Institut Henri Poincaré]], was photographed in 1934–1935 by [[W:Man Ray|Man Ray]], and formed part of two of his later "Shakesperean Equation" paintings.<ref>{{citation|title=Man Ray Human Equations: A journey from mathematics to Shakespeare|publisher=Hatje Cantz|editor1-first=Wendy A.|editor1-last=Grossman|editor2-first=Edouard|editor2-last=Sebline|year=2015}}. See in particular ''mathematical object mo-6.2'', p.&nbsp;58; ''Antony and Cleopatra'', SE-6, p.&nbsp;59; ''mathematical object mo-9'', p.&nbsp;64; ''Merchant of Venice'', SE-9, p.&nbsp;65, and "The Hexacosichoron", Philip Ordning, p.&nbsp;96.</ref> {| class=wikitable !colspan=2|Vertex-first projection |- |[[Image:600cell-perspective-vertex-first-multilayer-01.png|320px]] |This image shows a vertex-first perspective projection of the 600-cell into 3D. The 600-cell is scaled to a vertex-center radius of 1, and the 4D viewpoint is placed 5 units away. Then the following enhancements are applied: * The 20 tetrahedra meeting at the vertex closest to the 4D viewpoint are rendered in solid color. Their icosahedral arrangement is clearly shown. * The tetrahedra immediately adjoining these 20 cells are rendered in transparent yellow. * The remaining cells are rendered in edge-outline. * Cells facing away from the 4D viewpoint (those lying on the "far side" of the 600-cell) have been culled, to reduce visual clutter in the final image. |- !colspan=2|Cell-first projection |- |[[Image:600cell-perspective-cell-first-multilayer-02.png|320px]] |This image shows the 600-cell in cell-first perspective projection into 3D. Again, the 600-cell to a vertex-center radius of 1 and the 4D viewpoint is placed 5 units away. The following enhancements are then applied: * The nearest cell to the 4d viewpoint is rendered in solid color, lying at the center of the projection image. * The cells surrounding it (sharing at least 1 vertex) are rendered in transparent yellow. * The remaining cells are rendered in edge-outline. * Cells facing away from the 4D viewpoint have been culled for clarity. This particular viewpoint shows a nice outline of 5 tetrahedra sharing an edge, towards the front of the 3D image. |} === Animations=== {| class=wikitable width=540 !colspan=1|Coxeter section views |- |align=center|[[File:Cell120-OmniTruncated-Sections.webm|300px]]<br>Sections of an omnitrucated 4D 600/120-cell 97 frames (=48x2 L/R+1 Center) shown in 4D to 3D [[W:Flatland|Flatland]]er views. The center section is highlighted by also showing it as a combined set of convex hulls. |} == Diminished 600-cells == The [[W:Snub 24-cell|snub 24-cell]] may be obtained from the 600-cell by removing the vertices of an inscribed [[24-cell|24-cell]] and taking the [[W:Convex hull|convex hull]] of the remaining vertices.{{Sfn|Dechant|2021|pp=22-24|loc=§8. Snub 24-cell}} This process is a ''[[W:Diminishment (geometry)|diminishing]]'' of the 600-cell. The [[W:Grand antiprism|grand antiprism]] may be obtained by another diminishing of the 600-cell: removing 20 vertices that lie on two mutually orthogonal rings and taking the convex hull of the remaining vertices.{{Sfn|Dechant|2021|pp=20-22|loc=§7. The Grand Antiprism and H₂ × H₂}} A bi-24-diminished 600-cell, with all [[W:Tridiminished icosahedron|tridiminished icosahedron]] cells has 48 vertices removed, leaving 72 of 120 vertices of the 600-cell. The dual of a bi-24-diminished 600-cell, is a tri-24-diminished 600-cell, with 48 vertices and 72 hexahedron cells. There are a total of 314,248,344 diminishings of the 600-cell by non-adjacent vertices. All of these consist of regular tetrahedral and icosahedral cells.<ref>{{Cite journal|last1=Sikiric|first1=Mathieu|last2=Myrvold|first2=Wendy|date=2007|title=The special cuts of 600-cell|journal=Beiträge zur Algebra und Geometrie|volume=49|issue=1|arxiv=0708.3443}}</ref> {| class="wikitable collapsible" !colspan=12|Diminished 600-cells |- !Name !Tri-24-diminished 600-cell !Bi-24-diminished 600-cell ![[W:Snub 24-cell|Snub 24-cell]]<br>(24-diminished 600-cell) ![[W:Grand antiprism|Grand antiprism]]<br>(20-diminished 600-cell) !600-cell |- align=center !Vertices |48 |72 |96 |100 |120 |- align=center !Vertex figure<br>(Symmetry) |[[File:Dual tridiminished icosahedron.png|120px]]<br>dual of tridiminished icosahedron<br>([3], order 6) |[[File:Biicositetradiminished 600-cell vertex figure.png|120px]]<br>[[W:Hexahedron|tetragonal antiwedge]]<br>([2]⁺, order 2) |[[File:Snub 24-cell verf.png|120px]]<br>[[W:tridiminished icosahedron|tridiminished icosahedron]]<br>([3], order 6) |[[File:Grand antiprism verf.png|120px]]<br>[[W:Edge-contracted icosahedron|bidiminished icosahedron]]<br>([2], order 4) |[[File:600-cell verf.svg|120px]]<br>[[W:Icosahedron|icosahedron]]<br>([5,3], order 120) |- align=center !Symmetry |colspan=2|Order 144 (48×3 or 72×2) |[3⁺,4,3]<br>Order 576 (96×6) |[10,2⁺,10]<br>Order 400 (100×4) |[5,3,3]<br>Order 14400 (120×120) |- align=center !Net |[[File:Triicositetradiminished hexacosichoron net.png|100px]] |[[File:Biicositetradiminished hexacosichoron net.png|100px]] |[[File:Snub 24-cell-net.png|100px]] |[[File:Grand antiprism net.png|100px]] |[[File:600-cell net.png|100px]] |- align=center !Ortho<br>H₄ plane |[[File:Tridiminished 600-cell H4 Coxeter plane.svg|120px]] |[[File:bidex ortho-30-gon.png|120px]] |[[File:Snub 24-cell ortho30-gon.png|120px]] |[[File:Grand antiprism ortho-30-gon.png|120px]] |[[File:600-cell graph H4.svg|120px]] |- align=center !Ortho<br>F₄ plane |[[File:Tridiminished 600-cell F4 Coxeter plane.svg|120px]] |[[File:Bidex ortho 12-gon.png|120px]] |[[File:24-cell h01 F4.svg|120px]] |[[File:GrandAntiPrism-2D-F4.svg|120px]] |[[File:600-cell t0 F4.svg|120px]] |} == Related polytopes and honeycombs == The 600-cell is one of 15 regular and uniform polytopes with the same H₄ symmetry [3,3,5]:{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020}} {{H4_family}} It is similar to three [[W:Regular 4-polytope|regular 4-polytope]]s: the [[5-cell|5-cell]] {3,3,3}, [[16-cell|16-cell]] {3,3,4} of Euclidean 4-space, and the [[W:Order-6 tetrahedral honeycomb|order-6 tetrahedral honeycomb]] {3,3,6} of hyperbolic space. All of these have [[W:Tetrahedron|tetrahedral]] cells. {{Tetrahedral cell tessellations}} This 4-polytope is a part of a sequence of 4-polytope and honeycombs with [[W:Icosahedron|icosahedron]] vertex figures: {{Icosahedral vertex figure tessellations}} The [[W:regular complex polytope|regular complex polygons]] ₃{5}₃, {{Coxeter–Dynkin diagram|3node_1|5|3node}} and ₅{3}₅, {{Coxeter–Dynkin diagram|5node_1|3|5node}}, in <math>\mathbb{C}^2</math> have a real representation as ''600-cell'' in 4-dimensional space. Both have 120 vertices, and 120 edges. The first has [[W:Complex reflection group|complex reflection group]] ₃[5]₃, order 360, and the second has symmetry ₅[3]₅, order 600.{{Sfn|Coxeter|1991|pp=48-49}} {| class="wikitable collapsed collapsible" !colspan=3| Regular complex polytope in orthogonal projection of H₄ Coxeter plane{{Sfn|Dechant|2021|pp=18-20|loc=§6. The Coxeter Plane}} |- align=center |[[File:600-cell graph H4.svg|240px]]<br>{3,3,5}<br>Order 14400 |[[File:Complex polygon 3-5-3.png|240px]]<br>₃{5}₃<br>Order 360 |[[File:Complex polygon 5-3-5.png|240px]]<br>₅{3}₅<br>Order 600 |} == See also == * [[W:600-cell|Wikipedia:600-cell]], the article this article is an expanded version of * [[24-cell|24-cell]], the predecessor 4-polytope on which the 600-cell is based * [[120-cell|120-cell]], the dual 4-polytope to the 600-cell, and its successor * [[W:Uniform 4-polytope#The H4 family|Uniform 4-polytope family with [5,3,3] symmetry]] * [[W:Regular 4-polytope|Regular 4-polytope]] * [[W:Polytope|Polytope]] == Notes == {{Regular convex 4-polytopes Notelist|wiki=W:}} == Citations == {{Regular convex 4-polytopes Reflist|wiki=W:}} == References == {{Refbegin}} * {{Citation | last=Schläfli | first=Ludwig | author-link=W:Ludwig Schläfli |editor-first=Arthur | editor-last=Cayley | editor-link=W:Arthur Cayley | title=An attempt to determine the twenty-seven lines upon a surface of the third order, and to derive such surfaces in species, in reference to the reality of the lines upon the surface | url=http://resolver.sub.uni-goettingen.de/purl?PPN600494829_0002 | year=1858 | journal=Quarterly Journal of Pure and Applied Mathematics | volume=2 | pages=55–65, 110–120 }} {{Regular convex 4-polytopes Refs|wiki=W:}} * {{Cite journal | first1=F. | last1=Buekenhout | first2=M. | last2=Parker | title=The number of nets of the regular convex polytopes in dimension <= 4 | journal=[[W:Discrete Mathematics (journal)|Discrete Mathematics]] | volume=186 | issue=1–3 | date=15 May 1998 | pages=69–94| doi=10.1016/S0012-365X(97)00225-2 | doi-access=free | ref={{SfnRef|Buekenhout & Parker|1998}} }} * {{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}} }} * [http://www.polytope.de Four-dimensional Archimedean Polytopes] (German), Marco Möller, 2004 PhD dissertation [http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf] {{Webarchive|url=https://web.archive.org/web/20050322235615/http://www.sub.uni-hamburg.de/opus/volltexte/2004/2196/pdf/Dissertation.pdf |date=2005-03-22 }} * {{Cite journal | last=Oss | first=Salomon Levi van | title=Das regelmässige Sechshundertzell und seine selbstdeckenden Bewegungen | journal=Verhandelingen der Koninklijke (Nederlandse) Akademie van Wetenschappen, Sectie 1 (Afdeeling Natuurkunde) | volume=7 | issue=1 | pages=1–18 | place=Amsterdam | year=1899 | url=https://books.google.com/books?id=AfQ3AQAAMAAJ&pg=PA3 | ref={{SfnRef|van Oss|1899}} }} {{Refend}} == External links == * [https://bendwavy.org/klitzing/incmats/ex.htm ex], at [https://bendwavy.org/klitzing/home.htm Klitzing polytopes] * [https://polytope.miraheze.org/wiki/Hexacosichoron Hexacosichoron], at [https://polytope.miraheze.org/wiki/Main_Page Polytope wiki] * [http://hi.gher.space/wiki/Hydrochoron Hydrochoron], at [http://hi.gher.space/wiki/Main_Page Higher space] * [https://www.qfbox.info/4d/600-cell The 600-cell], at [https://www.qfbox.info/4d/index 4D Euclidean Space] [[Category:Geometry]] [[Category:Polyscheme]] 7qlerrfj27h0naav7dahpfvwvaa4cob 0 305659 2811989 2811968 2026-05-29T14:36:08Z Unitfreak 695864 /* First Set */ 2811989 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] === Time span covered by Bully timestamps === With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' *** [[w:Cosmic_inflation|Cosmic Inflation]] *** [[w:Baryogenesis|Baryogenesis]] *** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] ** Approximately: ''{{mono|0000 EA00 0000}}'' *** [[w:Decoupling_(cosmology)|Decoupling]] *** [[w:Recombination_(cosmology)|Recombination]] ** Approximately: ''{{mono|0100 0000 0000}}'' *** [[w:Star_formation|First Star Formation]] ** Approximately: ''{{mono|0297 0000 0000}}'' *** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> ** Approximately: ''{{mono|3B00 0000 0000}}'' *** [[w:Murchison_meteorite|Oldest Presolar Grains]] ** Approximately: ''{{mono|5720 9000 0000}}'' *** [[w:Hadean|Hadean Eon Begins]] ** Approximately: ''{{mono|5C2A 0000 0000}}'' *** [[w:Archean|Archean Eon Begins]] ** Approximately: ''{{mono|6A8C 0000 0000}}'' *** [[w:Proterozoic|Proterozoic Eon Begins]] ** Approximately: ''{{mono|7D56 0000 0000}}'' *** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> Metonic Cycle July 23 New Moon * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn more about the Metonic Cycle in Bully Metric]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Examples of contextualized vs decontextualized time]] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |The Bully Mnemonic]] [[Bully Mnemonic Extension |The Bully Mnemonic Extension]] = Realized vs. Estimated Bully timestamps = Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. == Realized Bully Time == [[Bully_Metric_Realized_Timestamps|Realized Bully Timestamps]] == Estimated Bully Time == == Future Bully Time == [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] ewnlb0y1a6s9fk7in2m4o6x9dcfw838 2811990 2811989 2026-05-29T14:36:31Z Unitfreak 695864 /* First Set */ 2811990 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] === Time span covered by Bully timestamps === With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> ** Approximately: ''{{mono|3B00 0000 0000}}'' *** [[w:Murchison_meteorite|Oldest Presolar Grains]] ** Approximately: ''{{mono|5720 9000 0000}}'' *** [[w:Hadean|Hadean Eon Begins]] ** Approximately: ''{{mono|5C2A 0000 0000}}'' *** [[w:Archean|Archean Eon Begins]] ** Approximately: ''{{mono|6A8C 0000 0000}}'' *** [[w:Proterozoic|Proterozoic Eon Begins]] ** Approximately: ''{{mono|7D56 0000 0000}}'' *** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> Metonic Cycle July 23 New Moon * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn more about the Metonic Cycle in Bully Metric]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Examples of contextualized vs decontextualized time]] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |The Bully Mnemonic]] [[Bully Mnemonic Extension |The Bully Mnemonic Extension]] = Realized vs. Estimated Bully timestamps = Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. == Realized Bully Time == [[Bully_Metric_Realized_Timestamps|Realized Bully Timestamps]] == Estimated Bully Time == == Future Bully Time == [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] otp59w0nl3tvv7ldv3fbwouewu3a25w 2811992 2811990 2026-05-29T14:37:01Z Unitfreak 695864 /* Second Set */ 2811992 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] === Time span covered by Bully timestamps === With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> Metonic Cycle July 23 New Moon * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn more about the Metonic Cycle in Bully Metric]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Examples of contextualized vs decontextualized time]] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |The Bully Mnemonic]] [[Bully Mnemonic Extension |The Bully Mnemonic Extension]] = Realized vs. Estimated Bully timestamps = Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. == Realized Bully Time == [[Bully_Metric_Realized_Timestamps|Realized Bully Timestamps]] == Estimated Bully Time == == Future Bully Time == [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] air5bwdzw84h9ia4re3jr9aw3s2jkjb 2811993 2811992 2026-05-29T14:37:45Z Unitfreak 695864 /* Third Set */ 2811993 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] === Time span covered by Bully timestamps === With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> Metonic Cycle July 23 New Moon * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn more about the Metonic Cycle in Bully Metric]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Examples of contextualized vs decontextualized time]] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |The Bully Mnemonic]] [[Bully Mnemonic Extension |The Bully Mnemonic Extension]] = Realized vs. Estimated Bully timestamps = Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. == Realized Bully Time == [[Bully_Metric_Realized_Timestamps|Realized Bully Timestamps]] == Estimated Bully Time == == Future Bully Time == [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] b573s6nkojcjenm1np2pqxtum435vij 2811994 2811993 2026-05-29T14:38:21Z Unitfreak 695864 /* Metonic Cycle */ 2811994 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] === Time span covered by Bully timestamps === With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn more about the Metonic Cycle in Bully Metric]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Examples of contextualized vs decontextualized time]] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |The Bully Mnemonic]] [[Bully Mnemonic Extension |The Bully Mnemonic Extension]] = Realized vs. Estimated Bully timestamps = Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. == Realized Bully Time == [[Bully_Metric_Realized_Timestamps|Realized Bully Timestamps]] == Estimated Bully Time == == Future Bully Time == [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] a8ru037pzlaqfyw46cko5u4in3uya4u 2811995 2811994 2026-05-29T14:42:18Z Unitfreak 695864 /* Realized vs. Estimated Bully timestamps */ 2811995 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] === Time span covered by Bully timestamps === With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn more about the Metonic Cycle in Bully Metric]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Examples of contextualized vs decontextualized time]] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |The Bully Mnemonic]] [[Bully Mnemonic Extension |The Bully Mnemonic Extension]] tk813dna0u0vcnhpoijnmoblgh5ptk8 2811996 2811995 2026-05-29T14:42:55Z Unitfreak 695864 /* Metonic Cycle */ 2811996 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] === Time span covered by Bully timestamps === With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn more about the Metonic Cycle in Bully Metric]] = Realized vs. Estimated Bully timestamps = Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. == Realized Bully Time == [[Bully_Metric_Realized_Timestamps|Realized Bully Timestamps]] == Estimated Bully Time == == Future Bully Time == [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Examples of contextualized vs decontextualized time]] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |The Bully Mnemonic]] [[Bully Mnemonic Extension |The Bully Mnemonic Extension]] 6qe0dnqi63bdwpuz1mn5egmumi38b1t 2811998 2811996 2026-05-29T14:43:44Z Unitfreak 695864 /* Realized vs. Estimated Bully timestamps */ 2811998 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] === Time span covered by Bully timestamps === With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn more about the Metonic Cycle in Bully Metric]] == Realized vs. Estimated Bully timestamps == Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. === Realized Bully Time === [[Bully_Metric_Realized_Timestamps|Realized Bully Timestamps]] === Estimated Bully Time === === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Examples of contextualized vs decontextualized time]] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |The Bully Mnemonic]] [[Bully Mnemonic Extension |The Bully Mnemonic Extension]] l2xv36srf20jqdssdu7gq1fcom7hqss 2811999 2811998 2026-05-29T14:44:53Z Unitfreak 695864 /* Realized vs. Estimated Bully timestamps */ 2811999 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] === Time span covered by Bully timestamps === With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn more about the Metonic Cycle in Bully Metric]] === Realized Bully Time === [[Bully_Metric_Realized_Timestamps|Realized Bully Timestamps]] === Estimated Bully Time === === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Examples of contextualized vs decontextualized time]] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |The Bully Mnemonic]] [[Bully Mnemonic Extension |The Bully Mnemonic Extension]] mvjzs5x39xlifmydwy1o69n7689jsfx 2812000 2811999 2026-05-29T14:45:35Z Unitfreak 695864 /* Third Set */ 2812000 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] === Time span covered by Bully timestamps === With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn more about the Metonic Cycle in Bully Metric]] === Realized Bully Time === [[Bully_Metric_Realized_Timestamps|Realized Bully Timestamps]] === Estimated Bully Time === === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Examples of contextualized vs decontextualized time]] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |The Bully Mnemonic]] [[Bully Mnemonic Extension |The Bully Mnemonic Extension]] gnw3ws668kybw5xqpxwhh0n3c5l03rz 2812001 2812000 2026-05-29T14:45:57Z Unitfreak 695864 /* Realized Bully Time */ 2812001 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] === Time span covered by Bully timestamps === With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn more about the Metonic Cycle in Bully Metric]] === Estimated Bully Time === === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Examples of contextualized vs decontextualized time]] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |The Bully Mnemonic]] [[Bully Mnemonic Extension |The Bully Mnemonic Extension]] dj9mjazj0yfhein1dt0v6xn3mmufxum 2812002 2812001 2026-05-29T14:47:14Z Unitfreak 695864 /* Realized vs. Estimated Bully timestamps */ 2812002 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] === Time span covered by Bully timestamps === With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn more about the Metonic Cycle in Bully Metric]] === Estimated Bully Time === === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Examples of contextualized vs decontextualized time]] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |The Bully Mnemonic]] [[Bully Mnemonic Extension |The Bully Mnemonic Extension]] 96737qc7md3rrv2s7vsc8ghmlotapzi 2812003 2812002 2026-05-29T14:49:39Z Unitfreak 695864 /* Realized vs. Estimated Bully timestamps */ 2812003 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] === Time span covered by Bully timestamps === With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... 2025 AD). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn more about the Metonic Cycle in Bully Metric]] === Estimated Bully Time === === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Examples of contextualized vs decontextualized time]] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |The Bully Mnemonic]] [[Bully Mnemonic Extension |The Bully Mnemonic Extension]] 5hkbio5es0suiwi48es5sgdtrdiaflo 2812004 2812003 2026-05-29T14:51:19Z Unitfreak 695864 /* Time span covered by Bully timestamps */ 2812004 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... 2025 AD). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn more about the Metonic Cycle in Bully Metric]] === Estimated Bully Time === === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Examples of contextualized vs decontextualized time]] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |The Bully Mnemonic]] [[Bully Mnemonic Extension |The Bully Mnemonic Extension]] 6s3lojdr706o2nxtdv0wuhb7ab0d4sp 2812005 2812004 2026-05-29T14:51:49Z Unitfreak 695864 2812005 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... 2025 AD). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn more about the Metonic Cycle in Bully Metric]] === Estimated Bully Time === === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Examples of contextualized vs decontextualized time]] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |The Bully Mnemonic]] [[Bully Mnemonic Extension |The Bully Mnemonic Extension]] efqcbzqu1zur53lhclb3cnkpchdervy 2812006 2812005 2026-05-29T14:57:00Z Unitfreak 695864 /* Metonic Cycle */ 2812006 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... 2025 AD). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Metric]] === Estimated Bully Time === === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Examples of contextualized vs decontextualized time]] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |The Bully Mnemonic]] [[Bully Mnemonic Extension |The Bully Mnemonic Extension]] k0s81molpwt8qysm4iik5x7gdm840l6 2812007 2812006 2026-05-29T14:57:47Z Unitfreak 695864 /* Estimated Bully Time */ 2812007 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... 2025 AD). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Metric]] === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Examples of contextualized vs decontextualized time]] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |The Bully Mnemonic]] [[Bully Mnemonic Extension |The Bully Mnemonic Extension]] 9e4hi4kh65gsgoclf8ezqhku56bpj3m 2812009 2812007 2026-05-29T14:59:28Z Unitfreak 695864 /* Contextualized vs. Decontextualized Time */ 2812009 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... 2025 AD). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Metric]] === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |The Bully Mnemonic]] [[Bully Mnemonic Extension |The Bully Mnemonic Extension]] da8vzgcxt1k4ukoa8qa369f3he2y1u6 2812010 2812009 2026-05-29T15:00:37Z Unitfreak 695864 /* Why do we need Bully timestamps? */ 2812010 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... 2025 AD). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Metric]] === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |The Bully Mnemonic]] [[Bully Mnemonic Extension |The Bully Mnemonic Extension]] 9d8xosyss3fllbsmwc49bg37ympf221 2812012 2812010 2026-05-29T15:01:39Z Unitfreak 695864 /* Metonic Cycle */ 2812012 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... 2025 AD). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |The Bully Mnemonic]] [[Bully Mnemonic Extension |The Bully Mnemonic Extension]] njx2ym96ihppwly70v7w43xb2yu1elx 2812013 2812012 2026-05-29T15:02:57Z Unitfreak 695864 /* The Foundations of Bully Metric */ 2812013 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... 2025 AD). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |The Bully Mnemonic]] [[Bully Mnemonic Extension |The Bully Mnemonic Extension]] 8xaz47rlvjq3n3moysn6xrqh8lfneqk 2812014 2812013 2026-05-29T15:03:47Z Unitfreak 695864 /* The Bully Mnemonic */ 2812014 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... 2025 AD). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. [[Bully Mnemonic |Learn More About The Bully Mnemonic]] [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] dj5eoo4yjijzre6bppy1a85wgtb7hyp 2812015 2812014 2026-05-29T15:04:13Z Unitfreak 695864 /* The Bully Mnemonic */ 2812015 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... 2025 AD). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System is derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] ehgleup2no4w5q7drrqtmvtydm2ze70 2812016 2812015 2026-05-29T15:04:49Z Unitfreak 695864 /* The Foundations of Bully Metric */ 2812016 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures the formative universe, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). Here are a few significant events from selected timestamps during the formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Measures cosmic look-back time, starting approximately 10.4 billion years in the past and ending at 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Here are a few significant events from selected timestamps during the presolar through geological eras: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... 2025 AD). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] g9z53mmabgr4kxw2xae2ea1mv59177n 2812023 2812016 2026-05-29T15:51:59Z Unitfreak 695864 /* Bully timestamp Divisions */ 2812023 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... 2025 AD). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] 8tyc3u31xftd0da56nz7fefla7tzh3a 2812024 2812023 2026-05-29T15:55:14Z Unitfreak 695864 /* Third Set */ 2812024 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired on Earth in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... 2025 AD). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] 9hslyequrof299pkduqcuvcx0e07mqj 2812025 2812024 2026-05-29T15:56:07Z Unitfreak 695864 /* Realized vs. Estimated Bully timestamps */ 2812025 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... 2025 AD). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] 2jd0fexzmjvnhtyh7od1jyqes73rruw 2812026 2812025 2026-05-29T15:57:14Z Unitfreak 695864 /* Realized vs. Estimated Bully timestamps */ 2812026 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] bf6gawllwad17p9mlf934fb28ugqytx 2812027 2812026 2026-05-29T15:59:28Z Unitfreak 695864 /* Metonic Cycle */ 2812027 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Future Bully Time === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] 6e58szlxqnhntxixc7d1wp0p46tqq9y 2812028 2812027 2026-05-29T16:00:55Z Unitfreak 695864 /* Future Bully Time */ 2812028 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] == Relativistic and Cosmological Considerations == [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] pa7t1k1ibtbsovz0xmc3jaei0fykxbp 2812029 2812028 2026-05-29T16:01:08Z Unitfreak 695864 /* Relativistic and Cosmological Considerations */ 2812029 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] ebmll2jzua1cj1o4o4pe2ritvxjvbdi 2812030 2812029 2026-05-29T16:08:02Z Unitfreak 695864 /* Relativistic and Cosmological Considerations */ 2812030 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] lk0xrn1vr1p2rbu1bjfoqpzejhse4l9 2812032 2812030 2026-05-29T16:16:23Z Unitfreak 695864 /* Relativistic and Cosmological Considerations */ 2812032 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The 'age of the universe' cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. [[Bully_Metric_CMB_Stabilized_Timestamps| CMB Stabilized Bully Timestamps]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] oppryf21ff7huya7xmys28l78l7ouia 2812035 2812032 2026-05-29T16:45:09Z Unitfreak 695864 /* Relativistic and Cosmological Considerations */ 2812035 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit hexadecimal "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set utilizing the CMB rest frame. Timestamps in the second and third sets are realized using atomic clocks at sea level on Earth. Due to gravitational time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. Consequently, all "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decayed at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] 0f9jmhdc5jds28714x9bqucikrwb9v5 2812036 2812035 2026-05-29T16:47:18Z Unitfreak 695864 2812036 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set utilizing the CMB rest frame. Timestamps in the second and third sets are realized using atomic clocks at sea level on Earth. Due to gravitational time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. Consequently, all "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decayed at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] 76tc8aag2m01s6ky7zsoqui89a8a9f6 2812037 2812036 2026-05-29T16:48:31Z Unitfreak 695864 /* Bully timestamp Divisions */ 2812037 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set utilizing the CMB rest frame. Timestamps in the second and third sets are realized using atomic clocks at sea level on Earth. Due to gravitational time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. Consequently, all "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decayed at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] e95ch6qfowi06frwuac8pem0xn1ihui 2812038 2812037 2026-05-29T16:50:12Z Unitfreak 695864 /* Third Set */ 2812038 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set utilizing the CMB rest frame. Timestamps in the second and third sets are realized using atomic clocks at sea level on Earth. Due to gravitational time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. Consequently, all "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decayed at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] iv56t9u781u4my4pm0iww0wufbsw1gf 2812039 2812038 2026-05-29T16:50:32Z Unitfreak 695864 /* Third Set */ 2812039 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the 66 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set utilizing the CMB rest frame. Timestamps in the second and third sets are realized using atomic clocks at sea level on Earth. Due to gravitational time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. Consequently, all "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decayed at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] mpyxe9pdpvxonokdop5rn925y2e6wac 2812040 2812039 2026-05-29T16:52:12Z Unitfreak 695864 /* Realized vs. Estimated Bully timestamps */ 2812040 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655,360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the more than 65 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set utilizing the CMB rest frame. Timestamps in the second and third sets are realized using atomic clocks at sea level on Earth. Due to gravitational time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. Consequently, all "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decayed at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] 13ewp5eusyfkzgrzwjj6ap4gd4wlygx 2812043 2812040 2026-05-29T16:57:36Z Unitfreak 695864 /* The Metonic Cycle */ 2812043 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655,360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the more than 65 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set utilizing the CMB rest frame. Timestamps in the second and third sets are realized using atomic clocks at sea level on Earth. Due to gravitational time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. Consequently, all "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decayed at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] cgwyaekwawnbiqosfkl9tj6uh2jf8to 2812044 2812043 2026-05-29T17:00:37Z Unitfreak 695864 /* Relativistic and Cosmological Considerations */ 2812044 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655,360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the more than 65 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set utilizing the CMB rest frame. Timestamps in the second and third sets are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. Consequently, all "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decayed at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] q7a2d3i1mmr7yz4crhs1ecju37ridz9 2812045 2812044 2026-05-29T17:02:41Z Unitfreak 695864 /* Contextualized vs. Decontextualized Time */ 2812045 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655,360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the more than 65 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set utilizing the CMB rest frame. Timestamps in the second and third sets are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. Consequently, all "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decayed at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 2''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 2: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] or0u357v29u7mpuwh7uqz1qgtcsyhi7 2812046 2812045 2026-05-29T17:08:21Z Unitfreak 695864 /* Contextualized vs. Decontextualized Time */ 2812046 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655,360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the more than 65 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set utilizing the CMB rest frame. Timestamps in the second and third sets are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. Consequently, all "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decayed at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 3''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 2''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 3: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] m0aglb2un26q4hoqnvjoqw7l301o2x1 2812047 2812046 2026-05-29T17:08:44Z Unitfreak 695864 /* Contextualized vs. Decontextualized Time */ 2812047 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang (see Figure 1 above). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 (see Figure 2 below). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655,360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the more than 65 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set utilizing the CMB rest frame. Timestamps in the second and third sets are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. Consequently, all "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decayed at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 3''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 3''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 3: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] pqzh473q9g0fwpsgvas81mziewayha4 2812048 2812047 2026-05-29T17:09:47Z Unitfreak 695864 /* Bully timestamp Divisions */ 2812048 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang ('''Figure 1'''). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 ('''Figure 2'''). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655,360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the more than 65 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set utilizing the CMB rest frame. Timestamps in the second and third sets are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. Consequently, all "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decayed at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 3''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 3''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 3: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 2''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] gdmo0b52takyt21ruqpvuvb9fp699z1 2812049 2812048 2026-05-29T17:10:44Z Unitfreak 695864 /* Contextualized vs. Decontextualized Time */ 2812049 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang ('''Figure 1'''). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 ('''Figure 2'''). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655,360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the more than 65 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set utilizing the CMB rest frame. Timestamps in the second and third sets are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. Consequently, all "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decayed at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 3''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 3''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 3: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 3''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 1''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 1: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] ff9vcdq3b42n2jejxh89wr20216k3yk 2812050 2812049 2026-05-29T17:11:12Z Unitfreak 695864 /* Why do we need Bully timestamps? */ 2812050 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang ('''Figure 1'''). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 ('''Figure 2'''). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655,360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the more than 65 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set utilizing the CMB rest frame. Timestamps in the second and third sets are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. Consequently, all "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decayed at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 3''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 3''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 3: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 3''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 4''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 4: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 1''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] q0ry1rasnjddulocrlc8ys4uj95iws9 2812051 2812050 2026-05-29T17:12:18Z Unitfreak 695864 /* Legacy Decontextualized Timestamps */ 2812051 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang ('''Figure 1'''). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 ('''Figure 2'''). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655,360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the more than 65 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set utilizing the CMB rest frame. Timestamps in the second and third sets are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. Consequently, all "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decayed at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 3''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 3''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 3: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 3''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 4''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 4: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 4''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 1''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] cnnmw8bb6ebpbb1vlq5i5z6fzgjf46d 2812052 2812051 2026-05-29T17:13:41Z Unitfreak 695864 /* Decontextualized Bully Timestamps */ 2812052 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang ('''Figure 1'''). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 ('''Figure 2'''). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655,360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the more than 65 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set utilizing the CMB rest frame. Timestamps in the second and third sets are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. Consequently, all "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decayed at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 3''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 3''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 3: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 3''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 4''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 4: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 4''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 4''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the larger [[Bully_Metric_Foundations|Great Weeks and Great Years]]. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] nsumwujmaqf0tqzwdbr6gb7xlhch5vw 2812053 2812052 2026-05-29T17:15:57Z Unitfreak 695864 /* The Foundations of Bully Metric */ 2812053 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] = Bully timestamp Divisions = The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang ('''Figure 1'''). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 ('''Figure 2'''). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655,360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the more than 65 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set utilizing the CMB rest frame. Timestamps in the second and third sets are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. Consequently, all "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decayed at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 3''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 3''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 3: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 3''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 4''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 4: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 4''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 4''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] rsdx0styuf22i1co2gu5ijw1c5kf8vl 2812069 2812053 2026-05-29T21:50:10Z Unitfreak 695864 2812069 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] == Bully timestamp Divisions == The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang ('''Figure 1'''). The following list highlights key events from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 ('''Figure 2'''). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655,360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the more than 65 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set utilizing the CMB rest frame. Timestamps in the second and third sets are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. Consequently, all "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decayed at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 3''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 3''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 3: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 3''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 4''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 4: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 4''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 4''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] rnjaajjdx3q9iun1nycv5af2uh7lam0 2812070 2812069 2026-05-29T21:57:07Z Unitfreak 695864 /* First Set */ 2812070 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] == Bully timestamp Divisions == The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events ('''Figure 1''') from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998 ('''Figure 2'''). Key milestones from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655,360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the more than 65 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set utilizing the CMB rest frame. Timestamps in the second and third sets are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. Consequently, all "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decayed at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 3''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 3''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 3: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 3''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 4''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 4: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 4''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 4''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] 3yydusw3p3elm6m0vt5w4opkebhtukq 2812071 2812070 2026-05-29T21:58:22Z Unitfreak 695864 /* Second Set */ 2812071 wikitext text/x-wiki <small>[[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> </small> In the '''Bully Timestamp System''', time is measured using 12-digit [[w:hexadecimal|hexadecimal]] "Bully timestamps," with a new timestamp realized every 3,055 SI seconds (TAI). With 12 hexadecimal digits, the system has a enough unique identifiers to span the entire history of the universe—from the Big Bang into the far-distant future. The total capacity of the system is: :<math>16^{12} \times 3,055 \text{ seconds} \approx 27.25 \text{ billion years}</math> [[File:History-of-the-Universe With Bully Timestamps.jpg|frame|center|text-bottom|Figure 1: History of the Universe with a few example Bully timestamps shown in red.]] == Bully timestamp Divisions == The Bully system's time range is divided into three distinct sets: === First Set === * ''{{mono|0000 0000 0000}}'' — ''{{mono|1FFF FFFF FFFF}}'': Measures time during the universe's formative period, spanning roughly 3 billion years beginning with the Big Bang. The following list highlights key events ('''Figure 1''') from selected timestamps during this formative era: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * First timestamp: ''{{mono|0000 0000 0000}}'' ** [[w:Cosmic_inflation|Cosmic Inflation]] ** [[w:Baryogenesis|Baryogenesis]] ** [[w:Big_Bang_nucleosynthesis|Nucleosynthesis]] * Approximately: ''{{mono|0000 EA00 0000}}'' ** [[w:Decoupling_(cosmology)|Decoupling]] ** [[w:Recombination_(cosmology)|Recombination]] * Approximately: ''{{mono|0100 0000 0000}}'' ** [[w:Star_formation|First Star Formation]] * Approximately: ''{{mono|0297 0000 0000}}'' ** [[w:MoM-z14|Oldest Observed Galaxy]] </div> === Second Set === * {{mono|2000 0000 0000}} — {{mono|8209 2800 0000}}: Tracks cosmic look-back time, spanning from approximately 10.4 billion years ago to exactly 12:00:00 TAI on June 21, 1998. Key milestones ('''Figure 2''') from the presolar through geological eras include: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> * Approximately: ''{{mono|3B00 0000 0000}}'' ** [[w:Murchison_meteorite|Oldest Presolar Grains]] * Approximately: ''{{mono|5720 9000 0000}}'' ** [[w:Hadean|Hadean Eon Begins]] * Approximately: ''{{mono|5C2A 0000 0000}}'' ** [[w:Archean|Archean Eon Begins]] * Approximately: ''{{mono|6A8C 0000 0000}}'' ** [[w:Proterozoic|Proterozoic Eon Begins]] * Approximately: ''{{mono|7D56 0000 0000}}'' ** [[w:Phanerozoic|Phanerozoic Eon Begins]] </div> [[File:Geologic time scale - spiral - ICS colours (light) - path text.svg|frame|center|text-bottom|alt=Geologic time scale proportionally represented as a log-spiral. The image also shows some notable events in Earth's history and the general evolution of life.|thumb|Figure 2: The geologic time scale, proportionally represented as a [[w:Logarithmic_spiral|log-spiral]] with some major events in Earth's history. A [[w:megaannum|megaannum]] (Ma) represents one million (10⁶) years.]] === Third Set === * {{mono|8209 2800 0000}} — {{mono|FFFF FFFF FFFF}}: Begins at precisely 12:00:00 TAI on June 21, 1998, and progresses forward for approximately 13.4 billion years. ==== Realized vs. Estimated Bully timestamps ==== Each Bully timestamp is realized exactly 3055 seconds TAI after the previous one. However, since atomic clocks did not exist prior to the 1950's, any assignment of Bully timestamps prior to 1958 should be viewed as an estimate of how elapsed time might have transpired in the past, rather than an actual realization of Bully time. Bully time should only be considered "realized" when time is measured with an accuracy of <math>{10}^{-10}</math>. There have been over 655,360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the more than 65 years of modern atomic time keeping (1958 AD ... present). [[Bully_Metric_Realized_Timestamps|Learn More About Realized Bully Timestamps]] ==== The Metonic Cycle ==== The '''Metonic cycle''' is a period of approximately 19 solar years, after which the moon's phases recur on the same days of the year. For example, a New Moon occurred on July 23 in 1998, and nineteen years later, in 2017, a New Moon again occurred on July 23. The last four hex digits of the Bully timestamp cycle approximately three times per Metonic cycle as illustrated in the following list: <div style="background-color: #f0f4f7; padding: 15px; border-left: 5px solid #009688;"> July 23 New Moon Metonic Cycle * July 23, 1998 on 8209 280'''0 038B''' * July 23, 2017 on 8209 280'''3 0238''' * July 23, 2036 on 8209 280'''6 00EA''' * July 23, 2055 on 8209 280'''8 FF9B''' * July 23, 2074 on 8209 280'''B FE45''' * July 23, 2093 on 8209 280'''E FCE6''' </div> [[Bully_Metric_Metonic_cycle|Learn More About the Metonic Cycle in Bully Timestamps]] === Relativistic and Cosmological Considerations === What does it mean when cosmologists state that the universe is approximately 13.8 billion years old? According to Einstein's theories of special and general relativity, time passes differently for each observer depending on their path through spacetime and the gravitational forces in their vicinity. How, then, can the universe have a single age? Shouldn't its age depend entirely on the observer's frame of reference? The "age of the universe" cited by cosmologists is actually its maximum possible age. Among all paths an observer could take through spacetime, one specific trajectory maximizes elapsed time. This privileged frame of reference belongs to an observer who remains at rest relative to the Cosmic Microwave Background (CMB) and resides in a region of space with negligible matter. We will refer to this as the "CMB rest frame." Importantly, Bully timestamps are divided into three distinct sets, with only the first set utilizing the CMB rest frame. Timestamps in the second and third sets are realized using atomic clocks at sea level on Earth. Due to relativistic time dilation, these terrestrial clocks run slower than identically constructed clocks placed at rest in empty space. Consequently, all "realized" Bully timestamps from 1958 to the present conform to Earth's sea-level frame of reference. Furthermore, the "estimated" Bully timestamps in the second set are typically derived from the radioactive decay of samples found on or within the Earth; thus, these samples decayed at a rate comparable to Earth's sea-level frame. The oldest timestamps in this second set come from presolar grains, which formed in different star systems prior to the emergence of our solar system. Because some of these samples may have traveled through space in frames of reference drastically different from Earth's current sea-level frame, the accuracy of these cosmic estimates is inherently limited. [[Bully_Metric_CMB_Stabilized_Timestamps| Learn More About Relativistic and Cosmological Considerations]] == Contextualized vs. Decontextualized Time == Local clocks and calendars reflect '''contextualized time''', which uses region-specific offsets from Coordinated Universal Time (UTC) to align with physical reality. This time is "contextual" because it provides an intuitive sense of conditions at some specific geographic location; for instance, a traveler arriving in London at 4:00 a.m. can instinctively expect darkness and quiet streets. To maintain this alignment with Earth's natural cycles, UTC requires periodic "leaps" (seconds and years). In '''Figure 3''', the light blue line represents Earth's irregular rotation ('''UT1'''), while the dark blue line shows '''UTC''', which is manually adjusted with leap seconds to track UT1. In contrast, standards such as International Atomic Time ('''TAI'''), Terrestrial Time ('''TT'''), and '''GPS time''' are '''decontextualized'''. They are independent of Earth's rotation, meaning they do not correspond to "true time" at any specific geographical location. Represented by the black lines in '''Figure 3''', these standards track a continuous, uniform interval measured by atomic clocks. This uninterrupted linearity is vital for scientific and technical systems, where the discontinuities introduced by leap seconds could lead to critical errors or system failures. [[File:Bully Timestamps in relation to modern time keeping.png|frame|center|text-bottom|Figure 3: Modern Time Keeping]] The various decontextualized standards currently in use are effectively "frozen" in the astronomical conditions present at the time of their deployment. Because long-term changes in Earth's motion are unpredictable, each system launched with a different initial offset. For example, when GPS was launched in 1980, the '''Delta T''' adjustment (TT-UTC) exceeded 51 seconds. In contrast, the 1972 LORAN-C upgrade began with an adjustment closer to 42 seconds. This historical discrepancy results in a permanent nine-second offset between GPS and LORAN-C. Similarly, LORAN-C remains offset from TAI (deployed in 1958) by exactly ten seconds. The Bully timestamp system, shown on the far-right axis of '''Figure 3''', follows the same uniform, decontextualized logic as TAI and TT but avoids this "legacy offset" confusion. Unlike existing standards, Bully timestamps are not linked to others by a constant, arbitrary time offset. This independence ensures they are uniquely recognizable and impossible to misinterpret. [[Bully_Metric_Timestamp_units|Learn More About Contextualized vs Decontextualized time]] == Why do we need Bully timestamps? == All the timestamps in '''Figure 4''' refer to one single, simultaneous moment in time. The left frame illustrates the fragmentation of Coordinated Universal Time (UTC) through time zones. For instance, on June 21, 1998, a UTC time of 11:59:29 a.m. in Accra, Ghana, was simultaneously 8:59:29 p.m. in Tokyo. These time zone offsets are not based on science, but on '''political mandates''' that have resulted in [https://en.wikipedia.org/wiki/List_of_UTC_offsets 38 distinct UTC offsets], including confusing half- and quarter-hour increments. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Figure 4: UTC Time Zones vs. Bully Timestamps. |- ! Selected UTC Time Zones !! [https://gssc.esa.int/navipedia/index.php/Transformations_between_Time_Systems Decontextualized timestamps] |- | rowspan = 3 | [[File:Timezone-boundary-builder_release_2023d.png|thumb|upright=1.0| June 21, 1998 at 8:59:29 pm (JST)</br> June 21, 1998 at 7:59:29 pm (CST)</br> June 21, 1998 at 2:59:29 pm (EEST)</br> June 21, 1998 at 12:59:29 pm (IST)</br> June 21, 1998 at 11:59:29 am (GMT)</br> June 21, 1998 at 8:59:29 am (BRT)</br> June 21, 1998 at 4:59:29 am (PDT)</br> June 21, 1998 at 1:59:29 am (HST)</br> ]] || [[File:WorldMap-Blank-Noborders.svg|thumb|<br/> 06/21/1998 12:00:32.184 (TT)<br/> 06/21/1998 12:00:00 (TAI)<br/> 06/21/1998 11:59:42 (GPS) ]] |- ! Bully Timestamp |- || [[File:WorldMap-Blank-Noborders.svg|thumb|8209 2800 0000 (+ 0.000 sec)]] |} ==== Legacy Decontextualized Timestamps ==== The decontextualized timestamps (TAI, TT, GPS) in the upper-right frame of '''Figure 4''' attempt to solve the UTC geographic fragmentation problem, yet they remain "cluttered" by Gregorian formatting. Applying a Gregorian date—which is built to track the Sun—to an atomic standard is a '''category error'''. Seeing three different timestamps share the same date while differing by several "leap" seconds is intellectually disorienting because the date has been stripped of its astronomical meaning. In these technical contexts, the Gregorian format is an artificial mask applied for convenience, hiding the true linear nature of time. For scientific and technical applications, TAI and TT are often expressed via '''Modified Julian Date (MJD)'''—a continuous count of SI days since a fixed epoch. While MJD avoids Gregorian irregularities, it remains "tethered" to the 86,400-second day, a unit that is astronomically meaningless when decontextualized. Similarly, '''GPS time''' relies on a week-based count (since January 6, 1980), forcing a technical system to conform to an arbitrary seven-day cycle. Both systems are cumbersome "hybrids" that attempt to measure linear time using units designed for Earth’s rotation. ==== Decontextualized Bully Timestamps ==== The '''Bully Timestamp''', shown in the lower-right frame of '''Figure 4''', breaks the Gregorian formatting tether. It is a single, unique identifier that applies simultaneously to all locations on Earth because it is never adjusted for geography or orbital drift. For example, Bully timestamp {{mono|8209 2800 0000}} was realized at the exact moment the UTC based clock read 11:59:29 a.m. in Accra and 8:59:29 p.m. in Tokyo. By discarding the baggage of weeks, days, and hours, the Bully timestamp emerges as the least ambiguous format for representing universal, decontextualized time. Click on the below links for a comparison of current time in six time standards (local, UTC, GPS, Loran, and TAI), all displayed using traditional Gregorian format: [http://www.leapsecond.com/m/gps.htm LeapSecond.com] [https://www.ipses.com/eng/in-depth-analysis/standard-of-time-definition ipses.com] [http://www.csgnetwork.com/multitimedisp.html csgnetwork.com] == The Foundations of Bully Metric == The Bully Timestamp System was derived from the orbital periods of major Solar System bodies. Specifically, the duration of Earth's '''sidereal year''' (~31,558,150 seconds) is roughly equal to <math>10,330 \times 3,055</math> SI seconds. This foundational constant—3,055 seconds—serves as the building block for the Bully timestamp system. The name "Bully" is a dual-reference to the massive astronomical objects that define our local spacetime. In an archaic sense, "bully" means '''"beautiful" or "excellent,"''' describing the celestial harmony of the cosmos. In the modern sense, it refers to the '''dominance and gravitational influence''' of "bullies" like [https://en.wikipedia.org/wiki/Sagittarius_A* Sagittarius A*], the [https://en.wikipedia.org/wiki/Sun Sun], and giant planets like Jupiter and Saturn. These massive bodies dictate the motion of everything around them, serving as the physical anchors for the Bully Metric system. * [[Bully_Metric_Foundations|Learn More About The Foundations of Bully Metric]] * [[Bully_Metric_Astronomical_Coordinates|Learn More About The Bully Metric Coordinate System]] == The Bully Mnemonic == <math display="block"> {1 \, Sidereal \, Year} = {31,558,150 \, Seconds} </math> <math display="block"> {1 \, Tropical \, Year} = {31,556,926 \, Seconds} </math> <math display="block"> 1 \, Great \, Year \approx 25,824 \, Sidereal \, Years \approx 25,825 \, Tropical \, Years </math> <math display="block">{1 \, Galactic \, Year} \approx 8264 \, Great \, Year \approx 213,417,800 \, Tropical \, Years </math> The '''Bully Mnemonic''' is a technique for remembering the exact number of seconds that occur in Earth's [https://en.wikipedia.org/wiki/Sidereal_year sidereal year] and [https://en.wikipedia.org/wiki/Tropical_year tropical year], a good approximation of the Earth's [https://en.wikipedia.org/wiki/Great_Year Great Year], and a rough approximation of the Solar System's [https://en.wikipedia.org/wiki/Galactic_year galactic year]. Click on the following link to learn more about the Bully Mnemonic and the role it plays in the mathematical foundation of Bully timestamps. * [[Bully Mnemonic |Learn More About The Bully Mnemonic]] * [[Bully Mnemonic Extension |Learn More About The Bully Mnemonic Extension]] ejy3y7rr244jb05n4c8723v8h54itc5 0 321521 2812113 2808928 2026-05-30T06:20:44Z James500 297601 /* Japanese and Japan */ Add 2812113 wikitext text/x-wiki {{Bibliography}} See [[s:Category:Music]] and [[w:Category:Music books]] This part of the [[Universal Bibliography]] is a bibliography of music. Bibliography *[[w:Bibliography of Music Literature|Bibliography of Music Literature]] *Green (ed). Foundations in Music Bibliography. 1993. [https://books.google.co.uk/books?id=rADdpZN9UhAC&pg=PR3#v=onepage&q&f=false] *Krummel. The Literature of Music Bibliography: An Account of the Writings on the History of Music Printing & Publishing. 2nd Ed: 1992. [https://books.google.com/books?id=3AZsiITI-IEC] *Bibliography of Music Bibliographies. 1967. [https://books.google.co.uk/books?id=d6YJAQAAMAAJ] *Bayne. A Guide to Library Research in Music. 2008. [https://books.google.co.uk/books?id=ExGbDqu9gPAC&pg=PP1#v=onepage&q&f=false] *A Selected Bibliography of Music Librarianship [https://books.google.co.uk/books?id=X5AeOl4O-osC] *Bradley. American Music Librarianship: A Research and Information Guide. [https://books.google.co.uk/books?id=VabcAAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Music Reference and Research Materials. 3rd Ed: 1974: [https://books.google.com/books?id=5Y1IAAAAMAAJ] *Agruss. Guide to Reference Books on Music. 1948. [https://books.google.co.uk/books?id=wX06AAAAIAAJ] *Haggerty. A Guide to Popular Music Reference Books: An Annotated Bibliography. 1995. [https://books.google.co.uk/books?id=2OnEEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Coover. A Bibliography of Music Dictionaries. 1952: [https://books.google.co.uk/books?id=NH06AAAAIAAJ]. Music Lexicography. 2nd Ed: 1958. Including a Study of Lacunae in Music Lexicography and a Bibliography of Music Dictionaries. 3rd Ed: 1971: [https://books.google.co.uk/books?id=jKMJAQAAMAAJ]. *A Bibliography of Books on Music and Collections of Music. 1948. [https://books.google.co.uk/books?id=vfvpnwWWlZwC] *Deakin. Musical Bibliography: A Catalogue of the Musical Works. 1892. [https://books.google.co.uk/books?id=-UgQAAAAYAAJ&pg=PP7#v=onepage&q&f=false] (England 15th to 18th century) *Matthew. The Literature of Music. 1896. [https://books.google.co.uk/books?id=fTQ6AAAAMAAJ&pg=PR3#v=onepage&q&f=false]. Reviews: [https://books.google.co.uk/books?id=bjdVAAAAYAAJ&pg=RA1-PA56#v=onepage&q&f=false] [https://books.google.co.uk/books?id=dzcZAAAAYAAJ&pg=PA22#v=onepage&q&f=false] [https://books.google.co.uk/books?id=R0gcAQAAMAAJ&pg=PA470#v=onepage&q&f=false] [https://books.google.co.uk/books?id=qK5OAQAAMAAJ&pg=PA55#v=onepage&q&f=false] [https://books.google.co.uk/books?id=1chZAAAAYAAJ&pg=PA155#v=onepage&q&f=false] [https://books.google.co.uk/books?id=ezszAQAAMAAJ] [https://books.google.co.uk/books?id=5h61TMyTmOMC] [https://books.google.co.uk/books?id=8k8wAQAAIAAJ] [https://books.google.co.uk/books?id=i8W8LKTuc0AC]. Author: [https://books.google.co.uk/books?id=awIQAAAAYAAJ&pg=PA275#v=onepage&q&f=false]. *Hoek. Analyses of Nineteenth- and Twentieth-Century Music, 1940-2000. 2007. [https://books.google.co.uk/books?id=CRG4AQAAQBAJ&pg=PP1#v=onepage&q&f=false] *RILM Abstracts of Music Literature. [https://books.google.co.uk/books?id=HxjjAAAAMAAJ] *Elliker. The Periodical Literature of Music: Trends from 1952 to 1987. 1996. [https://books.google.co.uk/books?id=T5ifAAAAMAAJ] *Forkel. Allgemeine Litteratur der Musik. 1792. [https://books.google.co.uk/books?id=VTRDAAAAcAAJ&pg=PR1#v=onepage&q&f=false] Review: [https://books.google.co.uk/books?id=3N8sAAAAYAAJ&pg=PA33#v=onepage&q&f=false] History and bibliography *Matthew. A Handbook of Musical History and Bibliography. 1898. [https://books.google.co.uk/books?id=V1g5AAAAIAAJ&pg=PR3#v=onepage&q&f=false] Review: [https://books.google.co.uk/books?id=P1lDAQAAMAAJ&pg=PA229#v=onepage&q&f=false] *Boyden. The History and Literature of Music: 1750 to the Present. 1959. [https://books.google.co.uk/books?id=XcAZAQAAIAAJ] *Brown. An Introduction to the History and Literature of Music in Western Culture. 2nd Ed: 2011. [https://books.google.co.uk/books?id=aKpGAAAACAAJ] Chronology, annuals, year books, years *Eisler. World Chronology of Music History. *Lowe. A Chronological Cyclopædia of Musicians and Musical Events. 1896. *Tokyo Ongaku Gakko. Kinsei Hogaku Nempyo. [Chronology of Japanese Music in Recent Ages.] Rokugatsu-Kan. Volume 1. 1912. Volume 2. 1914. Volume 3. 1927. [https://books.google.co.uk/books?id=drMQAQAAMAAJ] *Cossar. This Day in Music. 2005. 2010. *Glassman. The Year in Music. Columbia House. *[[w:Herman Klein|Hermann Klein]]. Musical Notes. Annual Critical Record of Important Musical Events. *[[w:Joseph Bennett (critic)|Bennett]]. The Musical Year. *Hinrichsen's Musical Year Book *The Musical Year Book of the United States **The Boston Musical Year Book *Billboard. Overview. 1982: [https://books.google.co.uk/books?id=YyQEAAAAMBAJ&pg=PT53#v=onepage&q&f=false]. *Billboard. The Year in Music. 1994: [https://books.google.co.uk/books?id=ZAgEAAAAMBAJ&pg=PA62#v=onepage&q&f=false]. 2003: [https://books.google.co.uk/books?id=bA8EAAAAMBAJ&pg=PA47#v=onepage&q&f=false]. **The Year in Music and Video. 1985: [https://books.google.co.uk/books?id=uyQEAAAAMBAJ&pg=PT50#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=tiQEAAAAMBAJ&pg=PA49#v=onepage&q&f=false]. *Jackson. 1965: The Most Revolutionary Year in Music. *Porter. A Musical Season: 1972-1973. **Music of Three Seasons: 1974-1977 **Music of Three More Seasons 1977-1980 **Musical Events: A Chronicle, 1980-1983. *[https://news.1242.com/article/tag/大人のmusic-calendar 【大人のMusic Calendar】]. Nippon Broadcasting System. [Articles from 2016 are included in [https://news.1242.com/article/author/toritani/page/42 NEWS ONLINE 編集部の記事一覧].] *[http://music-calendar.jp Music Calendar] History *"Recorded Sound: The First Century: 1877-1977". Billboard. 21 May 1977. pp [https://books.google.co.uk/books?id=XCMEAAAAMBAJ&pg=PT39#v=onepage&q&f=false RS-1] to RS-117. Encyclopedias See also [[w:List of encyclopedias by branch of knowledge/Music]] and [[w:Bibliography of encyclopedias#Music and dance]] *Encyclopedia of Music in the 20th Century [https://books.google.co.uk/books?id=m8W2AgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Moore. Complete Encyclopædia of Music. 1852. [https://books.google.co.uk/books?id=-QBFAQAAMAAJ&pg=PA1#v=onepage&q&f=false] Dictionaries *Apel. "Dictionaries of music". Harvard Dictionary of Music. 1969. pp [https://books.google.co.uk/books?id=TMdf1SioFk4C&pg=PA232#v=onepage&q&f=false 232] to 234. United Kingdom: *Billboard. Spotlight on the United Kingdom. 1978: [https://books.google.co.uk/books?id=TSQEAAAAMBAJ&pg=PT78#v=onepage&q&f=false]. 1979: [https://books.google.co.uk/books?id=MCUEAAAAMBAJ&pg=PT100#v=onepage&q&f=false]. Australia: *Billboard. Spotlight on Australia/New Zealand. 1982: [https://books.google.co.uk/books?id=GCQEAAAAMBAJ&pg=PT54#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=hiQEAAAAMBAJ&pg=PT29#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=UCQEAAAAMBAJ&pg=PA60#v=onepage&q&f=false]. **Live Talent of Australia: [https://books.google.co.uk/books?id=YyQEAAAAMBAJ&pg=PT94#v=onepage&q&f=false] New Zealand: *Harvey. A Bibliography of Writings about New Zealand Music Published to the End of 1983. 1985. [https://books.google.co.uk/books?id=B1ROA_sP-xsC&pg=PP1#v=onepage&q&f=false] *The Complete New Zealand Music Charts, 1966-2006: Singles, Albums, DVDs, Compilations. 2007. [https://books.google.co.uk/books?id=wyU5AQAAIAAJ] *Billboard. New Zealand. 2002: [https://books.google.co.uk/books?id=Rg0EAAAAMBAJ&pg=PA37#v=onepage&q&f=false] Canada: *Billboard. Spotlight on Canada. 1981: [https://books.google.co.uk/books?id=DSQEAAAAMBAJ&pg=PT50#v=onepage&q&f=false]. Scandanavia: *Billboard. Spotlight on Scandanavia. 1981: [https://books.google.co.uk/books?id=GCUEAAAAMBAJ&pg=PT86#v=onepage&q&f=false]. France: *Billboard. Spotlight on France. 1971: [https://books.google.co.uk/books?id=-wgEAAAAMBAJ&pg=PA35#v=onepage&q&f=false]. 1972: [https://books.google.co.uk/books?id=REUEAAAAMBAJ&pg=PA35#v=onepage&q&f=false]. 1982: [https://books.google.co.uk/books?id=AyQEAAAAMBAJ&pg=PT66#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=ICUEAAAAMBAJ&pg=PA41#v=onepage&q&f=false] Germany: *Billboard. Spotlight on West Germany. 1971: [https://books.google.co.uk/books?id=zQgEAAAAMBAJ&pg=PA45#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=-iMEAAAAMBAJ&pg=PT12#v=onepage&q&f=false]. **Spotlight on West Germany, Austria and Switzerland. 1986: [https://books.google.co.uk/books?id=CSUEAAAAMBAJ&pg=RA1-PA35#v=onepage&q&f=false] Italy: *Billboard. Spotlight on Italy. 1981: [https://books.google.co.uk/books?id=8iQEAAAAMBAJ&pg=PT3#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=3yQEAAAAMBAJ&pg=PT36#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=2SQEAAAAMBAJ&pg=PA38-IA1#v=onepage&q&f=false]. 1994: [https://books.google.co.uk/books?id=XQgEAAAAMBAJ&pg=PA67#v=onepage&q&f=false]. Spain: *Billboard. Spotlight on Spain. 1971: [https://books.google.co.uk/books?id=5Q8EAAAAMBAJ&pg=PA49#v=onepage&q&f=false] Philipines: *[https://billboardphilippines.com/culture/scenes/lost-history-how-filipino-music-was-documented-in-the-40s-to-2010s/ Lost History: How Filipino Music Was Documented In The ’40s To 2010s]. Billboard Philippines. 18 January 2024. *[[w:en:Billboard Philippines|Billboard Philippines]] Brazil: *Billboard. Spotlight on Brazil. 1996: [https://books.google.co.uk/books?id=NA0EAAAAMBAJ&pg=PA51#v=onepage&q&f=false]. United States *Krummel. Bibliographical Handbook of American Music. 1987. [https://books.google.co.uk/books?id=G4wcnkvFZl4C&pg=PP1#v=onepage&q&f=false] *Krummel. Resources of American Music History: A Directory of Source Materials from Colonial Times to World War II. 1981. [https://books.google.co.uk/books?id=bJcYAAAAIAAJ] Soviet *Aschmann. Current Soviet Music Bibliography. 1976. [https://books.google.co.uk/books?id=2i7jAAAAMAAJ] Decline of pop music: *[https://www.smithsonianmag.com/smart-news/science-proves-pop-music-has-actually-gotten-worse-8173368/ Science Proves: Pop Music Has Actually Gotten Worse]. [[w:Smithsonian (magazine)|Smithsonian]]. 27 July 2012. *[https://faroutmagazine.co.uk/new-study-discovers-pop-music-has-suffered-significant-decline-in-one-area/ New study discovers pop music has suffered “significant decline” in one area]. [[w:Far Out (website)|Far Out]]. 5 July 2024. *[https://www.globalnews.ca/news/9001083/why-older-music-more-popular-than-new-music/amp/ There is something very, very wrong with today’s music. It just may not be very good.] [[w:Global News|Global News]]. 24 July 2022. *[https://www.bbc.co.uk/music/articles/fb84bf19-29c9-4ed3-b6b6-953e8a083334 Has pop music lost its fun?]. BBC. 12 January 2018. *[https://www.spectator.co.uk/article/its-official-modern-music-is-bad/ It’s official: modern music is bad]. The Spectator. 13 February 2024. Homogeneity of pop music: *[https://www.theguardian.com/music/2012/jul/27/pop-music-sounds-same-survey-reveals Pop music these days: it all sounds the same, survey reveals]. The Guardian. 27 July 2012. *[https://www.nbcnews.com/id/wbna48356108 Pop Music All Sounds the Same Nowadays]. NBC News. 27 July 2012. *[https://www.independent.co.uk/voices/comment/why-does-today-s-pop-music-sound-the-same-because-the-same-people-make-it-8368714.html Why does today's pop music sound the same? Because the same people make it]. The Independent. 29 November 2012. *[https://www.reuters.com/article/lifestyle/science/pop-music-too-loud-and-all-sounds-the-same-official-idUSBRE86P0R9/ Pop music too loud and all sounds the same: official]. Reuters. 26 July 2012. *[https://theconversation.com/from-art-form-to-asset-our-study-found-popular-songs-are-becoming-more-generic-266097 From art form to asset: our study found popular songs are becoming more generic]. The Conversation. 3 October 2025. Conferences: *International Music Industry Conference. 1971: [https://books.google.co.uk/books?id=tggEAAAAMBAJ&pg=PA29#v=onepage&q&f=false] Laserdisc/Karaoke/CES *Billboard. Karaoke. 1992: [https://books.google.co.uk/books?id=jg8EAAAAMBAJ&pg=PA41-IA1#v=onepage&q&f=false] **CES and Karaoke. 1994. [https://books.google.co.uk/books?id=UggEAAAAMBAJ&pg=PA77#v=onepage&q&f=false] **Laserdisc. 1995. [https://books.google.co.uk/books?id=7AsEAAAAMBAJ&pg=PA67#v=onepage&q&f=false] **Laserdisc/Karaoke. 1996: [https://books.google.co.uk/books?id=iQ8EAAAAMBAJ&pg=PA59#v=onepage&q&f=false] Classical music *Billboard spotlights: 1995 [https://books.google.co.uk/books?id=1g0EAAAAMBAJ&pg=PA39#v=onepage&q&f=false] (9 September 1995) **"Classical Music Recording Market". Billboard. 12 April 1980. pp C-1 to C-12 and p 32. (A Billboard Spotlight). **"Classical Music: Discovering New Dimensions". Billboard. 10 September 1983. pp C-1 to C-18. (A Billboard Spotlight). *"Classical" section, and "Best Selling Classical LPs" chart, in Billboard Jazz *[[w:en:All About Jazz|All About Jazz]] Oldies *"Oldies stations find their place in radio market". Star-News. 13 March 1988. pp 1D & [https://books.google.co.uk/books?id=2OoyAAAAIBAJ&pg=PA16#v=onepage&q&f=false 6D]: "Oldies". *Billboard. 15 April 1972. [https://books.google.co.uk/books?id=a0UEAAAAMBAJ&pg=PT7#v=onepage&q&f=false p 47]. *Billboard. 17 April 1961, [https://books.google.co.uk/books?id=JiIEAAAAMBAJ&pg=PA1#v=onepage&q&f=false p 1]. *Billboard. 4 January 1960, [https://books.google.co.uk/books?id=Ch8EAAAAMBAJ&pg=PA1#v=onepage&q&f=false p 1] Nostalgia See also [[Universal Bibliography/Nostalgia]] *"A Perspective on the Future of Nostalgia". Billboard. 4 May 1974. pp [https://books.google.co.uk/books?id=cgkEAAAAMBAJ&pg=PA37#v=onepage&q&f=false N-1] to N-54 and two more pages. *Carr. Nostalgia, Song and the Quest for Home: Production, Text, Reception. 2025. [https://books.google.co.uk/books?id=xz1jEQAAQBAJ&pg=PP1#v=onepage&q&f=false] Charts *Carroll, " Did Billboard, Cash Box, and Record World Charts Tell the Same Story? Perception and Reality, 1960-1979"(2022) 9 Rock Music Studies [https://www.tandfonline.com/doi/full/10.1080/19401159.2022.2054107 199] Magazines See also [[w:Category:Music magazines]] *Billboard. Google: [https://books.google.co.uk/books/serial/ISSN:00062510?rview=1&lr=&sa=N&start=2770 1942] onwards ==Japanese and Japan== *The Ashgate Research Companion to Japanese Music. 2017. [https://books.google.co.uk/books?id=W2JTgQGc99EC&pg=PP1#v=onepage&q&f=false] [https://books.google.co.uk/books?id=4tINDgAAQBAJ&pg=PA2#v=onepage&q&f=false] *Billboard. Spotlight on Japan. 1970: 19 December 1970 [https://books.google.co.uk/books?id=mSkEAAAAMBAJ&pg=PA37#v=onepage&q&f=false]. 1971: 11 December 1971 [https://books.google.co.uk/books?id=Fg8EAAAAMBAJ&pg=PA39#v=onepage&q&f=false]. 1973: 17 February 1973 [https://books.google.co.uk/books?id=QEUEAAAAMBAJ&pg=PT25#v=onepage&q&f=false]. 1977: 30 April 1977 [https://books.google.co.uk/books?id=USMEAAAAMBAJ&pg=PT46#v=onepage&q&f=false]. 1979: [https://books.google.co.uk/books?id=_iQEAAAAMBAJ&pg=PT48#v=onepage&q&f=false]. 1982:[https://books.google.co.uk/books?id=byQEAAAAMBAJ&pg=PT38#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=1CQEAAAAMBAJ&pg=PT65#v=onepage&q&f=false]. 1986:[https://books.google.co.uk/books?id=-CMEAAAAMBAJ&pg=RA1-PA79#v=onepage&q&f=false]. 1993: 12 June 1993 [https://books.google.co.uk/books?id=9A8EAAAAMBAJ&pg=PA57#v=onepage&q&f=false]. 1995: 5 August 1995 [https://books.google.co.uk/books?id=xwsEAAAAMBAJ&pg=PA52-IA1#v=onepage&q&f=false]. 1996: 31 August 1996 [https://books.google.co.uk/books?id=vwcEAAAAMBAJ&pg=PA66#v=onepage&q&f=false]. 1997: 30 August 1997 [https://books.google.co.uk/books?id=_gkEAAAAMBAJ&pg=PA61#v=onepage&q&f=false]. 1998: 26 September 1998 [https://books.google.co.uk/books?id=GgoEAAAAMBAJ&pg=PA117#v=onepage&q&f=false]. 2000: 9 September 2000 [https://books.google.co.uk/books?id=aREEAAAAMBAJ&pg=PA65#v=onepage&q&f=false]. 2002: 7 September 2002 [https://books.google.co.uk/books?id=-QwEAAAAMBAJ&pg=PA53#v=onepage&q&f=false]. 2003: 5 July 2003 [https://books.google.co.uk/books?id=3w0EAAAAMBAJ&pg=PA45#v=onepage&q&f=false]. **"Japan in 1974: Business Bristles While Shortages Are Met". Billboard. 23 February 1974. pp J-1 to J-30. (A Billboard Spotlight). **"Made in Japan: A Dynamic Music Industry". Billboard. 1 March 1975. pp J-1 to J-23. (A Billboard Spotlight). **"Japan '76". Billboard. 17 April 1976. pp 36 to 59. (A Billboard Spotlight). **"Japanese Music: The Challenge of Recession". Billboard. 27 May 1978. pp J-1 to J-31. (A Billboard Spotlight). **"Music in Japan: Industry Views 1981 With Quiet Optimism". Billboard. 30 May 1981. pp J-1 to J-18. **"Japan: Where Technology Greets Tradition". (An International Market Profile). Billboard. 21 May 1983. pp J-1 to J-13. Follows p 34. **"Billboard Spotlight on Japan: VCRs and CDs Will Be Pacemakers". Billboard. 26 May 1984. pp J-1 to J-11. Follows p 38. **"Spotlight on Japan". Billboard. 6 June 1987. pp J-1 to J-12. **"Japan '88". Billboard. 9 July 1988. pp J-1 to J-11. (A Billboard International Spotlight). **"Japan". ("Japan '89"/"Spotlight on Japan"). Billboard. 3 June 1989. pp J-1 to J-20. (International Spotlight). **"Japan". ("International Spotlight"/"A Billboard Spotlight"). Billboard. 25 May 1991. pp J-1 to J-26. Follows p 50. Called "Japan '91" on front page. *[[w:The Best Ten|The Best Ten]] (ザ・ベストテン). [Television programme]. [https://www.tbs.co.jp/tbs-ch/special/the_bestten/ Episodes]. *[[w:ja:Music Station|Music Station]]. [Television programme]. Episodes: [https://www.tv-asahi.co.jp/music/contents/m_lineup/0003/index.html episode 1] etc. *Wade. Music in Japan: Experiencing Music, Expressing Culture. 2005. [https://books.google.co.uk/books?id=XXYIAQAAMAAJ] *Malm. Japanese Music & Musical Instruments. 1959. [https://books.google.com/books?id=QkTaAAAAMAAJ] *[[w:Francis Taylor Piggott|Pigott]]. The Music and Musical Instruments of Japan. 1893 [https://books.google.co.uk/books?id=ttKTUwmjzMwC&pg=PR3#v=onepage&q&f=false]. 1909. [https://books.google.co.uk/books?id=MAM5AAAAIAAJ] Bibliography *Tsuge. Japanese Music: An Annotated Bibliography. 1986. [https://books.google.com/books?id=YCsKAQAAMAAJ] *[[w:ja:三井徹|Tōru Mitsui]]. Popyurā Ongaku Kankei Tosho Mokuroku: Ryūkōka, Jazu, Rokku, J-poppu no Hyakunen. (Japanese: ポピュラー音楽関係図書目録: 流行歌、ジャズ、ロック、Jポップの百年). Nichigai Associates. 2009. [https://books.google.co.uk/books?id=dSAxAQAAIAAJ]. Catalogues: [https://search.worldcat.org/title/406243182] [https://cir.nii.ac.jp/crid/1970586434933272116] *[https://ndlsearch.ndl.go.jp/rnavi/avmaterials/post_572 音楽に関する文献を探すには（主題書誌）]. NDL. Dictionaries *[[w:ja:下中弥三郎|Shimonaka Yasaburo]] (ed). Ongaku Jiten. Heibonsha. Review: (1959) 18 Journal of Asian Studies 295 [https://www.cambridge.org/core/journals/journal-of-asian-studies/article/abs/ongaku-jiten-dictionary-of-music-ed-shimonaka-yasaburo-tokyo-heibonsha-195557-12-volumes-900-yen-per-volume/F3067B1CE61B5B2C647091E69CE8C8DD] [https://read.dukeupress.edu/journal-of-asian-studies/article-abstract/18/2/295/322980/Ongaku-jiten-Dictionary-of-Music?redirectedFrom=fulltext] History *Eta Harich-Schneider. A History of Japanese Music. 1973. [https://books.google.com/books?id=3AraAAAAMAAJ] *Koh-ichi Hattori. 123 Years of Japanese Music: The Culture of Japan Through a Look at Its Music. 2004. [https://books.google.com/books?id=znzsAAAAMAAJ] **Koh-ichi Hattori. 36,000 Days of Japanese Music: The Culture of Japan Through A Look At Its Music. Pacific Vision. Pierce, Southfield, Michigan. 1996. ISBN 0965364208. *Shinpan Nihon Ryūkōkashi. (Japanese: 新版日本流行歌史). [[w:ja:社会思想社|Shakaishisosha]]. 1994. Review: [https://books.google.co.uk/books?id=XQdIAAAAMAAJ]. Catalogue: [https://ndlsearch.ndl.go.jp/en/books/R100000002-I000002420287] **新版日本流行歌史: 1960-1994. [https://books.google.com/books?id=_b4pAQAAIAAJ] [https://books.google.co.uk/books?id=nb4pAQAAIAAJ]. **新版日本流行歌史: 1938-1959 **1867-1937 *Mehl. Music and the Making of Modern Japan: Joining the Global Concert. 2024. [https://books.google.co.uk/books?id=P3QMEQAAQBAJ&pg=PA2#v=onepage&q&f=false] Modern, contemporary, today *Johnson. Handbook of Japanese Music in the Modern Era. 2024. [https://books.google.co.uk/books?id=KNP7EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Matsue. Focus: Music in Contemporary Japan. 2016. [https://books.google.co.uk/books?id=AQgtCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Music of Japan Today. [https://books.google.co.uk/books?id=YZQYEAAAQBAJ&pg=PP1#v=onepage&q&f=false] Popular music *Mitsui (ed). Made in Japan: Studies in Popular Music. 2014. [https://books.google.co.uk/books?id=YWQKBAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Stevens. Japanese Popular Music: Culture, Authenticity and Power. 2008. [https://books.google.co.uk/books?id=OHMkdcL9DAMC&pg=PP1#v=onepage&q&f=false] *Mitsui. Popular Music in Japan: Transformation Inspired by the West. 2020. [https://books.google.co.uk/books?id=FpbqDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Nagahara. Tokyo Boogie-Woogie: Japan’s Pop Era and Its Discontents. 2017. [https://books.google.co.uk/books?id=iTxYDgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Patterson. Music and Words: Producing Popular Songs in Modern Japan, 1887–1952. 2019. [https://books.google.co.uk/books?id=P0FvDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *James Stanlaw. "Using English identity markers in Japanese Popular Music". English in East and South Asia. Chapter 14. [https://books.google.co.uk/books?id=88A1EAAAQBAJ&pg=PT109#v=onepage&q&f=false] *"Japanese Popular Music in Singapore". Asian Music. vol 34. No 1: Fall/Winter 2002/2003. p 1. [https://books.google.co.uk/books?id=_D4JAQAAMAAJ] *Steve McClure. Nipponpop. Tuttle Publishing. 1998. ISBN 9780804821070. ISBN 0804821070. [Sometimes called "Nippon Pop"]. Catalogue: [https://search.worldcat.org/title/Nipponpop/oclc/247384040] Review: (1998) [https://books.google.co.uk/books?id=f9egmeZ8YywC 245] The Publishers Weekly 2 From folk to J-pop *[[w:ja:富澤一誠|Issei Tomizawa]]. Ano subarashii kyoku o mō ichido: fōku kara J-poppu made. (Japanese: あの素晴しい曲をもう一度: フォークからJポップまで). [[w:Shinchosha|Shinchosha]]. 2010. [https://books.google.com/books?id=ju9MAQAAIAAJ]. Catalogue: [https://search.worldcat.org/title/501749494]. Commentary on book: [https://www.ytv.co.jp/michiura/time/2010/01/j2010110.html]. Review of the CD: [https://www.cdjournal.com/i/disc/great-agefree-music-forever-and-great-music-are-o/4109110788]. J-pop *Bourdaghs. Sayonara Amerika, Sayonara Nippon: A Geopolitical Prehistory of J-pop. 2012. [https://books.google.co.uk/books?id=K_y88JwibrMC&pg=PP1#v=onepage&q&f=false] *"The Rise of J-Pop in Asia and Its Impact" (2004) Japan Spotlight. vol 23. p 24. [https://books.google.co.uk/books?id=i7C0AAAAIAAJ] *Terence Lancashire. "J-pop's elusive J: Is Japanese popular music Japanese?" (2008) Perfect Beat. vol 9. No 1. p 38. [https://books.google.co.uk/books?id=5No4AQAAIAAJ] *Tetsu Misaki. J-poppu no Nihongo: kashiron. (Japanese: Jポップの日本語: 歌詞論). [[w:ja:彩流社|彩流社 (Sairyusha)]]. 2002. [https://books.google.com/books?id=dsMpAQAAIAAJ] [https://search.worldcat.org/ja/title/J-:/oclc/52005194] *[[w:ja:烏賀陽弘道|Hiromichi Ugaya]]. Jpoppu Towa Nanika: Kyodaikasuru Ongaku Sangyō. (Japanese: Jポップとは何か: 巨大化する音楽産業). 2005. [https://books.google.co.uk/books?id=TLlOAAAAMAAJ] catalogue [https://search.worldcat.org/ja/title/J-:/oclc/676652594] [https://ci.nii.ac.jp/ncid/BA71618018] Japanese rock *Takarajima Special Edition: Encyclopedia of Japanese Rock 1955-1990. Nihon rokku daihyakka: Rokabirī kara bando būmu made. (Japanese: 日本ロック大百科 [年表編] ロカビリーからバンド・ブームまで 1955〜1990). [[w:ja:JICC出版局|JICC Shuppankyoku]]. 1992. ISBN 9784796602907. ISBN 4796602909. Catalogues: [https://ci.nii.ac.jp/ncid/BN07889172] [https://catalogue.nla.gov.au/catalog/2263400]. *Japanese Rock: Standard: 1967-1985. 日本のロック名曲徹底ガイド: 名曲263決定盤846. CDJournal. 2008. ISBN 9784861710469. ISBN 4861710464. [https://www.cdjournal.com/Company/products/mook.php?mno=20081002]. Catalogue: [https://ci.nii.ac.jp/ncid/BA8932668X?l=en]. *Kojima Satoshi (Japanese: 小島智). 検証・80年代日本のロック. アルファベータブックス. 2024. ISBN 9784865981179. ISBN 4865981179. [https://books.google.com/books?id=0gbl0AEACAAJ]. Review: [https://mainichi.jp/articles/20241026/ddm/015/070/005000c]. Jazz *[[w:ja:スイングジャーナル|Swing Journal]] (1947 to 2010) Commentary: [https://www.allaboutjazz.com/news/swing-journal-long-standing-jazz-magazine-to-be-suspended-in-june/] Japanese fusion: *THE DIG presents 日本のフュージョン. Shinko Music Mook. Released 19 April 2013. Commentary: [https://www.cdjournal.com/news/casiopea/50967]. No II. Released 23 October 2014. Commentary: [https://www.cdjournal.com/news/takanaka-masayoshi/62225] Classical *[[w:ja:ぶらあぼ|Bravo]] (Japanese: ぶらあぼ) ebravo.jp *[[w:ja:音楽芸術 (雑誌)|Ongaku Geijutsu]] (Japanese: 音楽芸術) Magazines For Japanese music magazines, see [[w:ja:日本の音楽雑誌]]. *Music Periodicals in Japan — A Comprehensive List (1988) 35 Fontes Artis Musicae 116 [https://www.jstor.org/stable/23507222] [https://books.google.com/books?id=qHYWAAAAIAAJ] **Kishimoto, "Additional Corrections and Alphabetical Title Index" (1989) 36 Fontes Artis Musicae 38 [https://www.jstor.org/stable/23507313] [https://books.google.co.uk/books?id=7XYWAAAAIAAJ] *Special Bibliography: A Bibliography of Japanese Magazines and Music (1959) 3 Ethnomusicology 76 [https://www.jstor.org/stable/924290] *A Historical Survey of Music Periodicals in Japan: 1881—1920 (1989) 36 Fontes Artis Musicae 44 [https://www.jstor.org/stable/23507314] *[[w:ja:篠原章|Akira Shinohara]]. 日本ロック雑誌クロニクル. [[w:en:Ohta Publishing|Ohta Publishing]]. 2005. [https://books.google.co.uk/books?id=L8opAQAAIAAJ] *[[w:Oricon|Oricon]] (オリコン) **[https://web.archive.org/web/19970412131857/http://www.999.com/Oricon/index.html Oricon Music Site]. Commentary: [https://internet.watch.impress.co.jp/www/article/980309/oms.htm]. *[[w:Billboard Japan|Billboard Japan]] (ビルボード・ジャパン) **Music Labo (ミュージック・ラボ) (1970 to 1994) *Music Research (ミュージック・リサーチ) ["Weekly Music Magazine"]. Catalogue: [https://web.archive.org/web/20260319070908/https://ndlsearch.ndl.go.jp/books/R100000002-I000000039804]. *Rolling Stone Japan *新譜ジャーナル (Shinpu Journal). Catalogue: [https://ndlsearch.ndl.go.jp/books/R100000002-I000000012315]. Began 1968 [https://books.google.co.uk/books?id=L8opAQAAIAAJ], later called シンプジャーナル **シンプジャーナル *Myūjikku mansurī [ミュージック・マンスリー]  [https://ci.nii.ac.jp/ncid/AN00396190] *カセットライフ. (Cassette Life). [[w:ja:シンコーミュージック・エンタテイメント|Shinko Music Entertainment]] *[[w:ja:CDジャーナル|CDJournal]] *[[w:ja:Rockin'on Japan|Rockin'on Japan]]. (ロッキング・オン・ジャパン). (1986 onwards) *[[w:ja:Rooftop|Rooftop]] (1976 onwards) Columns in periodicals *"Japanese Newsnotes". Billboard. (eg 17 April 1961, [https://books.google.co.uk/books?id=JiIEAAAAMBAJ&pg=PA13#v=onepage&q&f=false p 13].) Websites *[[w:ja:ナタリー (ニュースサイト)|Natalie]] (ナタリー) *[[w:ja:BARKS|Barks]] *OKMusic. Commentary: [https://xtech.nikkei.com/it/article/NEWS/20120626/405442/]. Charts For Japanese music charts, see [[w:ja:日本の音楽チャート]] Chart books *Oricon Chart Book (Japanese: オリコンチャート・ブック) **1987 to 1998 Oricon Chart Book. All Albums. [https://books.google.co.uk/books?id=KvEoNwAACAAJ] **Album Chart Book Complete Edition 1970〜2005. Catalogue:[https://www.tosyokan.pref.shizuoka.jp/licsxp-opac/WOpacMsgNewListToTifTilDetailAction.do?tilcod=1000610247212] *澤山博之. ミュージック・ライフ 東京で1番売れていたレコード 1958~1966. Shinko Music Entertainment. 2019. [Charts published in Music Life from 1958 onwards]. Commentary: [https://mikiki.tokyo.jp/articles/-/20952 Mikiki] Number ones *Oricon No.1 Hits 500. Clubhouse (Japanese: クラブハウス). 1994. 1998. **[https://books.google.com/books?id=GlsnNwAACAAJ vol 1 (1968~1985)]. ISBN 9784906496129. **[https://books.google.com/books?id=icInNwAACAAJ vol 2 (1986~1994)]. ISBN 9784906496136. Awards Japan Record Awards *輝く!日本レコード大賞 公式データブック: 放送60回記念: TBS公認. Shinko Music Entertainment. ISBN 9784401647019. [https://books.google.co.uk/books?id=JcDqvwEACAAJ] [https://ci.nii.ac.jp/ncid/BB2773137X] Traditional, Hogaku *Malm. Traditional Japanese Music and Musical Instruments. [https://books.google.co.uk/books?id=Yn3VQbqywCsC&pg=PP1#v=onepage&q&f=false] *Miyuki Yoshikami. Japan's Musical Tradition: Hogaku from Prehistory to the Present. 2020. [https://books.google.co.uk/books?id=X3XTDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Hughes. Traditional Folk Song in Modern Japan: Sources, Sentiment and Society. 2008. [https://books.google.co.uk/books?id=yfV5DwAAQBAJ&pg=PR1#v=onepage&q&f=false] Koto: *Tokyo Academy of Music. Collection of Japanese Koto Music. 1888. [https://books.google.co.uk/books?id=RncQAAAAYAAJ&pg=PP13#v=onepage&q&f=false][https://babel.hathitrust.org/cgi/pt?id=hvd.32044040839565&seq=1] Exam guides: For the 音楽CD検定 exam on music CDs: *音楽CD検定公式ガイドブック. 2007. [[w:ja:音楽出版社 (企業)|Ongaku Shuppansha Co Ltd]] (音楽出版社). [https://books.google.co.uk/books?id=sbjdeDJMkQcC&pg=PP1#v=onepage&q&f=false vol 1]. [https://books.google.co.uk/books?id=AoFgIowII48C&pg=PP1#v=onepage&q&f=false vol 2]. Commentary: [https://www.cdjournal.com/i/news/-/15303] [https://www.oricon.co.jp/news/46065/full/] [https://allabout.co.jp/gm/gc/57723/] [https://www.oricon.co.jp/news/45388/full/]. Children's music *Elizabeth May. The Influence of the Meiji Period on Japanese Children's Music. University of California Press. 1963. [https://books.google.co.uk/books?id=54cHAQAAMAAJ] **Japanese Children's Music Before and After Contact with the West. University of California at Los Angeles. 1958. (doctoral dissertation). DJs *Masahiro Yasuda, "How Japanese DJs cut across Market Boundaries" (1999) [https://books.google.co.uk/books?id=H5QJAQAAMAAJ 4] Perfect Beat 45 [[Category:Music]] mu109mb3lsuhk4yc0oyfohie5ri3b9o 2812114 2812113 2026-05-30T06:24:10Z James500 297601 /* Japanese and Japan */ Add 2812114 wikitext text/x-wiki {{Bibliography}} See [[s:Category:Music]] and [[w:Category:Music books]] This part of the [[Universal Bibliography]] is a bibliography of music. Bibliography *[[w:Bibliography of Music Literature|Bibliography of Music Literature]] *Green (ed). Foundations in Music Bibliography. 1993. [https://books.google.co.uk/books?id=rADdpZN9UhAC&pg=PR3#v=onepage&q&f=false] *Krummel. The Literature of Music Bibliography: An Account of the Writings on the History of Music Printing & Publishing. 2nd Ed: 1992. [https://books.google.com/books?id=3AZsiITI-IEC] *Bibliography of Music Bibliographies. 1967. [https://books.google.co.uk/books?id=d6YJAQAAMAAJ] *Bayne. A Guide to Library Research in Music. 2008. [https://books.google.co.uk/books?id=ExGbDqu9gPAC&pg=PP1#v=onepage&q&f=false] *A Selected Bibliography of Music Librarianship [https://books.google.co.uk/books?id=X5AeOl4O-osC] *Bradley. American Music Librarianship: A Research and Information Guide. [https://books.google.co.uk/books?id=VabcAAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Music Reference and Research Materials. 3rd Ed: 1974: [https://books.google.com/books?id=5Y1IAAAAMAAJ] *Agruss. Guide to Reference Books on Music. 1948. [https://books.google.co.uk/books?id=wX06AAAAIAAJ] *Haggerty. A Guide to Popular Music Reference Books: An Annotated Bibliography. 1995. [https://books.google.co.uk/books?id=2OnEEAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Coover. A Bibliography of Music Dictionaries. 1952: [https://books.google.co.uk/books?id=NH06AAAAIAAJ]. Music Lexicography. 2nd Ed: 1958. Including a Study of Lacunae in Music Lexicography and a Bibliography of Music Dictionaries. 3rd Ed: 1971: [https://books.google.co.uk/books?id=jKMJAQAAMAAJ]. *A Bibliography of Books on Music and Collections of Music. 1948. [https://books.google.co.uk/books?id=vfvpnwWWlZwC] *Deakin. Musical Bibliography: A Catalogue of the Musical Works. 1892. [https://books.google.co.uk/books?id=-UgQAAAAYAAJ&pg=PP7#v=onepage&q&f=false] (England 15th to 18th century) *Matthew. The Literature of Music. 1896. [https://books.google.co.uk/books?id=fTQ6AAAAMAAJ&pg=PR3#v=onepage&q&f=false]. Reviews: [https://books.google.co.uk/books?id=bjdVAAAAYAAJ&pg=RA1-PA56#v=onepage&q&f=false] [https://books.google.co.uk/books?id=dzcZAAAAYAAJ&pg=PA22#v=onepage&q&f=false] [https://books.google.co.uk/books?id=R0gcAQAAMAAJ&pg=PA470#v=onepage&q&f=false] [https://books.google.co.uk/books?id=qK5OAQAAMAAJ&pg=PA55#v=onepage&q&f=false] [https://books.google.co.uk/books?id=1chZAAAAYAAJ&pg=PA155#v=onepage&q&f=false] [https://books.google.co.uk/books?id=ezszAQAAMAAJ] [https://books.google.co.uk/books?id=5h61TMyTmOMC] [https://books.google.co.uk/books?id=8k8wAQAAIAAJ] [https://books.google.co.uk/books?id=i8W8LKTuc0AC]. Author: [https://books.google.co.uk/books?id=awIQAAAAYAAJ&pg=PA275#v=onepage&q&f=false]. *Hoek. Analyses of Nineteenth- and Twentieth-Century Music, 1940-2000. 2007. [https://books.google.co.uk/books?id=CRG4AQAAQBAJ&pg=PP1#v=onepage&q&f=false] *RILM Abstracts of Music Literature. [https://books.google.co.uk/books?id=HxjjAAAAMAAJ] *Elliker. The Periodical Literature of Music: Trends from 1952 to 1987. 1996. [https://books.google.co.uk/books?id=T5ifAAAAMAAJ] *Forkel. Allgemeine Litteratur der Musik. 1792. [https://books.google.co.uk/books?id=VTRDAAAAcAAJ&pg=PR1#v=onepage&q&f=false] Review: [https://books.google.co.uk/books?id=3N8sAAAAYAAJ&pg=PA33#v=onepage&q&f=false] History and bibliography *Matthew. A Handbook of Musical History and Bibliography. 1898. [https://books.google.co.uk/books?id=V1g5AAAAIAAJ&pg=PR3#v=onepage&q&f=false] Review: [https://books.google.co.uk/books?id=P1lDAQAAMAAJ&pg=PA229#v=onepage&q&f=false] *Boyden. The History and Literature of Music: 1750 to the Present. 1959. [https://books.google.co.uk/books?id=XcAZAQAAIAAJ] *Brown. An Introduction to the History and Literature of Music in Western Culture. 2nd Ed: 2011. [https://books.google.co.uk/books?id=aKpGAAAACAAJ] Chronology, annuals, year books, years *Eisler. World Chronology of Music History. *Lowe. A Chronological Cyclopædia of Musicians and Musical Events. 1896. *Tokyo Ongaku Gakko. Kinsei Hogaku Nempyo. [Chronology of Japanese Music in Recent Ages.] Rokugatsu-Kan. Volume 1. 1912. Volume 2. 1914. Volume 3. 1927. [https://books.google.co.uk/books?id=drMQAQAAMAAJ] *Cossar. This Day in Music. 2005. 2010. *Glassman. The Year in Music. Columbia House. *[[w:Herman Klein|Hermann Klein]]. Musical Notes. Annual Critical Record of Important Musical Events. *[[w:Joseph Bennett (critic)|Bennett]]. The Musical Year. *Hinrichsen's Musical Year Book *The Musical Year Book of the United States **The Boston Musical Year Book *Billboard. Overview. 1982: [https://books.google.co.uk/books?id=YyQEAAAAMBAJ&pg=PT53#v=onepage&q&f=false]. *Billboard. The Year in Music. 1994: [https://books.google.co.uk/books?id=ZAgEAAAAMBAJ&pg=PA62#v=onepage&q&f=false]. 2003: [https://books.google.co.uk/books?id=bA8EAAAAMBAJ&pg=PA47#v=onepage&q&f=false]. **The Year in Music and Video. 1985: [https://books.google.co.uk/books?id=uyQEAAAAMBAJ&pg=PT50#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=tiQEAAAAMBAJ&pg=PA49#v=onepage&q&f=false]. *Jackson. 1965: The Most Revolutionary Year in Music. *Porter. A Musical Season: 1972-1973. **Music of Three Seasons: 1974-1977 **Music of Three More Seasons 1977-1980 **Musical Events: A Chronicle, 1980-1983. *[https://news.1242.com/article/tag/大人のmusic-calendar 【大人のMusic Calendar】]. Nippon Broadcasting System. [Articles from 2016 are included in [https://news.1242.com/article/author/toritani/page/42 NEWS ONLINE 編集部の記事一覧].] *[http://music-calendar.jp Music Calendar] History *"Recorded Sound: The First Century: 1877-1977". Billboard. 21 May 1977. pp [https://books.google.co.uk/books?id=XCMEAAAAMBAJ&pg=PT39#v=onepage&q&f=false RS-1] to RS-117. Encyclopedias See also [[w:List of encyclopedias by branch of knowledge/Music]] and [[w:Bibliography of encyclopedias#Music and dance]] *Encyclopedia of Music in the 20th Century [https://books.google.co.uk/books?id=m8W2AgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Moore. Complete Encyclopædia of Music. 1852. [https://books.google.co.uk/books?id=-QBFAQAAMAAJ&pg=PA1#v=onepage&q&f=false] Dictionaries *Apel. "Dictionaries of music". Harvard Dictionary of Music. 1969. pp [https://books.google.co.uk/books?id=TMdf1SioFk4C&pg=PA232#v=onepage&q&f=false 232] to 234. United Kingdom: *Billboard. Spotlight on the United Kingdom. 1978: [https://books.google.co.uk/books?id=TSQEAAAAMBAJ&pg=PT78#v=onepage&q&f=false]. 1979: [https://books.google.co.uk/books?id=MCUEAAAAMBAJ&pg=PT100#v=onepage&q&f=false]. Australia: *Billboard. Spotlight on Australia/New Zealand. 1982: [https://books.google.co.uk/books?id=GCQEAAAAMBAJ&pg=PT54#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=hiQEAAAAMBAJ&pg=PT29#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=UCQEAAAAMBAJ&pg=PA60#v=onepage&q&f=false]. **Live Talent of Australia: [https://books.google.co.uk/books?id=YyQEAAAAMBAJ&pg=PT94#v=onepage&q&f=false] New Zealand: *Harvey. A Bibliography of Writings about New Zealand Music Published to the End of 1983. 1985. [https://books.google.co.uk/books?id=B1ROA_sP-xsC&pg=PP1#v=onepage&q&f=false] *The Complete New Zealand Music Charts, 1966-2006: Singles, Albums, DVDs, Compilations. 2007. [https://books.google.co.uk/books?id=wyU5AQAAIAAJ] *Billboard. New Zealand. 2002: [https://books.google.co.uk/books?id=Rg0EAAAAMBAJ&pg=PA37#v=onepage&q&f=false] Canada: *Billboard. Spotlight on Canada. 1981: [https://books.google.co.uk/books?id=DSQEAAAAMBAJ&pg=PT50#v=onepage&q&f=false]. Scandanavia: *Billboard. Spotlight on Scandanavia. 1981: [https://books.google.co.uk/books?id=GCUEAAAAMBAJ&pg=PT86#v=onepage&q&f=false]. France: *Billboard. Spotlight on France. 1971: [https://books.google.co.uk/books?id=-wgEAAAAMBAJ&pg=PA35#v=onepage&q&f=false]. 1972: [https://books.google.co.uk/books?id=REUEAAAAMBAJ&pg=PA35#v=onepage&q&f=false]. 1982: [https://books.google.co.uk/books?id=AyQEAAAAMBAJ&pg=PT66#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=ICUEAAAAMBAJ&pg=PA41#v=onepage&q&f=false] Germany: *Billboard. Spotlight on West Germany. 1971: [https://books.google.co.uk/books?id=zQgEAAAAMBAJ&pg=PA45#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=-iMEAAAAMBAJ&pg=PT12#v=onepage&q&f=false]. **Spotlight on West Germany, Austria and Switzerland. 1986: [https://books.google.co.uk/books?id=CSUEAAAAMBAJ&pg=RA1-PA35#v=onepage&q&f=false] Italy: *Billboard. Spotlight on Italy. 1981: [https://books.google.co.uk/books?id=8iQEAAAAMBAJ&pg=PT3#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=3yQEAAAAMBAJ&pg=PT36#v=onepage&q&f=false]. 1986: [https://books.google.co.uk/books?id=2SQEAAAAMBAJ&pg=PA38-IA1#v=onepage&q&f=false]. 1994: [https://books.google.co.uk/books?id=XQgEAAAAMBAJ&pg=PA67#v=onepage&q&f=false]. Spain: *Billboard. Spotlight on Spain. 1971: [https://books.google.co.uk/books?id=5Q8EAAAAMBAJ&pg=PA49#v=onepage&q&f=false] Philipines: *[https://billboardphilippines.com/culture/scenes/lost-history-how-filipino-music-was-documented-in-the-40s-to-2010s/ Lost History: How Filipino Music Was Documented In The ’40s To 2010s]. Billboard Philippines. 18 January 2024. *[[w:en:Billboard Philippines|Billboard Philippines]] Brazil: *Billboard. Spotlight on Brazil. 1996: [https://books.google.co.uk/books?id=NA0EAAAAMBAJ&pg=PA51#v=onepage&q&f=false]. United States *Krummel. Bibliographical Handbook of American Music. 1987. [https://books.google.co.uk/books?id=G4wcnkvFZl4C&pg=PP1#v=onepage&q&f=false] *Krummel. Resources of American Music History: A Directory of Source Materials from Colonial Times to World War II. 1981. [https://books.google.co.uk/books?id=bJcYAAAAIAAJ] Soviet *Aschmann. Current Soviet Music Bibliography. 1976. [https://books.google.co.uk/books?id=2i7jAAAAMAAJ] Decline of pop music: *[https://www.smithsonianmag.com/smart-news/science-proves-pop-music-has-actually-gotten-worse-8173368/ Science Proves: Pop Music Has Actually Gotten Worse]. [[w:Smithsonian (magazine)|Smithsonian]]. 27 July 2012. *[https://faroutmagazine.co.uk/new-study-discovers-pop-music-has-suffered-significant-decline-in-one-area/ New study discovers pop music has suffered “significant decline” in one area]. [[w:Far Out (website)|Far Out]]. 5 July 2024. *[https://www.globalnews.ca/news/9001083/why-older-music-more-popular-than-new-music/amp/ There is something very, very wrong with today’s music. It just may not be very good.] [[w:Global News|Global News]]. 24 July 2022. *[https://www.bbc.co.uk/music/articles/fb84bf19-29c9-4ed3-b6b6-953e8a083334 Has pop music lost its fun?]. BBC. 12 January 2018. *[https://www.spectator.co.uk/article/its-official-modern-music-is-bad/ It’s official: modern music is bad]. The Spectator. 13 February 2024. Homogeneity of pop music: *[https://www.theguardian.com/music/2012/jul/27/pop-music-sounds-same-survey-reveals Pop music these days: it all sounds the same, survey reveals]. The Guardian. 27 July 2012. *[https://www.nbcnews.com/id/wbna48356108 Pop Music All Sounds the Same Nowadays]. NBC News. 27 July 2012. *[https://www.independent.co.uk/voices/comment/why-does-today-s-pop-music-sound-the-same-because-the-same-people-make-it-8368714.html Why does today's pop music sound the same? Because the same people make it]. The Independent. 29 November 2012. *[https://www.reuters.com/article/lifestyle/science/pop-music-too-loud-and-all-sounds-the-same-official-idUSBRE86P0R9/ Pop music too loud and all sounds the same: official]. Reuters. 26 July 2012. *[https://theconversation.com/from-art-form-to-asset-our-study-found-popular-songs-are-becoming-more-generic-266097 From art form to asset: our study found popular songs are becoming more generic]. The Conversation. 3 October 2025. Conferences: *International Music Industry Conference. 1971: [https://books.google.co.uk/books?id=tggEAAAAMBAJ&pg=PA29#v=onepage&q&f=false] Laserdisc/Karaoke/CES *Billboard. Karaoke. 1992: [https://books.google.co.uk/books?id=jg8EAAAAMBAJ&pg=PA41-IA1#v=onepage&q&f=false] **CES and Karaoke. 1994. [https://books.google.co.uk/books?id=UggEAAAAMBAJ&pg=PA77#v=onepage&q&f=false] **Laserdisc. 1995. [https://books.google.co.uk/books?id=7AsEAAAAMBAJ&pg=PA67#v=onepage&q&f=false] **Laserdisc/Karaoke. 1996: [https://books.google.co.uk/books?id=iQ8EAAAAMBAJ&pg=PA59#v=onepage&q&f=false] Classical music *Billboard spotlights: 1995 [https://books.google.co.uk/books?id=1g0EAAAAMBAJ&pg=PA39#v=onepage&q&f=false] (9 September 1995) **"Classical Music Recording Market". Billboard. 12 April 1980. pp C-1 to C-12 and p 32. (A Billboard Spotlight). **"Classical Music: Discovering New Dimensions". Billboard. 10 September 1983. pp C-1 to C-18. (A Billboard Spotlight). *"Classical" section, and "Best Selling Classical LPs" chart, in Billboard Jazz *[[w:en:All About Jazz|All About Jazz]] Oldies *"Oldies stations find their place in radio market". Star-News. 13 March 1988. pp 1D & [https://books.google.co.uk/books?id=2OoyAAAAIBAJ&pg=PA16#v=onepage&q&f=false 6D]: "Oldies". *Billboard. 15 April 1972. [https://books.google.co.uk/books?id=a0UEAAAAMBAJ&pg=PT7#v=onepage&q&f=false p 47]. *Billboard. 17 April 1961, [https://books.google.co.uk/books?id=JiIEAAAAMBAJ&pg=PA1#v=onepage&q&f=false p 1]. *Billboard. 4 January 1960, [https://books.google.co.uk/books?id=Ch8EAAAAMBAJ&pg=PA1#v=onepage&q&f=false p 1] Nostalgia See also [[Universal Bibliography/Nostalgia]] *"A Perspective on the Future of Nostalgia". Billboard. 4 May 1974. pp [https://books.google.co.uk/books?id=cgkEAAAAMBAJ&pg=PA37#v=onepage&q&f=false N-1] to N-54 and two more pages. *Carr. Nostalgia, Song and the Quest for Home: Production, Text, Reception. 2025. [https://books.google.co.uk/books?id=xz1jEQAAQBAJ&pg=PP1#v=onepage&q&f=false] Charts *Carroll, " Did Billboard, Cash Box, and Record World Charts Tell the Same Story? Perception and Reality, 1960-1979"(2022) 9 Rock Music Studies [https://www.tandfonline.com/doi/full/10.1080/19401159.2022.2054107 199] Magazines See also [[w:Category:Music magazines]] *Billboard. Google: [https://books.google.co.uk/books/serial/ISSN:00062510?rview=1&lr=&sa=N&start=2770 1942] onwards ==Japanese and Japan== *The Ashgate Research Companion to Japanese Music. 2017. [https://books.google.co.uk/books?id=W2JTgQGc99EC&pg=PP1#v=onepage&q&f=false] [https://books.google.co.uk/books?id=4tINDgAAQBAJ&pg=PA2#v=onepage&q&f=false] *Billboard. Spotlight on Japan. 1970: 19 December 1970 [https://books.google.co.uk/books?id=mSkEAAAAMBAJ&pg=PA37#v=onepage&q&f=false]. 1971: 11 December 1971 [https://books.google.co.uk/books?id=Fg8EAAAAMBAJ&pg=PA39#v=onepage&q&f=false]. 1973: 17 February 1973 [https://books.google.co.uk/books?id=QEUEAAAAMBAJ&pg=PT25#v=onepage&q&f=false]. 1977: 30 April 1977 [https://books.google.co.uk/books?id=USMEAAAAMBAJ&pg=PT46#v=onepage&q&f=false]. 1979: [https://books.google.co.uk/books?id=_iQEAAAAMBAJ&pg=PT48#v=onepage&q&f=false]. 1982:[https://books.google.co.uk/books?id=byQEAAAAMBAJ&pg=PT38#v=onepage&q&f=false]. 1985: [https://books.google.co.uk/books?id=1CQEAAAAMBAJ&pg=PT65#v=onepage&q&f=false]. 1986:[https://books.google.co.uk/books?id=-CMEAAAAMBAJ&pg=RA1-PA79#v=onepage&q&f=false]. 1993: 12 June 1993 [https://books.google.co.uk/books?id=9A8EAAAAMBAJ&pg=PA57#v=onepage&q&f=false]. 1995: 5 August 1995 [https://books.google.co.uk/books?id=xwsEAAAAMBAJ&pg=PA52-IA1#v=onepage&q&f=false]. 1996: 31 August 1996 [https://books.google.co.uk/books?id=vwcEAAAAMBAJ&pg=PA66#v=onepage&q&f=false]. 1997: 30 August 1997 [https://books.google.co.uk/books?id=_gkEAAAAMBAJ&pg=PA61#v=onepage&q&f=false]. 1998: 26 September 1998 [https://books.google.co.uk/books?id=GgoEAAAAMBAJ&pg=PA117#v=onepage&q&f=false]. 2000: 9 September 2000 [https://books.google.co.uk/books?id=aREEAAAAMBAJ&pg=PA65#v=onepage&q&f=false]. 2002: 7 September 2002 [https://books.google.co.uk/books?id=-QwEAAAAMBAJ&pg=PA53#v=onepage&q&f=false]. 2003: 5 July 2003 [https://books.google.co.uk/books?id=3w0EAAAAMBAJ&pg=PA45#v=onepage&q&f=false]. **"Japan in 1974: Business Bristles While Shortages Are Met". Billboard. 23 February 1974. pp J-1 to J-30. (A Billboard Spotlight). **"Made in Japan: A Dynamic Music Industry". Billboard. 1 March 1975. pp J-1 to J-23. (A Billboard Spotlight). **"Japan '76". Billboard. 17 April 1976. pp 36 to 59. (A Billboard Spotlight). **"Japanese Music: The Challenge of Recession". Billboard. 27 May 1978. pp J-1 to J-31. (A Billboard Spotlight). **"Music in Japan: Industry Views 1981 With Quiet Optimism". Billboard. 30 May 1981. pp J-1 to J-18. **"Japan: Where Technology Greets Tradition". (An International Market Profile). Billboard. 21 May 1983. pp J-1 to J-13. Follows p 34. **"Billboard Spotlight on Japan: VCRs and CDs Will Be Pacemakers". Billboard. 26 May 1984. pp J-1 to J-11. Follows p 38. **"Spotlight on Japan". Billboard. 6 June 1987. pp J-1 to J-12. **"Japan '88". Billboard. 9 July 1988. pp J-1 to J-11. (A Billboard International Spotlight). **"Japan". ("Japan '89"/"Spotlight on Japan"). Billboard. 3 June 1989. pp J-1 to J-20. (International Spotlight). **"Japan". ("International Spotlight"/"A Billboard Spotlight"). Billboard. 25 May 1991. pp J-1 to J-26. Follows p 50. Called "Japan '91" on front page. *[[w:The Best Ten|The Best Ten]] (ザ・ベストテン). [Television programme]. [https://www.tbs.co.jp/tbs-ch/special/the_bestten/ Episodes]. *[[w:ja:Music Station|Music Station]]. [Television programme]. Episodes: [https://www.tv-asahi.co.jp/music/contents/m_lineup/0003/index.html episode 1] etc. *Wade. Music in Japan: Experiencing Music, Expressing Culture. 2005. [https://books.google.co.uk/books?id=XXYIAQAAMAAJ] *Malm. Japanese Music & Musical Instruments. 1959. [https://books.google.com/books?id=QkTaAAAAMAAJ] *[[w:Francis Taylor Piggott|Pigott]]. The Music and Musical Instruments of Japan. 1893 [https://books.google.co.uk/books?id=ttKTUwmjzMwC&pg=PR3#v=onepage&q&f=false]. 1909. [https://books.google.co.uk/books?id=MAM5AAAAIAAJ] Bibliography *Tsuge. Japanese Music: An Annotated Bibliography. 1986. [https://books.google.com/books?id=YCsKAQAAMAAJ] *[[w:ja:三井徹|Tōru Mitsui]]. Popyurā Ongaku Kankei Tosho Mokuroku: Ryūkōka, Jazu, Rokku, J-poppu no Hyakunen. (Japanese: ポピュラー音楽関係図書目録: 流行歌、ジャズ、ロック、Jポップの百年). Nichigai Associates. 2009. [https://books.google.co.uk/books?id=dSAxAQAAIAAJ]. Catalogues: [https://search.worldcat.org/title/406243182] [https://cir.nii.ac.jp/crid/1970586434933272116] *[https://ndlsearch.ndl.go.jp/rnavi/avmaterials/post_572 音楽に関する文献を探すには（主題書誌）]. NDL. Dictionaries *[[w:ja:下中弥三郎|Shimonaka Yasaburo]] (ed). Ongaku Jiten. Heibonsha. Review: (1959) 18 Journal of Asian Studies 295 [https://www.cambridge.org/core/journals/journal-of-asian-studies/article/abs/ongaku-jiten-dictionary-of-music-ed-shimonaka-yasaburo-tokyo-heibonsha-195557-12-volumes-900-yen-per-volume/F3067B1CE61B5B2C647091E69CE8C8DD] [https://read.dukeupress.edu/journal-of-asian-studies/article-abstract/18/2/295/322980/Ongaku-jiten-Dictionary-of-Music?redirectedFrom=fulltext] History *Eta Harich-Schneider. A History of Japanese Music. 1973. [https://books.google.com/books?id=3AraAAAAMAAJ] *Koh-ichi Hattori. 123 Years of Japanese Music: The Culture of Japan Through a Look at Its Music. 2004. [https://books.google.com/books?id=znzsAAAAMAAJ] **Koh-ichi Hattori. 36,000 Days of Japanese Music: The Culture of Japan Through A Look At Its Music. Pacific Vision. Pierce, Southfield, Michigan. 1996. ISBN 0965364208. *Shinpan Nihon Ryūkōkashi. (Japanese: 新版日本流行歌史). [[w:ja:社会思想社|Shakaishisosha]]. 1994. Review: [https://books.google.co.uk/books?id=XQdIAAAAMAAJ]. Catalogue: [https://ndlsearch.ndl.go.jp/en/books/R100000002-I000002420287] **新版日本流行歌史: 1960-1994. [https://books.google.com/books?id=_b4pAQAAIAAJ] [https://books.google.co.uk/books?id=nb4pAQAAIAAJ]. **新版日本流行歌史: 1938-1959 **1867-1937 *Mehl. Music and the Making of Modern Japan: Joining the Global Concert. 2024. [https://books.google.co.uk/books?id=P3QMEQAAQBAJ&pg=PA2#v=onepage&q&f=false] Modern, contemporary, today *Johnson. Handbook of Japanese Music in the Modern Era. 2024. [https://books.google.co.uk/books?id=KNP7EAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Matsue. Focus: Music in Contemporary Japan. 2016. [https://books.google.co.uk/books?id=AQgtCgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Music of Japan Today. [https://books.google.co.uk/books?id=YZQYEAAAQBAJ&pg=PP1#v=onepage&q&f=false] Popular music *Mitsui (ed). Made in Japan: Studies in Popular Music. 2014. [https://books.google.co.uk/books?id=YWQKBAAAQBAJ&pg=PP1#v=onepage&q&f=false] *Stevens. Japanese Popular Music: Culture, Authenticity and Power. 2008. [https://books.google.co.uk/books?id=OHMkdcL9DAMC&pg=PP1#v=onepage&q&f=false] *Mitsui. Popular Music in Japan: Transformation Inspired by the West. 2020. [https://books.google.co.uk/books?id=FpbqDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Nagahara. Tokyo Boogie-Woogie: Japan’s Pop Era and Its Discontents. 2017. [https://books.google.co.uk/books?id=iTxYDgAAQBAJ&pg=PP1#v=onepage&q&f=false] *Patterson. Music and Words: Producing Popular Songs in Modern Japan, 1887–1952. 2019. [https://books.google.co.uk/books?id=P0FvDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *James Stanlaw. "Using English identity markers in Japanese Popular Music". English in East and South Asia. Chapter 14. [https://books.google.co.uk/books?id=88A1EAAAQBAJ&pg=PT109#v=onepage&q&f=false] *"Japanese Popular Music in Singapore". Asian Music. vol 34. No 1: Fall/Winter 2002/2003. p 1. [https://books.google.co.uk/books?id=_D4JAQAAMAAJ] *Steve McClure. Nipponpop. Tuttle Publishing. 1998. ISBN 9780804821070. ISBN 0804821070. [Sometimes called "Nippon Pop"]. Catalogue: [https://search.worldcat.org/title/Nipponpop/oclc/247384040] Review: (1998) [https://books.google.co.uk/books?id=f9egmeZ8YywC 245] The Publishers Weekly 2 From folk to J-pop *[[w:ja:富澤一誠|Issei Tomizawa]]. Ano subarashii kyoku o mō ichido: fōku kara J-poppu made. (Japanese: あの素晴しい曲をもう一度: フォークからJポップまで). [[w:Shinchosha|Shinchosha]]. 2010. [https://books.google.com/books?id=ju9MAQAAIAAJ]. Catalogue: [https://search.worldcat.org/title/501749494]. Commentary on book: [https://www.ytv.co.jp/michiura/time/2010/01/j2010110.html]. Review of the CD: [https://www.cdjournal.com/i/disc/great-agefree-music-forever-and-great-music-are-o/4109110788]. J-pop *Bourdaghs. Sayonara Amerika, Sayonara Nippon: A Geopolitical Prehistory of J-pop. 2012. [https://books.google.co.uk/books?id=K_y88JwibrMC&pg=PP1#v=onepage&q&f=false] *"The Rise of J-Pop in Asia and Its Impact" (2004) Japan Spotlight. vol 23. p 24. [https://books.google.co.uk/books?id=i7C0AAAAIAAJ] *Terence Lancashire. "J-pop's elusive J: Is Japanese popular music Japanese?" (2008) Perfect Beat. vol 9. No 1. p 38. [https://books.google.co.uk/books?id=5No4AQAAIAAJ] *Tetsu Misaki. J-poppu no Nihongo: kashiron. (Japanese: Jポップの日本語: 歌詞論). [[w:ja:彩流社|彩流社 (Sairyusha)]]. 2002. [https://books.google.com/books?id=dsMpAQAAIAAJ] [https://search.worldcat.org/ja/title/J-:/oclc/52005194] *[[w:ja:烏賀陽弘道|Hiromichi Ugaya]]. Jpoppu Towa Nanika: Kyodaikasuru Ongaku Sangyō. (Japanese: Jポップとは何か: 巨大化する音楽産業). 2005. [https://books.google.co.uk/books?id=TLlOAAAAMAAJ] catalogue [https://search.worldcat.org/ja/title/J-:/oclc/676652594] [https://ci.nii.ac.jp/ncid/BA71618018] Japanese rock *Takarajima Special Edition: Encyclopedia of Japanese Rock 1955-1990. Nihon rokku daihyakka: Rokabirī kara bando būmu made. (Japanese: 日本ロック大百科 [年表編] ロカビリーからバンド・ブームまで 1955〜1990). [[w:ja:JICC出版局|JICC Shuppankyoku]]. 1992. ISBN 9784796602907. ISBN 4796602909. Catalogues: [https://ci.nii.ac.jp/ncid/BN07889172] [https://catalogue.nla.gov.au/catalog/2263400]. *Japanese Rock: Standard: 1967-1985. 日本のロック名曲徹底ガイド: 名曲263決定盤846. CDJournal. 2008. ISBN 9784861710469. ISBN 4861710464. [https://www.cdjournal.com/Company/products/mook.php?mno=20081002]. Catalogue: [https://ci.nii.ac.jp/ncid/BA8932668X?l=en]. *Kojima Satoshi (Japanese: 小島智). 検証・80年代日本のロック. アルファベータブックス. 2024. ISBN 9784865981179. ISBN 4865981179. [https://books.google.com/books?id=0gbl0AEACAAJ]. Review: [https://mainichi.jp/articles/20241026/ddm/015/070/005000c]. Jazz *[[w:ja:スイングジャーナル|Swing Journal]] (1947 to 2010) Commentary: [https://www.allaboutjazz.com/news/swing-journal-long-standing-jazz-magazine-to-be-suspended-in-june/] Japanese fusion: *THE DIG presents 日本のフュージョン. Shinko Music Mook. Released 19 April 2013. Commentary: [https://www.cdjournal.com/news/casiopea/50967]. No II. Released 23 October 2014. Commentary: [https://www.cdjournal.com/news/takanaka-masayoshi/62225] Classical *[[w:ja:ぶらあぼ|Bravo]] (Japanese: ぶらあぼ) ebravo.jp *[[w:ja:音楽芸術 (雑誌)|Ongaku Geijutsu]] (Japanese: 音楽芸術) Magazines For Japanese music magazines, see [[w:ja:日本の音楽雑誌]]. *Music Periodicals in Japan — A Comprehensive List (1988) 35 Fontes Artis Musicae 116 [https://www.jstor.org/stable/23507222] [https://books.google.com/books?id=qHYWAAAAIAAJ] **Kishimoto, "Additional Corrections and Alphabetical Title Index" (1989) 36 Fontes Artis Musicae 38 [https://www.jstor.org/stable/23507313] [https://books.google.co.uk/books?id=7XYWAAAAIAAJ] *Special Bibliography: A Bibliography of Japanese Magazines and Music (1959) 3 Ethnomusicology 76 [https://www.jstor.org/stable/924290] *A Historical Survey of Music Periodicals in Japan: 1881—1920 (1989) 36 Fontes Artis Musicae 44 [https://www.jstor.org/stable/23507314] *[[w:ja:篠原章|Akira Shinohara]]. 日本ロック雑誌クロニクル. [[w:en:Ohta Publishing|Ohta Publishing]]. 2005. [https://books.google.co.uk/books?id=L8opAQAAIAAJ] *[[w:Oricon|Oricon]] (オリコン) **[https://web.archive.org/web/19970412131857/http://www.999.com/Oricon/index.html Oricon Music Site]. Commentary: [https://internet.watch.impress.co.jp/www/article/980309/oms.htm]. *[[w:Billboard Japan|Billboard Japan]] (ビルボード・ジャパン) **Music Labo (ミュージック・ラボ) (1970 to 1994) *Music Research (ミュージック・リサーチ) ["Weekly Music Magazine"]. Catalogue: [https://web.archive.org/web/20260319070908/https://ndlsearch.ndl.go.jp/books/R100000002-I000000039804]. *Rolling Stone Japan *新譜ジャーナル (Shinpu Journal). Catalogue: [https://ndlsearch.ndl.go.jp/books/R100000002-I000000012315]. Began 1968 [https://books.google.co.uk/books?id=L8opAQAAIAAJ], later called シンプジャーナル **シンプジャーナル *Myūjikku mansurī [ミュージック・マンスリー]  [https://ci.nii.ac.jp/ncid/AN00396190] *カセットライフ. (Cassette Life). [[w:ja:シンコーミュージック・エンタテイメント|Shinko Music Entertainment]] *[[w:ja:CDジャーナル|CDJournal]] *[[w:ja:Rockin'on Japan|Rockin'on Japan]]. (ロッキング・オン・ジャパン). (1986 onwards) *[[w:ja:Rooftop|Rooftop]] (1976 onwards) *[[w:ja:FOOL'S MATE|Fool's Mate]] Columns in periodicals *"Japanese Newsnotes". Billboard. (eg 17 April 1961, [https://books.google.co.uk/books?id=JiIEAAAAMBAJ&pg=PA13#v=onepage&q&f=false p 13].) Websites *[[w:ja:ナタリー (ニュースサイト)|Natalie]] (ナタリー) *[[w:ja:BARKS|Barks]] *OKMusic. Commentary: [https://xtech.nikkei.com/it/article/NEWS/20120626/405442/]. Charts For Japanese music charts, see [[w:ja:日本の音楽チャート]] Chart books *Oricon Chart Book (Japanese: オリコンチャート・ブック) **1987 to 1998 Oricon Chart Book. All Albums. [https://books.google.co.uk/books?id=KvEoNwAACAAJ] **Album Chart Book Complete Edition 1970〜2005. Catalogue:[https://www.tosyokan.pref.shizuoka.jp/licsxp-opac/WOpacMsgNewListToTifTilDetailAction.do?tilcod=1000610247212] *澤山博之. ミュージック・ライフ 東京で1番売れていたレコード 1958~1966. Shinko Music Entertainment. 2019. [Charts published in Music Life from 1958 onwards]. Commentary: [https://mikiki.tokyo.jp/articles/-/20952 Mikiki] Number ones *Oricon No.1 Hits 500. Clubhouse (Japanese: クラブハウス). 1994. 1998. **[https://books.google.com/books?id=GlsnNwAACAAJ vol 1 (1968~1985)]. ISBN 9784906496129. **[https://books.google.com/books?id=icInNwAACAAJ vol 2 (1986~1994)]. ISBN 9784906496136. Awards Japan Record Awards *輝く!日本レコード大賞 公式データブック: 放送60回記念: TBS公認. Shinko Music Entertainment. ISBN 9784401647019. [https://books.google.co.uk/books?id=JcDqvwEACAAJ] [https://ci.nii.ac.jp/ncid/BB2773137X] Traditional, Hogaku *Malm. Traditional Japanese Music and Musical Instruments. [https://books.google.co.uk/books?id=Yn3VQbqywCsC&pg=PP1#v=onepage&q&f=false] *Miyuki Yoshikami. Japan's Musical Tradition: Hogaku from Prehistory to the Present. 2020. [https://books.google.co.uk/books?id=X3XTDwAAQBAJ&pg=PP1#v=onepage&q&f=false] *Hughes. Traditional Folk Song in Modern Japan: Sources, Sentiment and Society. 2008. [https://books.google.co.uk/books?id=yfV5DwAAQBAJ&pg=PR1#v=onepage&q&f=false] Koto: *Tokyo Academy of Music. Collection of Japanese Koto Music. 1888. [https://books.google.co.uk/books?id=RncQAAAAYAAJ&pg=PP13#v=onepage&q&f=false][https://babel.hathitrust.org/cgi/pt?id=hvd.32044040839565&seq=1] Exam guides: For the 音楽CD検定 exam on music CDs: *音楽CD検定公式ガイドブック. 2007. [[w:ja:音楽出版社 (企業)|Ongaku Shuppansha Co Ltd]] (音楽出版社). [https://books.google.co.uk/books?id=sbjdeDJMkQcC&pg=PP1#v=onepage&q&f=false vol 1]. [https://books.google.co.uk/books?id=AoFgIowII48C&pg=PP1#v=onepage&q&f=false vol 2]. Commentary: [https://www.cdjournal.com/i/news/-/15303] [https://www.oricon.co.jp/news/46065/full/] [https://allabout.co.jp/gm/gc/57723/] [https://www.oricon.co.jp/news/45388/full/]. Children's music *Elizabeth May. The Influence of the Meiji Period on Japanese Children's Music. University of California Press. 1963. [https://books.google.co.uk/books?id=54cHAQAAMAAJ] **Japanese Children's Music Before and After Contact with the West. University of California at Los Angeles. 1958. (doctoral dissertation). DJs *Masahiro Yasuda, "How Japanese DJs cut across Market Boundaries" (1999) [https://books.google.co.uk/books?id=H5QJAQAAMAAJ 4] Perfect Beat 45 [[Category:Music]] et6w6k1e1ch7nzbb8yktzl2sslpr2gc 0 322040 2812041 2761049 2026-05-29T16:53:04Z Unitfreak 695864 2812041 wikitext text/x-wiki {| class=table style="width:100%;" |- | {{Original research}} | [https://physwiki.eeyabo.net/index.php/Main_Page <small>Development <br/>Area</small>] |} [[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> There have been over 655,360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the more than 65 years of modern atomic time keeping (1958 AD ... present). Given the availability of atomic clocks, it is anticipated that Bully timestamps will continue to be realized with great regularity for the foreseeable future. Each Bully timestamp should be considered "realized" after it occurs and is measured using precise clocks. === Leap Seconds (1972 - Present) === The below table (derived from the Wikipedia "Leap Second" article), lists all leap second insertions that have occurred since the introduction of leap seconds in 1972. For each leap second insertion, the below table lists the preceding Bully timestamp (that had been "realized" immediately prior to the leap second insertion), and the subsequent Bully timestamp (that was "realized" immediately after the leap second insertion). A few details are worth noting in the table. The TAI and UTC already differed by 10 seconds at the beginning of 1972 due to rubber seconds ([https://en.m.wikiversity.org/wiki/Bully_Metric_Realized_Timestamps#Rubber_Seconds_(1958_-_1971) see discussion below]), so when Bully Timestamp 8209 27FB E7FB was realized, the TAI time was 1972-06-30 23:34:45 TAI, whereas UTC time was 1972-06-30 23:34:35 UTC. An additional 27 leap seconds have been inserted into UTC during the fifty year period between 1972 and 2022, making a total of 37 leap seconds difference, so when Bully Timestamp 8209 2802 EBC0 was realized, the TAI time was 2017-01-01 00:32:00 TAI, whereas UTC time was 2017-01-01 00:31:23 UTC. You will also note that Bully timestamps are realized during TAI times with a seconds value ending in five or zero. The Bully timestamp and TAI both measure elapsed time as determined by atomic clocks, so these systems will always have this simple relationship. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Announced leap seconds to date |- ! Year !! 30 Jun !! 31 Dec !! Bully Timestamp !! International Atomic Time (TAI) !! Coordinated Universal Time (UTC) |- ! 1972 |bgcolor="lime"| +1 ||bgcolor="lime"| +1 || 8209 27FB E7FB <br /> 8209 27FB E7FC <br /> 8209 27FB FC4F <br /> 8209 27FB FC50 || 1972-06-30 23:34:45 TAI <br /> 1972-07-01 00:25:40 TAI <br /> 1972-12-31 23:45:05 TAI <br /> 1973-01-01 00:36:00 TAI || 1972-06-30 23:34:35 UTC <br /> 1972-07-01 00:25:29 UTC <br /> 1972-12-31 23:44:54 UTC <br /> 1973-01-01 00:35:48 UTC |- ! 1973 | 0 ||bgcolor="lime"| +1 || 8209 27FC 24A2 <br /> 8209 27FC 24A3 || 1973-12-31 23:57:50 TAI <br /> 1974-01-01 00:48:45 TAI || 1973-12-31 23:57:38 UTC <br /> 1974-01-01 00:48:32 UTC |- ! 1974 | 0 ||bgcolor="lime"| +1 || 8209 27FC 4CF4 <br /> 8209 27FC 4CF5 || 1974-12-31 23:19:40 TAI <br /> 1975-01-01 00:10:35 TAI || 1974-12-31 23:19:27 UTC <br /> 1975-01-01 00:10:21 UTC |- ! 1975 | 0 ||bgcolor="lime"| +1 || 8209 27FC 7547 <br /> 8209 27FC 7548 || 1975-12-31 23:32:25 TAI <br /> 1976-01-01 00:23:20 TAI || 1975-12-31 23:32:11 UTC <br /> 1976-01-01 00:23:05 UTC |- ! 1976 | 0 ||bgcolor="lime"| +1 || 8209 27FC 9DB6 <br /> 8209 27FC 9DB7 || 1976-12-31 23:30:50 TAI <br /> 1977-01-01 00:21:45 TAI || 1976-12-31 23:30:35 UTC <br /> 1977-01-01 00:21:29 UTC |- ! 1977 | 0 ||bgcolor="lime"| +1 || 8209 27FC C609 <br /> 8209 27FC C60A || 1977-12-31 23:43:35 TAI <br /> 1978-01-01 00:34:30 TAI || 1977-12-31 23:43:19 UTC <br /> 1978-01-01 00:34:13 UTC |- ! 1978 | 0 ||bgcolor="lime"| +1 || 8209 27FC EE5C <br /> 8209 27FC EE5D || 1978-12-31 23:56:20 TAI <br /> 1979-01-01 00:47:15 TAI || 1978-12-31 23:56:03 UTC <br /> 1979-01-01 00:46:57 UTC |- ! 1979 | 0 ||bgcolor="lime"| +1 || 8209 27FD 16AE <br /> 8209 27FD 16AF || 1979-12-31 23:18:10 TAI <br /> 1980-01-01 00:09:05 TAI || 1979-12-31 23:17:52 UTC <br /> 1980-01-01 00:08:46 UTC |- ! 1981 |bgcolor="lime"| +1 || 0 || 8209 27FD 531C <br /> 8209 27FD 531D || 1981-06-30 23:19:00 TAI <br /> 1981-07-01 00:09:55 TAI || 1981-06-30 23:18:41 UTC <br /> 1981-07-01 00:09:35 UTC |- ! 1982 |bgcolor="lime"| +1 || 0 || 8209 27FD 7B6F <br /> 8209 27FD 7B70 || 1982-06-30 23:31:45 TAI <br /> 1982-07-01 00:22:40 TAI || 1982-06-30 23:31:25 UTC <br /> 1982-07-01 00:22:19 UTC |- ! 1983 |bgcolor="lime"| +1 || 0 || 8209 27FD A3C2 <br /> 8209 27FD A3C3 || 1983-06-30 23:44:30 TAI <br /> 1983-07-01 00:35:25 TAI || 1983-06-30 23:44:09 UTC <br /> 1983-07-01 00:35:03 UTC |- ! 1985 |bgcolor="lime"| +1 || 0 || 8209 27FD F484 <br /> 8209 27FD F485 || 1985-06-30 23:55:40 TAI <br /> 1985-07-01 00:46:35 TAI || 1985-06-30 23:55:18 UTC <br /> 1985-07-01 00:46:12 UTC |- ! 1987 | 0 ||bgcolor="lime"| +1 || 8209 27FE 597D <br /> 8209 27FE 597E || 1987-12-31 23:40:35 TAI <br /> 1988-01-01 00:31:30 TAI || 1987-12-31 23:40:12 UTC <br /> 1988-01-01 00:31:06 UTC |- ! 1989 | 0 ||bgcolor="lime"| +1 || 8209 27FE AA3F <br /> 8209 27FE AA40 || 1989-12-31 23:51:45 TAI <br /> 1990-01-01 00:42:40 TAI || 1989-12-31 23:51:21 UTC <br /> 1990-01-01 00:42:15 UTC |- ! 1990 | 0 ||bgcolor="lime"| +1 || 8209 27FE D291 <br /> 8209 27FE D292 || 1990-12-31 23:13:35 TAI <br /> 1991-01-01 00:04:30 TAI || 1990-12-31 23:13:10 UTC <br /> 1991-01-01 00:04:04 UTC |- ! 1992 |bgcolor="lime"| +1 || 0 || 8209 27FF 0EFF <br /> 8209 27FF 0F00 || 1992-06-30 23:14:25 TAI <br /> 1992-07-01 00:05:20 TAI || 1992-06-30 23:13:59 UTC <br /> 1992-07-01 00:04:53 UTC |- ! 1993 |bgcolor="lime"| +1 || 0 || 8209 27FF 3752 <br /> 8209 27FF 3753 || 1993-06-30 23:27:10 TAI <br /> 1993-07-01 00:18:05 TAI || 1993-06-30 23:26:43 UTC <br /> 1993-07-01 00:17:37 UTC |- ! 1994 |bgcolor="lime"| +1 || 0 || 8209 27FF 5FA5 <br /> 8209 27FF 5FA6 || 1994-06-30 23:39:55 TAI <br /> 1994-07-01 00:30:50 TAI || 1994-06-30 23:39:27 UTC <br /> 1994-07-01 00:30:21 UTC |- ! 1995 | 0 ||bgcolor="lime"| +1 || 8209 27FF 9C4B <br /> 8209 27FF 9C4C || 1995-12-31 23:12:05 TAI <br /> 1996-01-01 00:03:00 TAI || 1995-12-31 23:11:36 UTC <br /> 1996-01-01 00:02:30 UTC |- ! 1997 |bgcolor="lime"| +1 || 0 || 8209 27FF D8B9 <br /> 8209 27FF D8BA || 1997-06-30 23:12:55 TAI <br /> 1997-07-01 00:03:50 TAI || 1997-06-30 23:12:25 UTC <br /> 1997-07-01 00:03:19 UTC |- ! 1998 | 0 ||bgcolor="lime"| +1 || 8209 2800 1560 <br /> 8209 2800 1561 || 1998-12-31 23:36:00 TAI <br /> 1999-01-01 00:26:55 TAI || 1998-12-31 23:35:29 UTC <br /> 1999-01-01 00:26:23 UTC |- ! 2005 | 0 ||bgcolor="lime"| +1 || 8209 2801 2FDC <br /> 8209 2801 2FDD || 2005-12-31 23:45:40 TAI <br /> 2006-01-01 00:36:35 TAI || 2005-12-31 23:45:08 UTC <br /> 2006-01-01 00:36:02 UTC |- ! 2008 | 0 ||bgcolor="lime"| +1 || 8209 2801 A8F0 <br /> 8209 2801 A8F1 || 2008-12-31 23:18:40 TAI <br /> 2009-01-01 00:09:35 TAI || 2008-12-31 23:18:07 UTC <br /> 2009-01-01 00:09:01 UTC |- ! 2012 |bgcolor="lime"| +1 || 0 || 8209 2802 3604 <br /> 8209 2802 3605 || 2012-06-30 23:45:00 TAI <br /> 2012-07-01 00:35:55 TAI || 2012-06-30 23:44:26 UTC <br /> 2012-07-01 00:35:20 UTC |- ! 2015 |bgcolor="lime"| +1 || 0 || 8209 2802 AEFC <br /> 8209 2802 AEFD || 2015-06-30 23:32:20 TAI <br /> 2015-07-01 00:23:15 TAI || 2015-06-30 23:31:45 UTC <br /> 2015-07-01 00:22:39 UTC |- ! 2016 | 0 ||bgcolor="lime"| +1 || 8209 2802 EBBF <br /> 8209 2802 EBC0 || 2016-12-31 23:41:05 TAI <br /> 2017-01-01 00:32:00 TAI || 2016-12-31 23:40:29 UTC <br /> 2017-01-01 00:31:23 UTC |} === Rubber Seconds (1958 - 1971) === [[File:Bully Timestamps in relation to rubber seconds.png|frame|center|text-bottom|Figure 2: Rubber Seconds]] Prior to 1972, the rate of UTC atomic clocks was offset from a pure atomic time scale by the BIH to remain synchronized with UT2, a practice known as the "rubber second" (see figure 2). The rate of UTC was decided at the start of each year. Alongside this shift in rate, an occasional 0.1&nbsp;s step (0.05&nbsp;s before 1963) was also implemented as needed. As shown in figure 2, for 1958-1961, the offset rate was −150 parts per 10{{sup|10}} (or 0.47 seconds per year). This stretching of UTC "rubber seconds" meant that fewer of them would occur during a Bully Timestamp. For example, during the 1958-1961 time period, each Bully timestamp was realized after exactly 3055 seconds TAI, which corresponded to 3054.999955264 seconds UTC. For 1962–63 the offset rate was set to −130 parts per 10{{sup|10}} (or 0.41 seconds per year, or 3054.999960285 seconds UTC per Bully timestamp), and then for 1964–65 the offset rate was returned to −150 parts per 10{{sup|10}}. The UTC rate of −150 parts per 10{{sup|10}} turned out to be notably inadequate during the 1964-1965 time period, and multiple 0.1&nbsp;s steps were needed (see figure 2). Beginning in 1966, the offset rate was set to −300 parts per 10{{sup|10}} (or 0.94 seconds per year, or 3054.99990835 seconds UTC per Bully timestamp), and this continued until the inauguration of Leap Seconds in 1972. At the beginning of 1958, the TAI and UTC clocks were in sync, with 1958-01-01 00:00:00.000 TAI occurring at the same time as 1958-01-01 00:00:00.000 UTC. By the end of 1972, the UTC clock had been adjusted (using rubber seconds and time steps) by ten leap seconds, so that 1972-01-01 00:00:10.003 TAI occurred at the same time as 1972-01-01 00:00:00.003 UTC. The following table illustrates the slow accumulation of leap seconds prior to 1972, resulting in this ten second difference. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Rubber Seconds and Accumulative (TAI - UTC) Time Delta |- ! Approximate Bully Timestamp <br /> Approximate International Atomic Time (TAI) <br /> Coordinated Universal Time (UTC) !! (ΔTAI - ΔUTC) !! Accumulative <br /> Difference |- ! 8209 27F9 9F04 (+2820.0 sec) . . . 8209 27F9 EFAA (+1290.9 sec) <br /> 1958-01-01 00:00:00.000 TAI . . . 1960-01-01 00:00:00.943 TAI <br /> 1958-01-01 00:00:00.002 UTC . . . 1960-01-01 00:00:00.000 UTC | 0.943 sec || 0.943 sec |- ! 8209 27F9 EFAA (+1290.9 sec) . . . 8209 27FA 1819 (+1386.4 sec) <br /> 1960-01-01 00:00:00.944 TAI . . . 1961-01-01 00:00:01.418 TAI <br /> 1960-01-01 00:00:00.000 UTC . . . 1961-01-01 00:00:00.000 UTC | 0.474 sec || 1.418 sec |- ! 8209 27FA 1819 (+1386.4 sec) <br /> 1961-01-01 00:00:01.418 TAI . . . 1961-01-01 00:00:01.423 TAI <br /> 1961-01-01 00:00:00.000 UTC | 0.005 sec || 1.423 sec |- ! 8209 27FA 1819 (+1386.4 sec) . . . 8209 27FA 2F85 (+406.6 sec) <br /> 1961-01-01 00:00:01.423 TAI . . . 1961-08-01 00:00:01.698 TAI <br /> 1961-01-01 00:00:00.000 UTC . . . 1961-08-01 00:00:00.000 UTC | 0.275 sec || 1.698 sec |- ! 8209 27FA 2F85 (+406.6 sec) <br /> 1961-08-01 00:00:01.698 TAI . . . 1961-08-01 00:00:01.648 TAI <br /> 1961-08-01 00:00:00.000 UTC | -0.050 sec || 1.648 sec |- ! 8209 27FA 2F85 (+406.6 sec) . . . 8209 27FA 406C (+621.8 sec) <br /> 1961-08-01 00:00:01.648 TAI . . . 1962-01-01 00:00:01.846 TAI <br /> 1961-08-01 00:00:00.000 UTC . . . 1962-01-01 00:00:00.000 UTC | 0.198 sec || 1.846 sec |- ! 8209 27FA 406C (+621.8 sec) . . . 8209 27FA 8A54 (+1622.5 sec) <br /> 1962-01-01 00:00:01.846 TAI . . . 1963-11-01 00:00:02.597 TAI <br /> 1962-01-01 00:00:00.000 UTC . . . 1963-11-01 00:00:00.000 UTC | 0.751 sec || 2.597 sec |- ! 8209 27FA 8A54 (+1622.6 sec) <br /> 1963-11-01 00:00:02.597 TAI . . . 1963-11-01 00:00:02.697 TAI <br /> 1963-11-01 00:00:00.000 UTC | 0.100 sec || 2.697 sec |- ! 8209 27FA 8A54 (+1622.6 sec) . . . 8209 27FA 9111 (+2147.7 sec) <br /> 1963-11-01 00:00:02.697 TAI . . . 1964-01-01 00:00:02.766 TAI <br /> 1963-11-01 00:00:00.000 UTC . . . 1964-01-01 00:00:00.000 UTC | 0.069 sec || 2.766 sec |- ! 8209 27FA 9111 (+2147.7 sec) . . . 8209 27FA 9B1F (+977.8 sec) <br /> 1964-01-01 00:00:02.766 TAI . . . 1964-04-01 00:00:02.884 TAI <br /> 1964-01-01 00:00:00.000 UTC . . . 1964-04-01 00:00:00.000 UTC | 0.118 sec || 2.884 sec |- ! 8209 27FA 9B1F (+977.9 sec) <br /> 1964-04-01 00:00:02.884 TAI . . . 1964-04-01 00:00:02.984 TAI <br /> 1964-04-01 00:00:00.000 UTC | 0.100 sec || 2.984 sec |- ! 8209 27FA 9B1F (+977.9 sec) . . . 8209 27FA AC06 (+1193.1 sec) <br /> 1964-04-01 00:00:02.984 TAI . . . 1964-09-01 00:00:03.182 TAI <br /> 1964-04-01 00:00:00.000 UTC . . . 1964-09-01 00:00:00.000 UTC | 0.198 sec || 3.182 sec |- ! 8209 27FA AC06 (+1193.2 sec) <br /> 1964-09-01 00:00:03.182 TAI . . . 1964-09-01 00:00:03.282 TAI <br /> 1964-09-01 00:00:00.000 UTC | 0.100 sec || 3.282 sec |- ! 8209 27FA AC06 (+1193.2 sec) . . . 8209 27FA B980 (+2243.4 sec) <br /> 1964-09-01 00:00:03.282 TAI . . . 1965-01-01 00:00:03.440 TAI <br /> 1964-09-01 00:00:00.000 UTC . . . 1965-01-01 00:00:00.000 UTC | 0.158 sec || 3.440 sec |- ! 8209 27FA B980 (+2243.5 sec) <br /> 1965-01-01 00:00:03.440 TAI . . . 1965-01-01 00:00:03.540 TAI <br /> 1965-01-01 00:00:00.000 UTC | 0.100 sec || 3.540 sec |- ! 8209 27FA B980 (+2243.5 sec) . . . 8209 27FA C005 (+1048.6 sec) <br /> 1965-01-01 00:00:03.540 TAI . . . 1965-03-01 00:00:03.617 TAI <br /> 1965-01-01 00:00:00.000 UTC . . . 1965-03-01 00:00:00.000 UTC | 0.076 sec || 3.617 sec |- ! 8209 27FA C005 (+1048.7 sec) <br /> 1965-03-01 00:00:03.617 TAI . . . 1965-03-01 00:00:03.717 TAI <br /> 1965-03-01 00:00:00.000 UTC | 0.100 sec || 3.717 sec |- ! 8209 27FA C005 (+1048.7 sec) . . . 8209 27FA CD7F (+2098.8 sec) <br /> 1965-03-01 00:00:03.717 TAI . . . 1965-07-01 00:00:03.875 TAI <br /> 1965-03-01 00:00:00.000 UTC . . . 1965-07-01 00:00:00.000 UTC | 0.158 sec || 3.875 sec |- ! 8209 27FA CD7F (+2098.9 sec) <br /> 1965-07-01 00:00:03.875 TAI . . . 1965-07-01 00:00:03.975 TAI <br /> 1965-07-01 00:00:00.000 UTC | 0.100 sec || 3.975 sec |- ! 8209 27FA CD7F (+2098.9 sec) . . . 8209 27FA D459 (+429.0 sec) <br /> 1965-07-01 00:00:03.975 TAI . . . 1965-09-01 00:00:04.055 TAI <br /> 1965-07-01 00:00:00.000 UTC . . . 1965-09-01 00:00:00.000 UTC | 0.080 sec || 4.055 sec |- ! 8209 27FA D459 (+429.1 sec) <br /> 1965-09-01 00:00:04.055 TAI . . . 1965-09-01 00:00:04.155 TAI <br /> 1965-09-01 00:00:00.000 UTC | 0.100 sec || 4.155 sec |- ! 8209 27FA D459 (+429.1 sec) . . . 8209 27FA E1D3 (+1479.3 sec) <br /> 1965-09-01 00:00:04.155 TAI . . . 1966-01-01 00:00:04.313 TAI <br /> 1965-09-01 00:00:00.000 UTC . . . 1966-01-01 00:00:00.000 UTC | 0.158 sec || 4.313 sec |- ! 8209 27FA E1D3 (+1479.3 sec) . . . 8209 27FB 35E5 (+2171.2 sec) <br /> 1966-01-01 00:00:04.313 TAI . . . 1968-02-01 00:00:06.286 TAI <br /> 1966-01-01 00:00:00.000 UTC . . . 1968-02-01 00:00:00.000 UTC | 1.973 sec || 6.286 sec |- ! 8209 27FB 35E5 (+2171.1 sec) <br /> 1968-02-01 00:00:06.286 TAI . . . 1968-02-01 00:00:06.186 TAI <br /> 1968-02-01 00:00:00.000 UTC | -0.100 sec || 6.186 sec |- ! 8209 27FB 35E5 (+2171.1 sec) . . . 8209 27FB D3E0 (+809.9 sec) <br /> 1968-02-01 00:00:06.186 TAI . . . 1972-01-01 00:00:09.891 TAI <br /> 1968-02-01 00:00:00.000 UTC . . . 1971-12-31 23:59:59.999 UTC | 3.707 sec || 9.892 sec |- ! 8209 27FB D3E0 (+809.9 sec) <br /> 1972-01-01 00:00:09.891 TAI . . . 1972-01-01 00:00:09.999 TAI <br /> 1971-12-31 23:59:59.999 UTC | 0.109 sec || 10.000 sec |} 7bb3tinffqor8m1w7wv4khjf03dsdmq 2812042 2812041 2026-05-29T16:54:28Z Unitfreak 695864 2812042 wikitext text/x-wiki {| class=table style="width:100%;" |- | {{Original research}} | [https://physwiki.eeyabo.net/index.php/Main_Page <small>Development <br/>Area</small>] |} [[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> There have been over 655,360 realized Bully timestamps (8209 27F9 0000 ... 8209 2804 0000) during the more than 65 years of modern atomic time keeping (1958 AD ... present). Given the availability of atomic clocks, it is anticipated that Bully timestamps will continue to be realized with great regularity for the foreseeable future. Each Bully timestamp should be considered "realized" after it occurs and is measured using precise clocks with an accuracy of <math>{10}^{-10}</math> or better. === Leap Seconds (1972 - Present) === The below table (derived from the Wikipedia "Leap Second" article), lists all leap second insertions that have occurred since the introduction of leap seconds in 1972. For each leap second insertion, the below table lists the preceding Bully timestamp (that had been "realized" immediately prior to the leap second insertion), and the subsequent Bully timestamp (that was "realized" immediately after the leap second insertion). A few details are worth noting in the table. The TAI and UTC already differed by 10 seconds at the beginning of 1972 due to rubber seconds ([https://en.m.wikiversity.org/wiki/Bully_Metric_Realized_Timestamps#Rubber_Seconds_(1958_-_1971) see discussion below]), so when Bully Timestamp 8209 27FB E7FB was realized, the TAI time was 1972-06-30 23:34:45 TAI, whereas UTC time was 1972-06-30 23:34:35 UTC. An additional 27 leap seconds have been inserted into UTC during the fifty year period between 1972 and 2022, making a total of 37 leap seconds difference, so when Bully Timestamp 8209 2802 EBC0 was realized, the TAI time was 2017-01-01 00:32:00 TAI, whereas UTC time was 2017-01-01 00:31:23 UTC. You will also note that Bully timestamps are realized during TAI times with a seconds value ending in five or zero. The Bully timestamp and TAI both measure elapsed time as determined by atomic clocks, so these systems will always have this simple relationship. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Announced leap seconds to date |- ! Year !! 30 Jun !! 31 Dec !! Bully Timestamp !! International Atomic Time (TAI) !! Coordinated Universal Time (UTC) |- ! 1972 |bgcolor="lime"| +1 ||bgcolor="lime"| +1 || 8209 27FB E7FB <br /> 8209 27FB E7FC <br /> 8209 27FB FC4F <br /> 8209 27FB FC50 || 1972-06-30 23:34:45 TAI <br /> 1972-07-01 00:25:40 TAI <br /> 1972-12-31 23:45:05 TAI <br /> 1973-01-01 00:36:00 TAI || 1972-06-30 23:34:35 UTC <br /> 1972-07-01 00:25:29 UTC <br /> 1972-12-31 23:44:54 UTC <br /> 1973-01-01 00:35:48 UTC |- ! 1973 | 0 ||bgcolor="lime"| +1 || 8209 27FC 24A2 <br /> 8209 27FC 24A3 || 1973-12-31 23:57:50 TAI <br /> 1974-01-01 00:48:45 TAI || 1973-12-31 23:57:38 UTC <br /> 1974-01-01 00:48:32 UTC |- ! 1974 | 0 ||bgcolor="lime"| +1 || 8209 27FC 4CF4 <br /> 8209 27FC 4CF5 || 1974-12-31 23:19:40 TAI <br /> 1975-01-01 00:10:35 TAI || 1974-12-31 23:19:27 UTC <br /> 1975-01-01 00:10:21 UTC |- ! 1975 | 0 ||bgcolor="lime"| +1 || 8209 27FC 7547 <br /> 8209 27FC 7548 || 1975-12-31 23:32:25 TAI <br /> 1976-01-01 00:23:20 TAI || 1975-12-31 23:32:11 UTC <br /> 1976-01-01 00:23:05 UTC |- ! 1976 | 0 ||bgcolor="lime"| +1 || 8209 27FC 9DB6 <br /> 8209 27FC 9DB7 || 1976-12-31 23:30:50 TAI <br /> 1977-01-01 00:21:45 TAI || 1976-12-31 23:30:35 UTC <br /> 1977-01-01 00:21:29 UTC |- ! 1977 | 0 ||bgcolor="lime"| +1 || 8209 27FC C609 <br /> 8209 27FC C60A || 1977-12-31 23:43:35 TAI <br /> 1978-01-01 00:34:30 TAI || 1977-12-31 23:43:19 UTC <br /> 1978-01-01 00:34:13 UTC |- ! 1978 | 0 ||bgcolor="lime"| +1 || 8209 27FC EE5C <br /> 8209 27FC EE5D || 1978-12-31 23:56:20 TAI <br /> 1979-01-01 00:47:15 TAI || 1978-12-31 23:56:03 UTC <br /> 1979-01-01 00:46:57 UTC |- ! 1979 | 0 ||bgcolor="lime"| +1 || 8209 27FD 16AE <br /> 8209 27FD 16AF || 1979-12-31 23:18:10 TAI <br /> 1980-01-01 00:09:05 TAI || 1979-12-31 23:17:52 UTC <br /> 1980-01-01 00:08:46 UTC |- ! 1981 |bgcolor="lime"| +1 || 0 || 8209 27FD 531C <br /> 8209 27FD 531D || 1981-06-30 23:19:00 TAI <br /> 1981-07-01 00:09:55 TAI || 1981-06-30 23:18:41 UTC <br /> 1981-07-01 00:09:35 UTC |- ! 1982 |bgcolor="lime"| +1 || 0 || 8209 27FD 7B6F <br /> 8209 27FD 7B70 || 1982-06-30 23:31:45 TAI <br /> 1982-07-01 00:22:40 TAI || 1982-06-30 23:31:25 UTC <br /> 1982-07-01 00:22:19 UTC |- ! 1983 |bgcolor="lime"| +1 || 0 || 8209 27FD A3C2 <br /> 8209 27FD A3C3 || 1983-06-30 23:44:30 TAI <br /> 1983-07-01 00:35:25 TAI || 1983-06-30 23:44:09 UTC <br /> 1983-07-01 00:35:03 UTC |- ! 1985 |bgcolor="lime"| +1 || 0 || 8209 27FD F484 <br /> 8209 27FD F485 || 1985-06-30 23:55:40 TAI <br /> 1985-07-01 00:46:35 TAI || 1985-06-30 23:55:18 UTC <br /> 1985-07-01 00:46:12 UTC |- ! 1987 | 0 ||bgcolor="lime"| +1 || 8209 27FE 597D <br /> 8209 27FE 597E || 1987-12-31 23:40:35 TAI <br /> 1988-01-01 00:31:30 TAI || 1987-12-31 23:40:12 UTC <br /> 1988-01-01 00:31:06 UTC |- ! 1989 | 0 ||bgcolor="lime"| +1 || 8209 27FE AA3F <br /> 8209 27FE AA40 || 1989-12-31 23:51:45 TAI <br /> 1990-01-01 00:42:40 TAI || 1989-12-31 23:51:21 UTC <br /> 1990-01-01 00:42:15 UTC |- ! 1990 | 0 ||bgcolor="lime"| +1 || 8209 27FE D291 <br /> 8209 27FE D292 || 1990-12-31 23:13:35 TAI <br /> 1991-01-01 00:04:30 TAI || 1990-12-31 23:13:10 UTC <br /> 1991-01-01 00:04:04 UTC |- ! 1992 |bgcolor="lime"| +1 || 0 || 8209 27FF 0EFF <br /> 8209 27FF 0F00 || 1992-06-30 23:14:25 TAI <br /> 1992-07-01 00:05:20 TAI || 1992-06-30 23:13:59 UTC <br /> 1992-07-01 00:04:53 UTC |- ! 1993 |bgcolor="lime"| +1 || 0 || 8209 27FF 3752 <br /> 8209 27FF 3753 || 1993-06-30 23:27:10 TAI <br /> 1993-07-01 00:18:05 TAI || 1993-06-30 23:26:43 UTC <br /> 1993-07-01 00:17:37 UTC |- ! 1994 |bgcolor="lime"| +1 || 0 || 8209 27FF 5FA5 <br /> 8209 27FF 5FA6 || 1994-06-30 23:39:55 TAI <br /> 1994-07-01 00:30:50 TAI || 1994-06-30 23:39:27 UTC <br /> 1994-07-01 00:30:21 UTC |- ! 1995 | 0 ||bgcolor="lime"| +1 || 8209 27FF 9C4B <br /> 8209 27FF 9C4C || 1995-12-31 23:12:05 TAI <br /> 1996-01-01 00:03:00 TAI || 1995-12-31 23:11:36 UTC <br /> 1996-01-01 00:02:30 UTC |- ! 1997 |bgcolor="lime"| +1 || 0 || 8209 27FF D8B9 <br /> 8209 27FF D8BA || 1997-06-30 23:12:55 TAI <br /> 1997-07-01 00:03:50 TAI || 1997-06-30 23:12:25 UTC <br /> 1997-07-01 00:03:19 UTC |- ! 1998 | 0 ||bgcolor="lime"| +1 || 8209 2800 1560 <br /> 8209 2800 1561 || 1998-12-31 23:36:00 TAI <br /> 1999-01-01 00:26:55 TAI || 1998-12-31 23:35:29 UTC <br /> 1999-01-01 00:26:23 UTC |- ! 2005 | 0 ||bgcolor="lime"| +1 || 8209 2801 2FDC <br /> 8209 2801 2FDD || 2005-12-31 23:45:40 TAI <br /> 2006-01-01 00:36:35 TAI || 2005-12-31 23:45:08 UTC <br /> 2006-01-01 00:36:02 UTC |- ! 2008 | 0 ||bgcolor="lime"| +1 || 8209 2801 A8F0 <br /> 8209 2801 A8F1 || 2008-12-31 23:18:40 TAI <br /> 2009-01-01 00:09:35 TAI || 2008-12-31 23:18:07 UTC <br /> 2009-01-01 00:09:01 UTC |- ! 2012 |bgcolor="lime"| +1 || 0 || 8209 2802 3604 <br /> 8209 2802 3605 || 2012-06-30 23:45:00 TAI <br /> 2012-07-01 00:35:55 TAI || 2012-06-30 23:44:26 UTC <br /> 2012-07-01 00:35:20 UTC |- ! 2015 |bgcolor="lime"| +1 || 0 || 8209 2802 AEFC <br /> 8209 2802 AEFD || 2015-06-30 23:32:20 TAI <br /> 2015-07-01 00:23:15 TAI || 2015-06-30 23:31:45 UTC <br /> 2015-07-01 00:22:39 UTC |- ! 2016 | 0 ||bgcolor="lime"| +1 || 8209 2802 EBBF <br /> 8209 2802 EBC0 || 2016-12-31 23:41:05 TAI <br /> 2017-01-01 00:32:00 TAI || 2016-12-31 23:40:29 UTC <br /> 2017-01-01 00:31:23 UTC |} === Rubber Seconds (1958 - 1971) === [[File:Bully Timestamps in relation to rubber seconds.png|frame|center|text-bottom|Figure 2: Rubber Seconds]] Prior to 1972, the rate of UTC atomic clocks was offset from a pure atomic time scale by the BIH to remain synchronized with UT2, a practice known as the "rubber second" (see figure 2). The rate of UTC was decided at the start of each year. Alongside this shift in rate, an occasional 0.1&nbsp;s step (0.05&nbsp;s before 1963) was also implemented as needed. As shown in figure 2, for 1958-1961, the offset rate was −150 parts per 10{{sup|10}} (or 0.47 seconds per year). This stretching of UTC "rubber seconds" meant that fewer of them would occur during a Bully Timestamp. For example, during the 1958-1961 time period, each Bully timestamp was realized after exactly 3055 seconds TAI, which corresponded to 3054.999955264 seconds UTC. For 1962–63 the offset rate was set to −130 parts per 10{{sup|10}} (or 0.41 seconds per year, or 3054.999960285 seconds UTC per Bully timestamp), and then for 1964–65 the offset rate was returned to −150 parts per 10{{sup|10}}. The UTC rate of −150 parts per 10{{sup|10}} turned out to be notably inadequate during the 1964-1965 time period, and multiple 0.1&nbsp;s steps were needed (see figure 2). Beginning in 1966, the offset rate was set to −300 parts per 10{{sup|10}} (or 0.94 seconds per year, or 3054.99990835 seconds UTC per Bully timestamp), and this continued until the inauguration of Leap Seconds in 1972. At the beginning of 1958, the TAI and UTC clocks were in sync, with 1958-01-01 00:00:00.000 TAI occurring at the same time as 1958-01-01 00:00:00.000 UTC. By the end of 1972, the UTC clock had been adjusted (using rubber seconds and time steps) by ten leap seconds, so that 1972-01-01 00:00:10.003 TAI occurred at the same time as 1972-01-01 00:00:00.003 UTC. The following table illustrates the slow accumulation of leap seconds prior to 1972, resulting in this ten second difference. {| class="wikitable" style="margin-right: 0; margin-left: 1em; text-align: center;" |+ Rubber Seconds and Accumulative (TAI - UTC) Time Delta |- ! Approximate Bully Timestamp <br /> Approximate International Atomic Time (TAI) <br /> Coordinated Universal Time (UTC) !! (ΔTAI - ΔUTC) !! Accumulative <br /> Difference |- ! 8209 27F9 9F04 (+2820.0 sec) . . . 8209 27F9 EFAA (+1290.9 sec) <br /> 1958-01-01 00:00:00.000 TAI . . . 1960-01-01 00:00:00.943 TAI <br /> 1958-01-01 00:00:00.002 UTC . . . 1960-01-01 00:00:00.000 UTC | 0.943 sec || 0.943 sec |- ! 8209 27F9 EFAA (+1290.9 sec) . . . 8209 27FA 1819 (+1386.4 sec) <br /> 1960-01-01 00:00:00.944 TAI . . . 1961-01-01 00:00:01.418 TAI <br /> 1960-01-01 00:00:00.000 UTC . . . 1961-01-01 00:00:00.000 UTC | 0.474 sec || 1.418 sec |- ! 8209 27FA 1819 (+1386.4 sec) <br /> 1961-01-01 00:00:01.418 TAI . . . 1961-01-01 00:00:01.423 TAI <br /> 1961-01-01 00:00:00.000 UTC | 0.005 sec || 1.423 sec |- ! 8209 27FA 1819 (+1386.4 sec) . . . 8209 27FA 2F85 (+406.6 sec) <br /> 1961-01-01 00:00:01.423 TAI . . . 1961-08-01 00:00:01.698 TAI <br /> 1961-01-01 00:00:00.000 UTC . . . 1961-08-01 00:00:00.000 UTC | 0.275 sec || 1.698 sec |- ! 8209 27FA 2F85 (+406.6 sec) <br /> 1961-08-01 00:00:01.698 TAI . . . 1961-08-01 00:00:01.648 TAI <br /> 1961-08-01 00:00:00.000 UTC | -0.050 sec || 1.648 sec |- ! 8209 27FA 2F85 (+406.6 sec) . . . 8209 27FA 406C (+621.8 sec) <br /> 1961-08-01 00:00:01.648 TAI . . . 1962-01-01 00:00:01.846 TAI <br /> 1961-08-01 00:00:00.000 UTC . . . 1962-01-01 00:00:00.000 UTC | 0.198 sec || 1.846 sec |- ! 8209 27FA 406C (+621.8 sec) . . . 8209 27FA 8A54 (+1622.5 sec) <br /> 1962-01-01 00:00:01.846 TAI . . . 1963-11-01 00:00:02.597 TAI <br /> 1962-01-01 00:00:00.000 UTC . . . 1963-11-01 00:00:00.000 UTC | 0.751 sec || 2.597 sec |- ! 8209 27FA 8A54 (+1622.6 sec) <br /> 1963-11-01 00:00:02.597 TAI . . . 1963-11-01 00:00:02.697 TAI <br /> 1963-11-01 00:00:00.000 UTC | 0.100 sec || 2.697 sec |- ! 8209 27FA 8A54 (+1622.6 sec) . . . 8209 27FA 9111 (+2147.7 sec) <br /> 1963-11-01 00:00:02.697 TAI . . . 1964-01-01 00:00:02.766 TAI <br /> 1963-11-01 00:00:00.000 UTC . . . 1964-01-01 00:00:00.000 UTC | 0.069 sec || 2.766 sec |- ! 8209 27FA 9111 (+2147.7 sec) . . . 8209 27FA 9B1F (+977.8 sec) <br /> 1964-01-01 00:00:02.766 TAI . . . 1964-04-01 00:00:02.884 TAI <br /> 1964-01-01 00:00:00.000 UTC . . . 1964-04-01 00:00:00.000 UTC | 0.118 sec || 2.884 sec |- ! 8209 27FA 9B1F (+977.9 sec) <br /> 1964-04-01 00:00:02.884 TAI . . . 1964-04-01 00:00:02.984 TAI <br /> 1964-04-01 00:00:00.000 UTC | 0.100 sec || 2.984 sec |- ! 8209 27FA 9B1F (+977.9 sec) . . . 8209 27FA AC06 (+1193.1 sec) <br /> 1964-04-01 00:00:02.984 TAI . . . 1964-09-01 00:00:03.182 TAI <br /> 1964-04-01 00:00:00.000 UTC . . . 1964-09-01 00:00:00.000 UTC | 0.198 sec || 3.182 sec |- ! 8209 27FA AC06 (+1193.2 sec) <br /> 1964-09-01 00:00:03.182 TAI . . . 1964-09-01 00:00:03.282 TAI <br /> 1964-09-01 00:00:00.000 UTC | 0.100 sec || 3.282 sec |- ! 8209 27FA AC06 (+1193.2 sec) . . . 8209 27FA B980 (+2243.4 sec) <br /> 1964-09-01 00:00:03.282 TAI . . . 1965-01-01 00:00:03.440 TAI <br /> 1964-09-01 00:00:00.000 UTC . . . 1965-01-01 00:00:00.000 UTC | 0.158 sec || 3.440 sec |- ! 8209 27FA B980 (+2243.5 sec) <br /> 1965-01-01 00:00:03.440 TAI . . . 1965-01-01 00:00:03.540 TAI <br /> 1965-01-01 00:00:00.000 UTC | 0.100 sec || 3.540 sec |- ! 8209 27FA B980 (+2243.5 sec) . . . 8209 27FA C005 (+1048.6 sec) <br /> 1965-01-01 00:00:03.540 TAI . . . 1965-03-01 00:00:03.617 TAI <br /> 1965-01-01 00:00:00.000 UTC . . . 1965-03-01 00:00:00.000 UTC | 0.076 sec || 3.617 sec |- ! 8209 27FA C005 (+1048.7 sec) <br /> 1965-03-01 00:00:03.617 TAI . . . 1965-03-01 00:00:03.717 TAI <br /> 1965-03-01 00:00:00.000 UTC | 0.100 sec || 3.717 sec |- ! 8209 27FA C005 (+1048.7 sec) . . . 8209 27FA CD7F (+2098.8 sec) <br /> 1965-03-01 00:00:03.717 TAI . . . 1965-07-01 00:00:03.875 TAI <br /> 1965-03-01 00:00:00.000 UTC . . . 1965-07-01 00:00:00.000 UTC | 0.158 sec || 3.875 sec |- ! 8209 27FA CD7F (+2098.9 sec) <br /> 1965-07-01 00:00:03.875 TAI . . . 1965-07-01 00:00:03.975 TAI <br /> 1965-07-01 00:00:00.000 UTC | 0.100 sec || 3.975 sec |- ! 8209 27FA CD7F (+2098.9 sec) . . . 8209 27FA D459 (+429.0 sec) <br /> 1965-07-01 00:00:03.975 TAI . . . 1965-09-01 00:00:04.055 TAI <br /> 1965-07-01 00:00:00.000 UTC . . . 1965-09-01 00:00:00.000 UTC | 0.080 sec || 4.055 sec |- ! 8209 27FA D459 (+429.1 sec) <br /> 1965-09-01 00:00:04.055 TAI . . . 1965-09-01 00:00:04.155 TAI <br /> 1965-09-01 00:00:00.000 UTC | 0.100 sec || 4.155 sec |- ! 8209 27FA D459 (+429.1 sec) . . . 8209 27FA E1D3 (+1479.3 sec) <br /> 1965-09-01 00:00:04.155 TAI . . . 1966-01-01 00:00:04.313 TAI <br /> 1965-09-01 00:00:00.000 UTC . . . 1966-01-01 00:00:00.000 UTC | 0.158 sec || 4.313 sec |- ! 8209 27FA E1D3 (+1479.3 sec) . . . 8209 27FB 35E5 (+2171.2 sec) <br /> 1966-01-01 00:00:04.313 TAI . . . 1968-02-01 00:00:06.286 TAI <br /> 1966-01-01 00:00:00.000 UTC . . . 1968-02-01 00:00:00.000 UTC | 1.973 sec || 6.286 sec |- ! 8209 27FB 35E5 (+2171.1 sec) <br /> 1968-02-01 00:00:06.286 TAI . . . 1968-02-01 00:00:06.186 TAI <br /> 1968-02-01 00:00:00.000 UTC | -0.100 sec || 6.186 sec |- ! 8209 27FB 35E5 (+2171.1 sec) . . . 8209 27FB D3E0 (+809.9 sec) <br /> 1968-02-01 00:00:06.186 TAI . . . 1972-01-01 00:00:09.891 TAI <br /> 1968-02-01 00:00:00.000 UTC . . . 1971-12-31 23:59:59.999 UTC | 3.707 sec || 9.892 sec |- ! 8209 27FB D3E0 (+809.9 sec) <br /> 1972-01-01 00:00:09.891 TAI . . . 1972-01-01 00:00:09.999 TAI <br /> 1971-12-31 23:59:59.999 UTC | 0.109 sec || 10.000 sec |} quc29lin8g2dyi4f7xumq2lsk5rp6f1 1 325527 2812120 2809175 2026-05-30T11:26:28Z Saltrabook 1417466 /* Purpose of Page */ Reply 2812120 wikitext text/x-wiki == Purpose of Page == Hi {{ping|Saltrabook}} what is the intent of this page? There's no link to this page either. Thanks. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 18:20, 14 November 2025 (UTC) :we are an international group of medical researchers in preventing prediabetes, diabetes mellitus and cardiovaskular heart- and diseases. The maritime health research has been closed and now we only work in the diabetes and the cadiovaskular area (Saltrabook greetings) [[User:Saltrabook|Saltrabook]] ([[User talk:Saltrabook|discusión]] • [[Special:Contributions/Saltrabook|contribs.]]) 13:50, 14 May 2026 (UTC) ::Reason for this : The Maritime Health research is not the main function any longer institute in Esbjerg, Denmark and the international research centre in Cardif, have been closed. The health research in the maritime specific areas is now done in other Public Health Institutes (Saltrabook) [[User:Saltrabook|Saltrabook]] ([[User talk:Saltrabook|discusión]] • [[Special:Contributions/Saltrabook|contribs.]]) 13:52, 14 May 2026 (UTC) :this is a teaching and research project with different target groups, [[User:Saltrabook|Saltrabook]] ([[User talk:Saltrabook|discusión]] • [[Special:Contributions/Saltrabook|contribs.]]) 11:26, 30 May 2026 (UTC) 8pmp9seomeeltkin305564905cfkj50 2 326765 2812017 2811946 2026-05-29T15:23:36Z Dc.samizdat 2856930 /* The 600-cell */ 2812017 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Note that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a 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 0.76537,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.84776,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes two completely orthogonal turns simultaneously, effectively <math>i</math> 90° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] great circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is <math>4 \pi</math>, and it occurs in either a left or right chiral form. We shall refer to the helical geodesic circle as an ''isocline'', and to the skew {8/3} octagram 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 central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. 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> It is the maximal regular construct of triangles and squares (with no pentagons). It is its own [[W:Dual polytope|dual polytope]]. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}</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> The 24-cell possesses only chords of these four lengths. Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord makes two simultaneous completely orthogonal 60° turns, effectively five zig-zag consecutive 60° turns. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering: 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> The <math>r_5</math> chord, equal to the radius, is an edge of an inscribed hexagon. When the 120 vertices of a 600-cell are assembled from 4 disjoint planar {30}-gons by skewing the {30}-gons, the rigid hexagon edges become edges of 24-cells and tesseracts inscribed in the 600-cell. The skewed {30}-gons assume these chord lengths: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> The 600-cell possesses only chords of these eight lengths. Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} hfddt99v4bjhds3ga1g0m1nrdvvwah2 2812018 2812017 2026-05-29T15:25:30Z Dc.samizdat 2856930 /* The 24-cell */ 2812018 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Note that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a 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 0.76537,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.84776,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes two completely orthogonal turns simultaneously, effectively <math>i</math> 90° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] great circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is <math>4 \pi</math>, and it occurs in either a left or right chiral form. We shall refer to the helical geodesic circle as an ''isocline'', and to the skew {8/3} octagram 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 central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. 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> It is the maximal regular construct of triangles and squares (with no pentagons). It is its own [[W:Dual polytope|dual polytope]]. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}</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> The 24-cell possesses only chords of these four lengths. Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes two completely orthogonal turns simultaneously, effectively <math>i</math> 60° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord makes two simultaneous completely orthogonal 60° turns, effectively five zig-zag consecutive 60° turns. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering: 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> The <math>r_5</math> chord, equal to the radius, is an edge of an inscribed hexagon. When the 120 vertices of a 600-cell are assembled from 4 disjoint planar {30}-gons by skewing the {30}-gons, the rigid hexagon edges become edges of 24-cells and tesseracts inscribed in the 600-cell. The skewed {30}-gons assume these chord lengths: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> The 600-cell possesses only chords of these eight lengths. Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} h0xj28eh3j1lruxrxvs82mhy96e7u61 2812019 2812018 2026-05-29T15:27:09Z Dc.samizdat 2856930 /* The 600-cell */ 2812019 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Note that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a 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 0.76537,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.84776,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes two completely orthogonal turns simultaneously, effectively <math>i</math> 90° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] great circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is <math>4 \pi</math>, and it occurs in either a left or right chiral form. We shall refer to the helical geodesic circle as an ''isocline'', and to the skew {8/3} octagram 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 central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. 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> It is the maximal regular construct of triangles and squares (with no pentagons). It is its own [[W:Dual polytope|dual polytope]]. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}</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> The 24-cell possesses only chords of these four lengths. Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes two completely orthogonal turns simultaneously, effectively <math>i</math> 60° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord makes two simultaneous completely orthogonal 60° turns, effectively five zig-zag consecutive 60° turns. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering: 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> The <math>r_5</math> chord, equal to the radius, is an edge of an inscribed hexagon. When the 120 vertices of a 600-cell are assembled from 4 disjoint planar {30}-gons by skewing the {30}-gons, the rigid hexagon edges become edges of 24-cells and tesseracts inscribed in the 600-cell. The skewed {30}-gons assume these chord lengths: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> The 600-cell possesses only chords of these eight lengths. Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns in a simple 2-dimensional rotation. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} qnmvenzsuqjc61n9f0rqmfdlngos2vv 2812020 2812019 2026-05-29T15:29:07Z Dc.samizdat 2856930 /* The 600-cell */ 2812020 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Note that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a 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 0.76537,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.84776,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes two completely orthogonal turns simultaneously, effectively <math>i</math> 90° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] great circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is <math>4 \pi</math>, and it occurs in either a left or right chiral form. We shall refer to the helical geodesic circle as an ''isocline'', and to the skew {8/3} octagram 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 central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. 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> It is the maximal regular construct of triangles and squares (with no pentagons). It is its own [[W:Dual polytope|dual polytope]]. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}</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 indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes two completely orthogonal turns simultaneously, effectively <math>i</math> 60° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord makes two simultaneous completely orthogonal 60° turns, effectively five zig-zag consecutive 60° turns. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering: 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> The <math>r_5</math> chord, equal to the radius, is an edge of an inscribed hexagon. When the 120 vertices of a 600-cell are assembled from 4 disjoint planar {30}-gons by skewing the {30}-gons, the rigid hexagon edges become edges of 24-cells and tesseracts inscribed in the 600-cell. The skewed {30}-gons assume these chord lengths: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns in a simple 2-dimensional rotation. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} 3jd7sf4xdielyfwqc26yqsz5b6gmo8g 2812063 2812020 2026-05-29T21:13:45Z Dc.samizdat 2856930 /* The 600-cell */ 2812063 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.84776,r_3=1+\sqrt{2} \approx 2.41421,r_4=\sqrt{4 + \sqrt{8}} \approx 2.61313</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.41421</math> Note that <math>1/r_3=\sqrt{2}-1=r_3-2</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a 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 0.76537,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.84776,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes two completely orthogonal turns simultaneously, effectively <math>i</math> 90° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] great circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is <math>4 \pi</math>, and it occurs in either a left or right chiral form. We shall refer to the helical geodesic circle as an ''isocline'', and to the skew {8/3} octagram 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 central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. 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> It is the maximal regular construct of triangles and squares (with no pentagons). It is its own [[W:Dual polytope|dual polytope]]. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}</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 indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes two completely orthogonal turns simultaneously, effectively <math>i</math> 60° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord makes two simultaneous completely orthogonal 60° turns, effectively five zig-zag consecutive 60° turns. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering: 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> The <math>r_5</math> chord, equal to the radius, is an edge of an inscribed hexagon. When the 120 vertices of a 600-cell are assembled from 4 disjoint planar {30}-gons by skewing the {30}-gons, the rigid hexagon edges become edges of 24-cells and tesseracts inscribed in the 600-cell. The skewed {30}-gons assume these chord lengths: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns in a simple 2-dimensional rotation. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} 7mcuipdq5y824lkzezda3kh7ztf4cky 2812074 2812063 2026-05-29T22:53:02Z Dc.samizdat 2856930 /* The 600-cell */ 2812074 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=1+\sqrt{2} \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes two completely orthogonal turns simultaneously, effectively <math>i</math> 90° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] great circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is <math>4 \pi</math>, and it occurs in either a left or right chiral form. We shall refer to such a helical geodesic circle 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 central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. 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> It is the maximal regular construct of triangles and squares (with no pentagons). It is its own [[W:Dual polytope|dual polytope]]. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}</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 indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord makes two simultaneous completely orthogonal 60° turns, effectively five zig-zag consecutive 60° turns. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> The <math>r_5</math> chord, equal to the radius, is an edge of an inscribed hexagon. When the 120 vertices of a 600-cell are assembled from 4 disjoint planar {30}-gons by skewing the {30}-gons, the rigid hexagon edges become edges of 24-cells and tesseracts inscribed in the 600-cell. The skewed {30}-gons assume these chord lengths: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns in a simple 2-dimensional rotation. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} 935t4b7jpv27mxhts2n7ep9jqfj5gha 2812075 2812074 2026-05-29T22:59:36Z Dc.samizdat 2856930 /* Hypercubes */ 2812075 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=1+\sqrt{2} \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes two completely orthogonal turns simultaneously, effectively <math>i</math> 90° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] great circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is <math>4 \pi</math>, and it occurs in either a left or right chiral form. We shall refer to such a helical geodesic circle 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 central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint {8/3} octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. 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> It is the maximal regular construct of triangles and squares (with no pentagons). It is its own [[W:Dual polytope|dual polytope]]. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}</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 indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord makes two simultaneous completely orthogonal 60° turns, effectively five zig-zag consecutive 60° turns. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> The <math>r_5</math> chord, equal to the radius, is an edge of an inscribed hexagon. When the 120 vertices of a 600-cell are assembled from 4 disjoint planar {30}-gons by skewing the {30}-gons, the rigid hexagon edges become edges of 24-cells and tesseracts inscribed in the 600-cell. The skewed {30}-gons assume these chord lengths: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns in a simple 2-dimensional rotation. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} h2cjipnw0y9vnlyct053dfay3lubeb6 2812076 2812075 2026-05-29T23:10:00Z Dc.samizdat 2856930 /* The 24-cell */ 2812076 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=1+\sqrt{2} \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes two completely orthogonal turns simultaneously, effectively <math>i</math> 90° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] great circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is <math>4 \pi</math>, and it occurs in either a left or right chiral form. We shall refer to such a helical geodesic circle 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 central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint {8/3} octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is its own [[W:Dual polytope|dual polytope]]. 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> It is the maximal regular construct of triangles and squares (with no pentagons). [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}</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 indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord makes two simultaneous completely orthogonal 60° turns, effectively five zig-zag consecutive 60° turns. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> The <math>r_5</math> chord, equal to the radius, is an edge of an inscribed hexagon. When the 120 vertices of a 600-cell are assembled from 4 disjoint planar {30}-gons by skewing the {30}-gons, the rigid hexagon edges become edges of 24-cells and tesseracts inscribed in the 600-cell. The skewed {30}-gons assume these chord lengths: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns in a simple 2-dimensional rotation. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} 8j5hspmm4rwe2e5wdu3hg1s433b0ul8 2812077 2812076 2026-05-29T23:11:55Z Dc.samizdat 2856930 Undid revision [[Special:Diff/2812076|2812076]] by [[Special:Contributions/Dc.samizdat|Dc.samizdat]] ([[User talk:Dc.samizdat|talk]]) 2812077 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=1+\sqrt{2} \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes two completely orthogonal turns simultaneously, effectively <math>i</math> 90° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] great circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is <math>4 \pi</math>, and it occurs in either a left or right chiral form. We shall refer to such a helical geodesic circle 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 central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint {8/3} octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. 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> It is the maximal regular construct of triangles and squares (with no pentagons). It is its own [[W:Dual polytope|dual polytope]]. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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}},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}},r_6=\sqrt{4}</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 indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord makes two simultaneous completely orthogonal 60° turns, effectively five zig-zag consecutive 60° turns. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> The <math>r_5</math> chord, equal to the radius, is an edge of an inscribed hexagon. When the 120 vertices of a 600-cell are assembled from 4 disjoint planar {30}-gons by skewing the {30}-gons, the rigid hexagon edges become edges of 24-cells and tesseracts inscribed in the 600-cell. The skewed {30}-gons assume these chord lengths: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns in a simple 2-dimensional rotation. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} h2cjipnw0y9vnlyct053dfay3lubeb6 2812079 2812077 2026-05-29T23:19:48Z Dc.samizdat 2856930 /* The 24-cell */ 2812079 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=1+\sqrt{2} \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes two completely orthogonal turns simultaneously, effectively <math>i</math> 90° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] great circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is <math>4 \pi</math>, and it occurs in either a left or right chiral form. We shall refer to such a helical geodesic circle 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 central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint {8/3} octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is its own [[W:Dual polytope|dual polytope]]. 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> It is the maximal regular construct of triangles and squares (with no pentagons). [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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> 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 indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord makes two simultaneous completely orthogonal 60° turns, effectively five zig-zag consecutive 60° turns. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> The <math>r_5</math> chord, equal to the radius, is an edge of an inscribed hexagon. When the 120 vertices of a 600-cell are assembled from 4 disjoint planar {30}-gons by skewing the {30}-gons, the rigid hexagon edges become edges of 24-cells and tesseracts inscribed in the 600-cell. The skewed {30}-gons assume these chord lengths: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns in a simple 2-dimensional rotation. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} mzb5fyiv422kbxkd6wjseptszhhmwfu 2812080 2812079 2026-05-29T23:27:15Z Dc.samizdat 2856930 /* The 8-point regular polytopes */ 2812080 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=1+\sqrt{2} \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes two completely orthogonal turns simultaneously, effectively <math>i</math> 90° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal invariant <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] great circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is <math>4 \pi</math>, and it occurs in either a left or right chiral form. We shall refer to such a helical geodesic circle 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 central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint {8/3} octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is its own [[W:Dual polytope|dual polytope]]. 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> It is the maximal regular construct of triangles and squares (with no pentagons). [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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> 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 indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord makes two simultaneous completely orthogonal 60° turns, effectively five zig-zag consecutive 60° turns. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> The <math>r_5</math> chord, equal to the radius, is an edge of an inscribed hexagon. When the 120 vertices of a 600-cell are assembled from 4 disjoint planar {30}-gons by skewing the {30}-gons, the rigid hexagon edges become edges of 24-cells and tesseracts inscribed in the 600-cell. The skewed {30}-gons assume these chord lengths: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns in a simple 2-dimensional rotation. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} ejy0asn51cwtm0blr0xj1ax30x9k3f8 2812081 2812080 2026-05-29T23:35:47Z Dc.samizdat 2856930 /* The 8-point regular polytopes */ 2812081 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=1+\sqrt{2} \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes two completely orthogonal turns simultaneously, effectively <math>i</math> 90° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal invariant <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] great circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is not <math>2\pi</math> (it is <math>4\pi</math> in this case), and it occurs in either a left or right chiral form. We shall refer to such a helical geodesic circle 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 central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint {8/3} octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is its own [[W:Dual polytope|dual polytope]]. 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> It is the maximal regular construct of triangles and squares (with no pentagons). [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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> 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 indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord makes two simultaneous completely orthogonal 60° turns, effectively five zig-zag consecutive 60° turns. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> The <math>r_5</math> chord, equal to the radius, is an edge of an inscribed hexagon. When the 120 vertices of a 600-cell are assembled from 4 disjoint planar {30}-gons by skewing the {30}-gons, the rigid hexagon edges become edges of 24-cells and tesseracts inscribed in the 600-cell. The skewed {30}-gons assume these chord lengths: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns in a simple 2-dimensional rotation. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} 1ggfgsyweb49nkcxltsais83vn8iwjp 2812085 2812081 2026-05-29T23:53:23Z Dc.samizdat 2856930 /* The 600-cell */ 2812085 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=1+\sqrt{2} \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes two completely orthogonal turns simultaneously, effectively <math>i</math> 90° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal invariant <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] great circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is not <math>2\pi</math> (it is <math>4\pi</math> in this case), and it occurs in either a left or right chiral form. We shall refer to such a helical geodesic circle 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 central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint {8/3} octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is its own [[W:Dual polytope|dual polytope]]. 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> It is the maximal regular construct of triangles and squares (with no pentagons). [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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> 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 indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord makes two simultaneous completely orthogonal 60° turns, effectively five zig-zag consecutive 60° turns. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> The <math>r_5</math> chord, equal to the radius, is an edge of an inscribed hexagon. When the 120 vertices of a 600-cell are assembled from 4 disjoint planar {30}-gons by skewing the {30}-gons, the rigid hexagon edges become edges of 24-cells and tesseracts inscribed in the 600-cell. The skewed {30}-gons assume these chord lengths: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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{4\pi}{5}/2) \approx 1.902</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns in a simple 2-dimensional rotation. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} 024z15g29v67l4aimfmh52ap7cg5enc 2812087 2812085 2026-05-30T00:01:53Z Dc.samizdat 2856930 /* The 600-cell */ 2812087 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=1+\sqrt{2} \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes two completely orthogonal turns simultaneously, effectively <math>i</math> 90° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal invariant <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] great circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is not <math>2\pi</math> (it is <math>4\pi</math> in this case), and it occurs in either a left or right chiral form. We shall refer to such a helical geodesic circle 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 central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint {8/3} octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is its own [[W:Dual polytope|dual polytope]]. 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> It is the maximal regular construct of triangles and squares (with no pentagons). [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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> 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 indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord makes two simultaneous completely orthogonal 60° turns, effectively five zig-zag consecutive 60° turns. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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) \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{4\pi}{5}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> The <math>r_5</math> chord, equal to the radius, is an edge of an inscribed hexagon. When the 120 vertices of a 600-cell are assembled from 4 disjoint planar {30}-gons by skewing the {30}-gons, the rigid hexagon edges become edges of 24-cells and tesseracts inscribed in the 600-cell. The skewed {30}-gons assume these chord lengths: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2) \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{4\pi}{5}/2) \approx 1.902</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns in a simple 2-dimensional rotation. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} 18vo34xvq93tbxml9hsmw6jhcpi23iz 2812090 2812087 2026-05-30T01:14:09Z Dc.samizdat 2856930 /* The 600-cell */ 2812090 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=1+\sqrt{2} \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes two completely orthogonal turns simultaneously, effectively <math>i</math> 90° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal invariant <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] great circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is not <math>2\pi</math> (it is <math>4\pi</math> in this case), and it occurs in either a left or right chiral form. We shall refer to such a helical geodesic circle 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 central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint {8/3} octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is its own [[W:Dual polytope|dual polytope]]. 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> It is the maximal regular construct of triangles and squares (with no pentagons). [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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> 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 indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord makes two simultaneous completely orthogonal 60° turns, effectively five zig-zag consecutive 60° turns. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol {3,3,5}. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Note the <math>r_5</math> chord, equal to the radius, the edge of an inscribed hexagon. The 600-cell may be constructed from 4 disjoint planar {30}-gons with rigid <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>r_9</math>, <math>r_{10}</math>, and <math>r_{12}</math> chords, but no rigid <math>r_1</math> edges, by skewing the {30}-gons. The <math>r_3</math> chords become 600-cell edges, <math>r_5</math> chords become edges of 24-cells and tesseracts, and new <math>r_7</math> and <math>r_8</math> chords of length <math>\sqrt{2}</math> emerge to become edges of 16-cells. The skewed {30}-gons assume these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</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{4\pi}{5}/2) \approx 1.902</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) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} 8go18eapaoclwf3ip5mywsun0coesg8 2812091 2812090 2026-05-30T01:39:02Z Dc.samizdat 2856930 /* The 600-cell */ 2812091 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=1+\sqrt{2} \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes two completely orthogonal turns simultaneously, effectively <math>i</math> 90° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal invariant <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] great circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is not <math>2\pi</math> (it is <math>4\pi</math> in this case), and it occurs in either a left or right chiral form. We shall refer to such a helical geodesic circle 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 central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint {8/3} octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is its own [[W:Dual polytope|dual polytope]]. 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> It is the maximal regular construct of triangles and squares (with no pentagons). [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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> 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 indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord makes two simultaneous completely orthogonal 60° turns, effectively five zig-zag consecutive 60° turns. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol {3,3,5}. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, and <math>r_{15}</math> chord lengths, and one other chord length <math>\sqrt{2}</math>, occur in the 600-cell. The 600-cell may be constructed by skewing 4 disjoint planar {30}-gons with these rigid chords but without rigid <math>r_1</math> edges. The skewed {30}-gons assume these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</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{4\pi}{5}/2) \approx 1.902</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) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} mtf7nw6lebi20m56e51mfilx2hjsokr 2812092 2812091 2026-05-30T01:41:35Z Dc.samizdat 2856930 /* The 600-cell */ 2812092 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=1+\sqrt{2} \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns in a simple 2-dimensional rotation. In an isoclinic 4-dimensional rotation it makes two completely orthogonal turns simultaneously, effectively <math>i</math> 90° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal invariant <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] great circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is not <math>2\pi</math> (it is <math>4\pi</math> in this case), and it occurs in either a left or right chiral form. We shall refer to such a helical geodesic circle 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 central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint {8/3} octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is its own [[W:Dual polytope|dual polytope]]. 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> It is the maximal regular construct of triangles and squares (with no pentagons). [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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> 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 indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord makes two simultaneous completely orthogonal 60° turns, effectively five zig-zag consecutive 60° turns. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol {3,3,5}. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, and <math>r_{15}</math> chord lengths, and one other chord length <math>\sqrt{2}</math>, occur in the 600-cell. A 600-cell may be constructed by skewing 4 disjoint planar {30}-gons with these rigid chords but without rigid <math>r_1</math> edges. The skewed {30}-gons assume these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</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{4\pi}{5}/2) \approx 1.902</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) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} kfbdtnjwcu39ow00p0hae1avjm9ynlt 2812094 2812092 2026-05-30T01:58:09Z Dc.samizdat 2856930 /* The 8-point regular polytopes */ 2812094 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=1+\sqrt{2} \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in ''both'' of the two completely orthogonal invariant <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is not <math>2\pi</math> (it is <math>4\pi</math> in this case), and it occurs in either a left or right chiral form. We shall refer to such a helical geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint {8/3} octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is its own [[W:Dual polytope|dual polytope]]. 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> It is the maximal regular construct of triangles and squares (with no pentagons). [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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> 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 indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of green <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord makes two simultaneous completely orthogonal 60° turns, effectively five zig-zag consecutive 60° turns. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol {3,3,5}. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, and <math>r_{15}</math> chord lengths, and one other chord length <math>\sqrt{2}</math>, occur in the 600-cell. A 600-cell may be constructed by skewing 4 disjoint planar {30}-gons with these rigid chords, but without rigid <math>r_1</math> edges. The skewed {30}-gons assume these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</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{4\pi}{5}/2) \approx 1.902</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) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} kev0vql6lalc1s2bpi471b2ue1fihlu 2812106 2812094 2026-05-30T04:40:21Z Dc.samizdat 2856930 /* The 8-point regular polytopes */ 2812106 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=1+\sqrt{2} \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is not <math>2\pi</math> (it is <math>4\pi</math> in this case), and it occurs in either a left or right chiral form. We shall refer to such a helical geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint {8/3} octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is its own [[W:Dual polytope|dual polytope]]. 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> It is the maximal regular construct of triangles and squares (with no pentagons). [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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> 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 indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord turns 150°. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol {3,3,5}. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, and <math>r_{15}</math> chord lengths, and one other chord length <math>\sqrt{2}</math>, occur in the 600-cell. A 600-cell may be constructed by skewing 4 disjoint planar {30}-gons with these rigid chords, but without rigid <math>r_1</math> edges. The skewed {30}-gons assume these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</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{4\pi}{5}/2) \approx 1.902</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) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} hs9fdyhn8dm62p930api63l0h0bwdpy 2812108 2812106 2026-05-30T04:48:40Z Dc.samizdat 2856930 /* The 600-cell */ 2812108 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=1+\sqrt{2} \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is not <math>2\pi</math> (it is <math>4\pi</math> in this case), and it occurs in either a left or right chiral form. We shall refer to such a helical geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint {8/3} octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is its own [[W:Dual polytope|dual polytope]]. 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> It is the maximal regular construct of triangles and squares (with no pentagons). [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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> 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 indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord turns 150°. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol {3,3,5}. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, and <math>r_{15}</math> chord lengths, and one other chord length <math>\sqrt{2}</math>, occur in the 600-cell. The <math>r_5</math> chord, equal to the radius, is the edge of the inscribed hexagons. A 600-cell may be constructed by skewing 4 disjoint planar {30}-gons with rigid <math>r_5</math> chords, but without rigid <math>r_1</math> edges. The skewed {30}-gons have these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</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{4\pi}{5}/2) \approx 1.902</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) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} d0e5ojpahluk0p1vczpx11q66c928rt 2812109 2812108 2026-05-30T04:55:10Z Dc.samizdat 2856930 /* The 600-cell */ 2812109 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=1+\sqrt{2} \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is not <math>2\pi</math> (it is <math>4\pi</math> in this case), and it occurs in either a left or right chiral form. We shall refer to such a helical geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint {8/3} octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is its own [[W:Dual polytope|dual polytope]]. 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> It is the maximal regular construct of triangles and squares (with no pentagons). [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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> 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 indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord turns 150°. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol {3,3,5}. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, and <math>r_{15}</math> chord lengths, and one other chord length <math>\sqrt{2}</math>, occur in the 600-cell. The <math>r_5</math> chord, equal to the radius, is the edge of the inscribed hexagons. A 600-cell may be constructed by skewing 4 disjoint planar {30}-gons that have rigid <math>r_5</math> chords, but no <math>r_1</math> edges. The skewed {30}-gons have these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</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{4\pi}{5}/2) \approx 1.902</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) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} h6ecx86imey3e7h9kc9ejb1hp29qn4n 2812110 2812109 2026-05-30T05:04:45Z Dc.samizdat 2856930 /* The 600-cell */ 2812110 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=1+\sqrt{2} \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is not <math>2\pi</math> (it is <math>4\pi</math> in this case), and it occurs in either a left or right chiral form. We shall refer to such a helical geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint {8/3} octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is its own [[W:Dual polytope|dual polytope]]. 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> It is the maximal regular construct of triangles and squares (with no pentagons). [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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> 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 indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord turns 150°. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol {3,3,5}. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, and <math>r_{15}</math> chord lengths, and one other chord length <math>\sqrt{2}</math>, occur in the 600-cell. The <math>r_5</math> chord, equal to the radius, is the edge of the inscribed hexagons. A 600-cell may be constructed by skewing 4 disjoint planar {30}-gons that have rigid <math>r_3</math> chords (but no <math>r_1</math> edges) so that the <math>r_3</math> chords become 600-cell edges. The skewed {30}-gons have these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</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{4\pi}{5}/2) \approx 1.902</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) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} p0i81v8uuv9ugbrg4s2uccdrh8vwidz 2812111 2812110 2026-05-30T05:08:09Z Dc.samizdat 2856930 /* The 600-cell */ 2812111 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=1+\sqrt{2} \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is not <math>2\pi</math> (it is <math>4\pi</math> in this case), and it occurs in either a left or right chiral form. We shall refer to such a helical geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint {8/3} octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is its own [[W:Dual polytope|dual polytope]]. 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> It is the maximal regular construct of triangles and squares (with no pentagons). [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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> 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 indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord turns 150°. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol {3,3,5}. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, and <math>r_{15}</math> chord lengths, and one other chord length <math>\sqrt{2}</math>, occur in the 600-cell. A 600-cell may be constructed by skewing 4 disjoint planar {30}-gons that have rigid <math>r_3</math> chords (but no <math>r_1</math> edges) so that the <math>r_3</math> chords become 600-cell edges. The skewed {30}-gons have these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</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{4\pi}{5}/2) \approx 1.902</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) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} k40jracgajtah3zj4jb3onki6sc8w3r 2812112 2812111 2026-05-30T05:11:17Z Dc.samizdat 2856930 /* The 600-cell */ 2812112 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 - May 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} ... The list of 30 chords can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973}} where Coxeter identified each row with a distinct polyhedral section of the 120-cell beginning with a vertex. The pairs of 180° complement chords radiate from each 120-cell vertex. In the curved 3-dimensional space <math>\mathbb{S}^3</math>, every vertex is the center of a set of concentric polyhedra of increasing radii that nest like Russian dolls. The smallest polyhedral section of radius <math>r_1</math> is a dodecahedron cell, and the largest, central section of radius <math>r_{30}</math> is a non-uniform rhombicosidodecahedron. ... == The 8-point regular polytopes == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=1+\sqrt{2} \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=1+\sqrt{2}</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. If we embed this planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, 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 two disjoint squares lie in completely orthogonal central planes.]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {3,3,4}. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in ''opposite'' planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are indistinguishable except in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a double rotation in pairs of completely orthogonal planes. 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 invariant planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel ''and'' perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by the same angle, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The circular helix is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its circumference is not <math>2\pi</math> (it is <math>4\pi</math> in this case), and it occurs in either a left or right chiral form. We shall refer to such a helical geodesic orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the central planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once on the same circular helix geodesic isocline, 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. After 360° of rotation each vertex 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 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> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. == Hypercubes == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] {4,3,3}. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The skew octagon geodesic orbits of the 16 vertices lie on two disjoint {8/3} octagram circular helix isoclines of the same chirality. Two [[w:Clifford_parallel|Clifford parallel]] skew octagon geodesic orbits over <math>\sqrt{2}</math> chords form a circular double helix. 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 completely orthogonal 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.|alt=]] 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 4 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol {3,4,3}. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is its own [[W:Dual polytope|dual polytope]]. 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> It is the maximal regular construct of triangles and squares (with no pentagons). [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations.]] The 24-cell 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 cube 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> 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 indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 30° turns. We can rotate the 24-cell isoclinically the way we rotated the 16-cell, by 90° in two completely orthogonal invariant square central planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. Three Clifford parallel skew octagon geodesic orbits of circumference <math>4\pi</math> over <math>\sqrt{2}</math> chords form a circular triple helix. [[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 3D surface made of 24 octahedra is visible.]] We can also rotate the 24-cell isoclinically by 60° in a hexagonal invariant central plane and its completely orthogonal invariant central plane. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The Clifford polygon of the hexagonal rotation is a skew {12/5} dodecagram of <math>r_5</math> chords, visible in the orthogonal projection. The rotational curve over each <math>r_5</math> chord turns 150°. Two Clifford parallel skew dodecagon geodesic orbits of circumference <math>8\pi</math> over <math>\sqrt{3}</math> chords form a circular double helix. In the 24-cell an isoclinic rotation by 60° in any hexagonal invariant central plane and its completely orthogonal invariant central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in a different 16-cell. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> chords within one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol {3,3,5}. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices between the 24-cell's existing 24 vertices, in effect adding twenty-four more 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3D surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular triacontagon {30}. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> A 600-cell may be constructed by skewing 4 disjoint planar {30}-gons that have rigid <math>r_3</math> chords (but no <math>r_1</math> edges) so that the <math>r_3</math> chords become 600-cell edges. Only the <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, and <math>r_{15}</math> chord lengths, and one other chord length <math>\sqrt{2}</math>, occur in the 600-cell. The skewed {30}-gons have these chords: :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{5}/2)=1/\phi \approx 0.618</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{4\pi}{5}/2) \approx 1.902</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) \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2) \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are indistinguishable except in the context of a rotation. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 12° turns. ... == Finally the 120-cell == ... == Conclusions == Fontaine and Hurley's discovery is more than a formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the distinct isoclinic rotation characteristic of a ''d''-dimensional polytope. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in higher-dimensional spaces demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden polygon chord sequences, to chord sequences in isoclinic rotation helixes, 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}} ntlm0bns9c0gl9av2hrfeqb7zaq90h3 0 327557 2811987 2801770 2026-05-29T14:30:12Z DavidMCEddy 218607 move Fig. 1 to the top 2811987 wikitext text/x-wiki [[File:MaddisonDataLeaders1349-2022.svg|thumb|''Figure 1. World leaders in GDP per capita 1349-2022 (NLD, GBR, USA, SGP).<ref>See Bolt and van Zanden (2024) for the Maddison Data generally, van Zanden and van Leeuwen (2012) for the data on Holland 1348–1807, Smits et al. (2000) for the data on the Netherlands 1808-1913, Broadberry et al. (2015) for the data on England 1252–1700 and on Great Britain until 1870, and Sugimoto (2011) for Singapore to 2007.</ref>'']] :''I am entitled to my [[Wiktionary:cockamamie|cockamamie]] ideas, and you are entitled to yours.''<ref>Humor is important but must be offered in a way that does not offend others. If others are offended, they may be less interested in dialogue. The term "cockamamie" is used here, hoping that this style of [[w:Self-deprecation|self-deprecation]] might be more inviting for dialogue.</ref> :This book invites you to improve your understanding of the role of the media in society and help you be more effective in talking with others and building consensus for action on the issues that most concern you. You are also invited to help improve this chapter and book. ==Introduction== Acemoglu and Johnson (2023) suggest that the stability of poverty and the [[w:Malthusianism|Malthusian trap]] in hierarchical societies ''prior'' to the Industrial Revolution was enforced by "societies of orders" consisting of those who (1) fought, (2) prayed, and (3) worked. Those who prayed convinced those who worked to live in poverty while giving increasing shares of what they produced so those who prayed and fought could live in leisure and opulence. This seems to describe the construction of the pyramids in Egypt and the cathedrals, castles and manor homes that dot Europe today.<ref>Acemoglu and Johnson (2023, esp. ch. 4).</ref> During the reign of [[w:James VI and I|King James of the King James bible]] pamphlets and newspapers began to compete with the church for helping peasants understand their role in society. That contributed to the [[w:English Civil War|English Civil War]] (1642-1651) during which James' son, [[w: Charles I of England|Charles I]], was decapitated for abuse of power. The new government allowed enough commoners to become entrepreneurs that it increased the rate of economic growth visible in Figure 1. The number of independent media organizations per million population continued to grow, especially in the US where they were subsidized by the US [[w:Postal Service Act|Postal Service Act]] of 1792, [[Media concentration per Columbia History Professor Richard John|until the mid nineteenth century]], when high-speed rotary presses substantially reduced the per copy cost of printing while increasing the cost of starting a newspaper.<ref>John (1995); John and Silberstein-Loeb (2015).</ref> That contributed to the expansion of voting rights, in the US officially to all adults by 1920. During the [[w:Great Depression|Great Depression]] with over 20 percent of the US workforce unemployed, conservative arguments that blamed the poor for their poverty did not sell newspapers. That helped US President [[w:Franklin D. Roosevelt|Franklin Roosevelt]] get the political support needed for his [[w:New Deal|New Deal]] program and [[The Media, the Great Depression, and our future|wage and price controls that enabled unprecedented economic growth]] during [[w:World War II|World War II]] by dramatically limiting price gouging that had generated inflation and stifled economic growth during previous major wars in US history. However, the consolidation of ownership of the major media since World War II limited the increases in inequality and then reversed them starting around the time that [[w:Ronald Reagan|Ronald Reagan]] became president of the US in 1981.<ref>Acemoglu and Johnson (2023).</ref> During the [[w:Great Recession|Great Recession]] (2007-2009) Fox featured "experts" who insisted that Franklin Roosevelt's New Deal made the Great Depression ''worse'', not better. That editorial distortion helped prevent the US Congress from protecting the victims of [[w:Stated income loan|liar loans]]. But the money had to be injected back into the economy, so the US Congress funded bonuses of over a million dollars each to over 5,000 finance industry leaders, some of whom should have been in prison, according to Acemoglu and Johnson (2023, ch. 3), who shared the 2024 Nobel Memorial Prize in Economics with [[w:James A. Robinson|James A. Robinson]]. You, dear reader, are invited to offer contrary evidence or questions regarding the evidence presented here. This chapter identifies the leading countries in [[w:List of countries by GDP (PPP) per capita|GDP per capita for each year]] in the [[w:Maddison Project|Maddison Historical Statistics Project]], which collates [[w:Gross domestic product|GDP]] per capita and population estimates for years 1 [[w:Common Era|CE]] to the present from all credible sources the project team has found. The MaddisonData package for R makes these data available as an R data object with companion functions to facilitate computing the leaders for each year with optional exclusions to facilitate identifying the technology leaders and with a function ggplotPath to make it relatively easy to plot and annotate the image as desired.<ref>Graves (2025).</ref> The analysis then narrows the focus to apparent technology leaders since 1349, when the data identify Holland as the leader. This analysis identifies 11 different countries with the highest GDP per capita for at least one year. However, only 5 lead for more than 10 years: Holland (NLD), England / Great Britain / the United Kingdom (GBR), Australia (AUS), the United States (USA), and Singapore (SGP).<ref>The population of [[w:Singapore|Singapore]] is roughly 6 million. If you think that's too small to be included in this analysis, you can either ignore it or, better, repeat this analysis without it. A tutorial on how to do that using free and open-source software is available in [[Most productive countries since 1349#Appendix. Companion R Markdown vignette|an R Markdown vignette]] supplied as an appendix to the Wikiversity article on "[[Most productive countries since 1349]]".</ref> Australia led for only 17 of the 39 years between 1853 and 1891 as the US was overtaking the UK as the technology leader. A plot of the leaders is then made without Australia, because its inclusion would seem to add more complexity than clarity to the message. That display is a [[w:Semi-log|semilog]] plot, because it makes a constant percentage increase look like a straight line. The resulting plot of GDP per capita suggests the Industrial Revolution began in England around 1649 when King Charles I was beheaded. == World leaders in GDP per capita == Table 1 lists all the countries with the highest GDP per capita for at least one year in MaddisonData. {| class="wikitable" |+ ''Table 1. all the countries with the highest GDP per capita for at least one year in the MaddisonData.'' |- ! rowspan=2 | country !! rowspan=2 | ISO !! colspan=2 | year !! rowspan=2 | n years !! rowspan=2 | percent of years |- ! begin !! end |- | [[w:Ancient Rome|Italy (ancient Rome)]] | ITA || style="text-align:right | 1 || 1501 || style="text-align:right | 3 || style="text-align:right | 0.2% |- | [[w:Iraq|Iraq]] || IRQ || style="text-align:right | 730 || 1000 || style="text-align:right | 271 || style="text-align:right | 100% |- | [[w:China|China]] || CHN || style="text-align:right | 1090 || 1150 || style="text-align:right | 61 || style="text-align:right | 100% |- | [[w:England|England]] / [[w:Great Britain|Great Britain]] / [[w:United Kingdom|United Kingdom]] || GBR || style="text-align:right | 1252 || 1898 || style="text-align:right | 91 || style="text-align:right | 14% |- | [[w:France|France]] || FRA || style="text-align:right | 1276 || 1374 || style="text-align:right | 19 || style="text-align:right | 19% |- | [[w:Spain|Spain]] || ESP || style="text-align:right | 1278 || 1348 || style="text-align:right | 50 || style="text-align:right | 70% |- | [[w:Sweden|Sweden]] || SWE || style="text-align:right | 1304 || 1509 || style="text-align:right | 13 || style="text-align:right | 6% |- | [[w:Holland|Holland]] / [[w:Netherlands|Netherlands]] || NLD || style="text-align:right | 1349 || 1807 || style="text-align:right | 447 || style="text-align:right | 97% |- | [[w:Belgium|Belgium]] || BEL || style="text-align:right | 1500 || 1500 || style="text-align:right | 1 || style="text-align:right | 100% |- | [[w:Australia|Australia]] || AUS || style="text-align:right | 1853 || 1891 || style="text-align:right | 17 || style="text-align:right | 44% |- | [[w:New Zealand|New Zealand]] || NZL || style="text-align:right | 1873 || 1874 || style="text-align:right | 2 || style="text-align:right | 100% |- | [[w:United States|United States]] || USA || style="text-align:right | 1882 || 1990 || style="text-align:right | 58 || style="text-align:right | 53% |- | [[w:Switzerland|Switzerland]] || CHE || style="text-align:right | 1931 || 1934 || style="text-align:right | 4 || style="text-align:right | 100% |- | [[w:Qatar|Qatar]] || QAT || style="text-align:right | 1950 || 2022 || style="text-align:right | 45 || style="text-align:right | 62% |- | [[w:Kuwait|Kuwait]] || KWT || style="text-align:right | 1953 || 1957 || style="text-align:right | 5 || style="text-align:right | 100% |- |[[w:United Arab Emirates|United Arab Emirates]] || ARE || style="text-align:right | 1965 || 1984 || style="text-align:right | 5 || style="text-align:right | 25% |- | [[w:Luxembourg|Luxembourg]] || LUX || style="text-align:right | 1991 || 1995 || style="text-align:right | 5 || style="text-align:right | 100% |- | [[w:Norway|Norway]] || NOR || style="text-align:right | 1996 || 2002 || style="text-align:right | 7 || style="text-align:right | 100% |} For simplicity, we focus on the data since 1349, when Holland had the highest GDP per capita, omitting Qatar (QAT), Kuwait (KWT), United Arab Emirates (ARE), and Norway (NOR), whose wealth may be more due to petroleum than to broad technology leadership. We also delete Luxenbourg (LUX), whose population is under a million and therefore may be too small to use for general conclusions about technology leadership. Table 2 is similar to Table 1 with these adjustments. {| class="wikitable" |+ ''Table 2. Highest GDP per capita among broad-based economies since 1349.'' |- ! rowspan=2 | country !! rowspan=2 | ISO !! colspan=2 | year !! rowspan=2 | n years !! rowspan=2 | percent of years |- ! begin !! end |- | [[w:Holland|Holland]] / [[w:Netherlands|Netherlands]] || NLD || 1349 || 1807 || style="text-align:right | 447 || style="text-align:right | 97% |- | [[w:France|France]] || FRA || 1357 || 1374 || style="text-align:right | 7 || style="text-align:right | 39% |- | [[w:Italy|Italy]] || ITA || 1451 || 1501 || style="text-align:right | 2 || style="text-align:right | 4% |- | [[w:Sweden|Sweden]] || SWE || 1468 || 1509 || style="text-align:right | 2 || style="text-align:right | 5% |- | [[w:Belgium|Belgium]] || BEL || 1500 || 1500 || style="text-align:right | 1 || style="text-align:right | 100% |- | [[w:England|England]] / [[w:Great Britain|Great Britain]] / [[w:United Kingdom|United Kingdom]] || GBR || 1808 || 1898 || style="text-align:right | 67 || style="text-align:right | 74% |- | [[w:Australia|Australia]] || AUS || 1853 || 1891 || style="text-align:right | 17 || style="text-align:right | 44% |- | [[w:New Zealand|New Zealand]] || NZL || 1873 || 1874 || style="text-align:right | 2 || style="text-align:right | 100% |- | [[w:United States|United States]] || USA || 1882 || 2005 || style="text-align:right | 107 || style="text-align:right | 86% |- | [[w:Switzerland|Switzerland]] || CHE || 1931 || 2009 || style="text-align:right | 9 || style="text-align:right | 11% |- | [[w:Singapore|Singapore]] || SGP || 2010 || 2022 || style="text-align:right | 13 || style="text-align:right | 100% |} Singapore (SGP) has replaced Norway as the current leader, according to the Maddison project data. The Wikipedia article on "[[w:List of countries by GDP (PPP) per capita|List of countries by GDP (PPP) per capita]]"<ref>accessed 2025-01-01</ref> notes that data from the US [[w:Central Intelligence Agency|Central Intelligence Agency]] report GDP per capita numbers for [[w:Monaco| Monaco]] (MCO) and [[w:Liechtenstein|Liechtenstein]] (LIE) higher than for Singapore and Norway. However, they are tiny countries with populations roughly 40,000 each and are not included in MaddisonData. [[w:Holland|Holland]] (NLD) was the leader for 97 percent of the years between 1349 and 1807, according to MaddisonData. Then between 1807 and 1808, GDP per capita for NLD fell by 32 percent -- almost a third. That change can be attributed at least in part to a change in the definition of "NLD": Up to 1807, NLD represented Holland, per van Zanden and van Leeuwen (2012). Beginning in 1808, the data are for the [[w:Netherlands|Netherlands]], per Smits et al. (2000), of which Holland is only part. Those years were also during the [[w:Napoleonic Wars|Napoleonic Wars]], and the Netherlands were part of France for part of that period. To understand this drop better, we would need to consult experts on that history. After that change, Holland / the Netherlands was replaced as the leader in GDP per capita by England / Great Britain / the United Kingdom (GBR), which led for 74 percent of the 91 years between 1808 and 1898. Then the US led for 84 percent of the years between 1882 and 1990 with Australia (AUS), New Zealand (NZL) and Switzerland (CHE) leading for the remaining 16 percent of those years. Luxembourg (LUX) led between 1991 and 2008, then Switzerland (CHE) led for 2009, then Singapore (SGP) between 2010 and 2022. The next section discusses a plot of the data for NLD, GBR, USA, and SGP. Others countries are omitted, because their leadership was so short, according to these data, that including them might add more complexity than information and make it harder to understand the big picture. == Plot broad-based leaders == Figure 1 is a [[w:Semi-log|semilog]] plot of GDP per capita for NLD, GBR, USA, and SGP between 1349 and 2022. A semilog plot like this makes a constant percentage increase appear as a straight line. Annotations document some of the potentially most important events during this period: * The orange line represents Holland through 1807 and the Netherlands starting in 1808. * The English Civil War (1642-1652), during which King Charles I was decapitated (1649). * The War of 1912 (1812-1815). * The American Civil War (1861-1865). * [[w:World War I|WW1]] (1914-1918). * The presidency of Herbert Hoover (1929-1933). * The presidency of Franklin Roosevelt (1933-1945). * [[w:World War II|WW2]] (1939-1945). * The presidency of Ronald Reagan (1981-1989). * The first presidency of Donald Trump (2017-2021). * The presidency of Joe Biden (2021-2025). The orange NLD line includes a drop of 32 percent between 1807 and 1808 as the data changed from representing only Holland to representing the Netherlands, as mentioned above. A key feature of a semilog plot is that a constant percentage increase appears as a straight line with the slope being proportional to the rate of growth. A fairly obvious feature of Figure 1 is that GDP per capita started increasing for England very close to 1649, which King Charles I lost his head. England combined with Scotland to become Great Britain by the [[w:Acts of Union 1707|Acts of Union of 1707]] during the reign of [[w:Anne, Queen of Great Britain|Queen Ann]], which was accompanied by economic turbulence visible in Figure 1. After she died, the economy began growing again but at a slower rate. Great Britain merged with Ireland by the [[w:Acts of Union 1800|Acts of Union of 1800]] to become the United Kingdom (UK) during the reign of George III. The creation of the UK was quickly followed by the [[w:Napoleonic Wars|Napoleonic Wars]] (1803-1815), which included the [[w:War of 1812|War of 1812]], which is marked on this plot. Those wars were followed by a brief decline in the UK GDP per capita, but it quickly started growing again at a faster rate. Both World Wars had negative impacts on the UK economy, visible in Figure 1. GDP per capita for the US started well below that of the UK, to the extent that the Maddison data are accurate, but grew faster and overtook the UK between 1882 and 1898, according to Table 2 above. The most spectacular features in Figure 1 are the unprecedented decline of the US economy during the administration of Herbert Hoover followed an even more unprecedented increase during the administration of FDR. The rate of growth in GDP per capita in the US is visibly slowing before Singapore takes the lead at the beginning of the Great Recession. Thomas Piketty, the world's leading expert on inequality, has attributed that slowing of the US economy to the increase in inequality since Reagan became US president in 1981, documented in Figures 4 and 5 of the Wikipedia article on "[[The Media, the Great Depression, and our future]]". That article includes a section on the "[[The Media, the Great Depression, and our future#Role of the media|Role of the media]]", which cites research suggesting that both the increase in inequality and the slowing of the rate of economic growth can be attributed to the increased concentration of ownership of the major media including for-profit social media, which make money increasing political polarization and violence. == Caveat == Lindert and Williamson insist that Maddison's data are deficient, at least regarding the relative position of the 13 colonies that became the US: {{quote| American world leadership in income per person has waxed and waned for centuries. Before the twentieth century, the period in which Americans most clearly led Britain and all of western Europe in purchasing power per capita was during colonial times—that is, when North Americans were still British. They were already ahead by the late seventeenth century. America lost that lead in the Revolutionary War and the Articles of Confederation years, gained it back by 1860, lost most of it again in the Civil War decade, gained it back once more by 1900, and briefly lost it again in the Great Depression of the 1930s. ''Angus Maddison’s claim that American income per capita did not catch up to that of Britain until the start of the twentieth century seems to be off the mark by at least two centuries.'' Over the whole span of over 360 years since the mid-seventeenth century, America’s income advantage over Britain has not increased and may have decreased slightly. The only historical moment in which the United States soared far ahead of the rest of the world in average income came at the end of World War II. Since then, western Europe and Japan have been growing faster than the United States in terms of incomes per person. (emphasis added.)}}<ref>Lindert and Williamson (2016, pp. 8-9).</ref> This challenge to the numbers in the current analysis is important for some purposes but irrelevant to the main point of this book, that media play a major role in helping humans understand what they should do to advance their interests. == Exercise == Share with others your thoughts on the issues raised in this discussion and summarize those discussions on the "Discuss" page associated with this chapter. Focus especially on how you managed your emotions and your relationships with the humans with whom you spoke as well as the strengths and weaknesses in the content of this chapter, challenges that should be addressed, and suggestions for improvement. == See also == * [[Most productive countries since 1349]] * [[The Media, the Great Depression, and our future]] == Notes == {{reflist}} == Bibliography == * [[d: Q125292212|Daron Acemoğlu and Simon Johnson (2023) ''Power and Progress: Our Thousand-Year Struggle Over Technology and Prosperity'' (PublicAffairs)]]. * [[d:Q126723821|Jutta Bolt and Jan Luiten van Zanden (2024) "Maddison style estimates of the evolution of the world economy: A new 2023 update", Journal of Economic Surveys, 1-41]]. * [[d:Q57945943|S. N. Broadberry, B. Campbell, A. Klein, M. Overton and B. van Leeuwen (2015) ''British Economic Growth 1270-1870'' (Cambridge University Press)]]. * [[d:Q137660377|Susan B. Carter, S. S. Gartner, M. R. Haineset (2006) ''Historical Statistics of the United States: Earliest Time to the Present'' (Cambridge University Press)]]. * [[d:Q137660514|Spencer Graves (2025-11-25) "MaddisonData: Maddison Project Data" software available from the Comprehensive R Archive Network (CRAN) and GitHub]]. * [[d:Q54641943|Richard R. John (1995) ''Spreading the News: The American Postal System from Franklin to Morse'' (Harvard University Press)]]. * [[d:Q131468166|Richard R. John; Jonathan Silberstein-Loeb, eds. (2015) ''Making News: The Political Economy of Journalism in Britain and America from the Glorious Revolution to the Internet'' (Oxford University Press)]]. * [[d:Q137669937|John J. McCusker (2006) "Colonial Statistics", Carter et al. (2006, V-671)]]. * [[d:Q135527962|Reece Peck (2016) "Usurping the usable past: How Fox News remembered the Great Depression during the Great Recession", ''Journalism'', 18(6)]]. * [[d:Q55878109|W. Scheidel and S. J. Friesen (2009) "The size of the economy and the distribution of income in the Roman Empire", ''Journal of Roman Studies'', 99, pp. 61–91]]. * [[d:Q137669960|J.P. Smits, E. Horlings and J.L. van Zanden (2000) Dutch GDP and its Components 1800-1913 (Groningen Growth and Development Centre)]]. * [[d:Q137669987|I. Sugimoto (2011) ''Economic growth of Singapore in the twentieth century: historical GDP estimates and empirical investigations'' ( World Scientific Publishing) * [[d: Q137670038|R. Sutch (2006). National Income and Product. Carter et al. (2006, III-23-25)]]. * [[d: Q137670058|J. L. van Zanden and B. van Leeuwen (2012), ‘Persistent but not consistent: the growth of national income in Holland 1347–1807’, Explorations in Economic History, 49, pp. 119–30]]. [[Category:Media literacy]] [[Category:Freedom and abundance]] [[Category:Communication]] [[Category:Political science]] [[Category:Law]] [[Category:Education]] [[Category:Economics]] [[Category:Self improvement]] <!-- https://en.wikiversity.org/wiki/Category_Review --> bpt43wyjg7vyu8tqv0sl3ft6nakn8i7 2811991 2811987 2026-05-29T14:36:34Z DavidMCEddy 218607 correct accidental changes 2811991 wikitext text/x-wiki [[File:MaddisonDataLeaders1349-2022.svg|thumb|''Figure 1. World leaders in GDP per capita 1349-2022 (NLD, GBR, USA, SGP).<ref>See Bolt and van Zanden (2024) for the Maddison Data generally, van Zanden and van Leeuwen (2012) for the data on Holland 1348–1807, Smits et al. (2000) for the data on the Netherlands 1808-1913, Broadberry et al. (2015) for the data on England 1252–1700 and on Great Britain until 1870, and Sugimoto (2011) for Singapore to 2007.</ref>'']] :''I am entitled to my [[Wiktionary:cockamamie|cockamamie]] ideas, and you are entitled to yours.''<ref>Humor is important but must be offered in a way that does not offend others. If others are offended, they may be less interested in dialogue. The term "cockamamie" is used here, hoping that this style of [[w:Self-deprecation|self-deprecation]] might be more inviting for dialogue.</ref> :This book invites you to improve your understanding of the role of the media in society and help you be more effective in talking with others and building consensus for action on the issues that most concern you. You are also invited to help improve this chapter and book. ==Introduction== Acemoglu and Johnson (2023) suggest that the stability of poverty and the [[w:Malthusianism|Malthusian trap]] in hierarchical societies ''prior'' to the Industrial Revolution was enforced by "societies of orders" consisting of those who (1) fought, (2) prayed, and (3) worked. Those who prayed convinced those who worked to live in poverty while giving increasing shares of what they produced so those who prayed and fought could live in leisure and opulence. This seems to describe the construction of the pyramids in Egypt and the cathedrals, castles and manor homes that dot Europe today.<ref>Acemoglu and Johnson (2023, esp. ch. 4).</ref> During the reign of [[w:James VI and I|King James of the King James bible]] pamphlets and newspapers began to compete with the church for helping peasants understand their role in society. That contributed to the [[w:English Civil War|English Civil War]] (1642-1651) during which James' son, [[w: Charles I of England|Charles I]], was decapitated for abuse of power. The new government allowed enough commoners to become entrepreneurs that it increased the rate of economic growth visible in Figure 1. The number of independent media organizations per million population continued to grow, especially in the US where they were subsidized by the US [[w:Postal Service Act|Postal Service Act]] of 1792, [[Media concentration per Columbia History Professor Richard John|until the mid nineteenth century]], when high-speed rotary presses substantially reduced the per copy cost of printing while increasing the cost of starting a newspaper.<ref>John (1995); John and Silberstein-Loeb (2015).</ref> That contributed to the expansion of voting rights, in the US officially to all adults by 1920. During the [[w:Great Depression|Great Depression]] with over 20 percent of the US workforce unemployed, conservative arguments that blamed the poor for their poverty did not sell newspapers. That helped US President [[w:Franklin D. Roosevelt|Franklin Roosevelt]] get the political support needed for his [[w:New Deal|New Deal]] program and [[The Media, the Great Depression, and our future|wage and price controls that enabled unprecedented economic growth]] during [[w:World War II|World War II]] by dramatically limiting price gouging that had generated inflation and stifled economic growth during previous major wars in US history. However, the consolidation of ownership of the major media since World War II limited the increases in inequality and then reversed them starting around the time that [[w:Ronald Reagan|Ronald Reagan]] became president of the US in 1981.<ref>Acemoglu and Johnson (2023).</ref> During the [[w:Great Recession|Great Recession]] (2007-2009) Fox featured "experts" who insisted that Franklin Roosevelt's New Deal made the Great Depression ''worse'', not better. That editorial distortion helped prevent the US Congress from protecting the victims of [[w:Stated income loan|liar loans]]. But the money had to be injected back into the economy, so the US Congress funded bonuses of over a million dollars each to over 5,000 finance industry leaders, some of whom should have been in prison, according to Acemoglu and Johnson (2023, ch. 3), who shared the 2024 Nobel Memorial Prize in Economics with [[w:James A. Robinson|James A. Robinson]]. A major contributor to the Great Recession has reportedly been the success of the finance industry in getting sufficient control of the major media that they have been able to dramatically reduce financial regulation codified in the [[w:Glass–Steagall legislation|Glass–Steagall legislation]] of 1933 and block other reforms like a [[w:Tobin tax|Tobin tax]] to reduce market swings from extremely short term trading, an idea for which [[w:James Tobin|James Tobin]] won the [[w:List of Nobel Memorial Prize laureates in Economic Sciences|1981 Nobel memorial prize in economics]]. You, dear reader, are invited to offer contrary evidence or questions regarding the evidence presented here. This chapter identifies the leading countries in [[w:List of countries by GDP (PPP) per capita|GDP per capita for each year]] in the [[w:Maddison Project|Maddison Historical Statistics Project]], which collates [[w:Gross domestic product|GDP]] per capita and population estimates for years 1 [[w:Common Era|CE]] to the present from all credible sources the project team has found. The MaddisonData package for R makes these data available as an R data object with companion functions to facilitate computing the leaders for each year with optional exclusions to facilitate identifying the technology leaders and with a function ggplotPath to make it relatively easy to plot and annotate the image as desired.<ref>Graves (2025).</ref> The analysis then narrows the focus to apparent technology leaders since 1349, when the data identify Holland as the leader. This analysis identifies 11 different countries with the highest GDP per capita for at least one year. However, only 5 lead for more than 10 years: Holland (NLD), England / Great Britain / the United Kingdom (GBR), Australia (AUS), the United States (USA), and Singapore (SGP).<ref>The population of [[w:Singapore|Singapore]] is roughly 6 million. If you think that's too small to be included in this analysis, you can either ignore it or, better, repeat this analysis without it. A tutorial on how to do that using free and open-source software is available in [[Most productive countries since 1349#Appendix. Companion R Markdown vignette|an R Markdown vignette]] supplied as an appendix to the Wikiversity article on "[[Most productive countries since 1349]]".</ref> Australia led for only 17 of the 39 years between 1853 and 1891 as the US was overtaking the UK as the technology leader. A plot of the leaders is then made without Australia, because its inclusion would seem to add more complexity than clarity to the message. That display is a [[w:Semi-log|semilog]] plot, because it makes a constant percentage increase look like a straight line. The resulting plot of GDP per capita suggests the Industrial Revolution began in England around 1649 when King Charles I was beheaded. == World leaders in GDP per capita == Table 1 lists all the countries with the highest GDP per capita for at least one year in MaddisonData. {| class="wikitable" |+ ''Table 1. all the countries with the highest GDP per capita for at least one year in the MaddisonData.'' |- ! rowspan=2 | country !! rowspan=2 | ISO !! colspan=2 | year !! rowspan=2 | n years !! rowspan=2 | percent of years |- ! begin !! end |- | [[w:Ancient Rome|Italy (ancient Rome)]] | ITA || style="text-align:right | 1 || 1501 || style="text-align:right | 3 || style="text-align:right | 0.2% |- | [[w:Iraq|Iraq]] || IRQ || style="text-align:right | 730 || 1000 || style="text-align:right | 271 || style="text-align:right | 100% |- | [[w:China|China]] || CHN || style="text-align:right | 1090 || 1150 || style="text-align:right | 61 || style="text-align:right | 100% |- | [[w:England|England]] / [[w:Great Britain|Great Britain]] / [[w:United Kingdom|United Kingdom]] || GBR || style="text-align:right | 1252 || 1898 || style="text-align:right | 91 || style="text-align:right | 14% |- | [[w:France|France]] || FRA || style="text-align:right | 1276 || 1374 || style="text-align:right | 19 || style="text-align:right | 19% |- | [[w:Spain|Spain]] || ESP || style="text-align:right | 1278 || 1348 || style="text-align:right | 50 || style="text-align:right | 70% |- | [[w:Sweden|Sweden]] || SWE || style="text-align:right | 1304 || 1509 || style="text-align:right | 13 || style="text-align:right | 6% |- | [[w:Holland|Holland]] / [[w:Netherlands|Netherlands]] || NLD || style="text-align:right | 1349 || 1807 || style="text-align:right | 447 || style="text-align:right | 97% |- | [[w:Belgium|Belgium]] || BEL || style="text-align:right | 1500 || 1500 || style="text-align:right | 1 || style="text-align:right | 100% |- | [[w:Australia|Australia]] || AUS || style="text-align:right | 1853 || 1891 || style="text-align:right | 17 || style="text-align:right | 44% |- | [[w:New Zealand|New Zealand]] || NZL || style="text-align:right | 1873 || 1874 || style="text-align:right | 2 || style="text-align:right | 100% |- | [[w:United States|United States]] || USA || style="text-align:right | 1882 || 1990 || style="text-align:right | 58 || style="text-align:right | 53% |- | [[w:Switzerland|Switzerland]] || CHE || style="text-align:right | 1931 || 1934 || style="text-align:right | 4 || style="text-align:right | 100% |- | [[w:Qatar|Qatar]] || QAT || style="text-align:right | 1950 || 2022 || style="text-align:right | 45 || style="text-align:right | 62% |- | [[w:Kuwait|Kuwait]] || KWT || style="text-align:right | 1953 || 1957 || style="text-align:right | 5 || style="text-align:right | 100% |- |[[w:United Arab Emirates|United Arab Emirates]] || ARE || style="text-align:right | 1965 || 1984 || style="text-align:right | 5 || style="text-align:right | 25% |- | [[w:Luxembourg|Luxembourg]] || LUX || style="text-align:right | 1991 || 1995 || style="text-align:right | 5 || style="text-align:right | 100% |- | [[w:Norway|Norway]] || NOR || style="text-align:right | 1996 || 2002 || style="text-align:right | 7 || style="text-align:right | 100% |} For simplicity, we focus on the data since 1349, when Holland had the highest GDP per capita, omitting Qatar (QAT), Kuwait (KWT), United Arab Emirates (ARE), and Norway (NOR), whose wealth may be more due to petroleum than to broad technology leadership. We also delete Luxenbourg (LUX), whose population is under a million and therefore may be too small to use for general conclusions about technology leadership. Table 2 is similar to Table 1 with these adjustments. {| class="wikitable" |+ ''Table 2. Highest GDP per capita among broad-based economies since 1349.'' |- ! rowspan=2 | country !! rowspan=2 | ISO !! colspan=2 | year !! rowspan=2 | n years !! rowspan=2 | percent of years |- ! begin !! end |- | [[w:Holland|Holland]] / [[w:Netherlands|Netherlands]] || NLD || 1349 || 1807 || style="text-align:right | 447 || style="text-align:right | 97% |- | [[w:France|France]] || FRA || 1357 || 1374 || style="text-align:right | 7 || style="text-align:right | 39% |- | [[w:Italy|Italy]] || ITA || 1451 || 1501 || style="text-align:right | 2 || style="text-align:right | 4% |- | [[w:Sweden|Sweden]] || SWE || 1468 || 1509 || style="text-align:right | 2 || style="text-align:right | 5% |- | [[w:Belgium|Belgium]] || BEL || 1500 || 1500 || style="text-align:right | 1 || style="text-align:right | 100% |- | [[w:England|England]] / [[w:Great Britain|Great Britain]] / [[w:United Kingdom|United Kingdom]] || GBR || 1808 || 1898 || style="text-align:right | 67 || style="text-align:right | 74% |- | [[w:Australia|Australia]] || AUS || 1853 || 1891 || style="text-align:right | 17 || style="text-align:right | 44% |- | [[w:New Zealand|New Zealand]] || NZL || 1873 || 1874 || style="text-align:right | 2 || style="text-align:right | 100% |- | [[w:United States|United States]] || USA || 1882 || 2005 || style="text-align:right | 107 || style="text-align:right | 86% |- | [[w:Switzerland|Switzerland]] || CHE || 1931 || 2009 || style="text-align:right | 9 || style="text-align:right | 11% |- | [[w:Singapore|Singapore]] || SGP || 2010 || 2022 || style="text-align:right | 13 || style="text-align:right | 100% |} Singapore (SGP) has replaced Norway as the current leader, according to the Maddison project data. The Wikipedia article on "[[w:List of countries by GDP (PPP) per capita|List of countries by GDP (PPP) per capita]]"<ref>accessed 2025-01-01</ref> notes that data from the US [[w:Central Intelligence Agency|Central Intelligence Agency]] report GDP per capita numbers for [[w:Monaco| Monaco]] (MCO) and [[w:Liechtenstein|Liechtenstein]] (LIE) higher than for Singapore and Norway. However, they are tiny countries with populations roughly 40,000 each and are not included in MaddisonData. [[w:Holland|Holland]] (NLD) was the leader for 97 percent of the years between 1349 and 1807, according to MaddisonData. Then between 1807 and 1808, GDP per capita for NLD fell by 32 percent -- almost a third. That change can be attributed at least in part to a change in the definition of "NLD": Up to 1807, NLD represented Holland, per van Zanden and van Leeuwen (2012). Beginning in 1808, the data are for the [[w:Netherlands|Netherlands]], per Smits et al. (2000), of which Holland is only part. Those years were also during the [[w:Napoleonic Wars|Napoleonic Wars]], and the Netherlands were part of France for part of that period. To understand this drop better, we would need to consult experts on that history. After that change, Holland / the Netherlands was replaced as the leader in GDP per capita by England / Great Britain / the United Kingdom (GBR), which led for 74 percent of the 91 years between 1808 and 1898. Then the US led for 84 percent of the years between 1882 and 1990 with Australia (AUS), New Zealand (NZL) and Switzerland (CHE) leading for the remaining 16 percent of those years. Luxembourg (LUX) led between 1991 and 2008, then Switzerland (CHE) led for 2009, then Singapore (SGP) between 2010 and 2022. The next section discusses a plot of the data for NLD, GBR, USA, and SGP. Others countries are omitted, because their leadership was so short, according to these data, that including them might add more complexity than information and make it harder to understand the big picture. == Plot broad-based leaders == Figure 1 is a [[w:Semi-log|semilog]] plot of GDP per capita for NLD, GBR, USA, and SGP between 1349 and 2022. A semilog plot like this makes a constant percentage increase appear as a straight line. Annotations document some of the potentially most important events during this period: * The orange line represents Holland through 1807 and the Netherlands starting in 1808. * The English Civil War (1642-1652), during which King Charles I was decapitated (1649). * The War of 1912 (1812-1815). * The American Civil War (1861-1865). * [[w:World War I|WW1]] (1914-1918). * The presidency of Herbert Hoover (1929-1933). * The presidency of Franklin Roosevelt (1933-1945). * [[w:World War II|WW2]] (1939-1945). * The presidency of Ronald Reagan (1981-1989). * The first presidency of Donald Trump (2017-2021). * The presidency of Joe Biden (2021-2025). The orange NLD line includes a drop of 32 percent between 1807 and 1808 as the data changed from representing only Holland to representing the Netherlands, as mentioned above. A key feature of a semilog plot is that a constant percentage increase appears as a straight line with the slope being proportional to the rate of growth. A fairly obvious feature of Figure 1 is that GDP per capita started increasing for England very close to 1649, which King Charles I lost his head. England combined with Scotland to become Great Britain by the [[w:Acts of Union 1707|Acts of Union of 1707]] during the reign of [[w:Anne, Queen of Great Britain|Queen Ann]], which was accompanied by economic turbulence visible in Figure 1. After she died, the economy began growing again but at a slower rate. Great Britain merged with Ireland by the [[w:Acts of Union 1800|Acts of Union of 1800]] to become the United Kingdom (UK) during the reign of George III. The creation of the UK was quickly followed by the [[w:Napoleonic Wars|Napoleonic Wars]] (1803-1815), which included the [[w:War of 1812|War of 1812]], which is marked on this plot. Those wars were followed by a brief decline in the UK GDP per capita, but it quickly started growing again at a faster rate. Both World Wars had negative impacts on the UK economy, visible in Figure 1. GDP per capita for the US started well below that of the UK, to the extent that the Maddison data are accurate, but grew faster and overtook the UK between 1882 and 1898, according to Table 2 above. The most spectacular features in Figure 1 are the unprecedented decline of the US economy during the administration of Herbert Hoover followed an even more unprecedented increase during the administration of FDR. The rate of growth in GDP per capita in the US is visibly slowing before Singapore takes the lead at the beginning of the Great Recession. Thomas Piketty, the world's leading expert on inequality, has attributed that slowing of the US economy to the increase in inequality since Reagan became US president in 1981, documented in Figures 4 and 5 of the Wikipedia article on "[[The Media, the Great Depression, and our future]]". That article includes a section on the "[[The Media, the Great Depression, and our future#Role of the media|Role of the media]]", which cites research suggesting that both the increase in inequality and the slowing of the rate of economic growth can be attributed to the increased concentration of ownership of the major media including for-profit social media, which make money increasing political polarization and violence. == Caveat == Lindert and Williamson insist that Maddison's data are deficient, at least regarding the relative position of the 13 colonies that became the US: {{quote| American world leadership in income per person has waxed and waned for centuries. Before the twentieth century, the period in which Americans most clearly led Britain and all of western Europe in purchasing power per capita was during colonial times—that is, when North Americans were still British. They were already ahead by the late seventeenth century. America lost that lead in the Revolutionary War and the Articles of Confederation years, gained it back by 1860, lost most of it again in the Civil War decade, gained it back once more by 1900, and briefly lost it again in the Great Depression of the 1930s. ''Angus Maddison’s claim that American income per capita did not catch up to that of Britain until the start of the twentieth century seems to be off the mark by at least two centuries.'' Over the whole span of over 360 years since the mid-seventeenth century, America’s income advantage over Britain has not increased and may have decreased slightly. The only historical moment in which the United States soared far ahead of the rest of the world in average income came at the end of World War II. Since then, western Europe and Japan have been growing faster than the United States in terms of incomes per person. (emphasis added.)<ref>Lindert and Williamson (2016, pp. 8-9).</ref>}} This challenge to the numbers in the current analysis is vital for understanding the impact of armed conflict on the economy but may be irrelevant to the main point of this book, that media play a major role in helping humans understand what they should do to advance their interests. == Exercise == Share with others your thoughts on the issues raised in this discussion and summarize those discussions on the "Discuss" page associated with this chapter. Focus especially on how you managed your emotions and your relationships with the humans with whom you spoke as well as the strengths and weaknesses in the content of this chapter, challenges that should be addressed, and suggestions for improvement. == See also == * [[Most productive countries since 1349]] * [[The Media, the Great Depression, and our future]] == Notes == {{reflist}} == Bibliography == * [[d: Q125292212|Daron Acemoğlu and Simon Johnson (2023) ''Power and Progress: Our Thousand-Year Struggle Over Technology and Prosperity'' (PublicAffairs)]]. * [[d:Q126723821|Jutta Bolt and Jan Luiten van Zanden (2024) "Maddison style estimates of the evolution of the world economy: A new 2023 update", Journal of Economic Surveys, 1-41]]. * [[d:Q57945943|S. N. Broadberry, B. Campbell, A. Klein, M. Overton and B. van Leeuwen (2015) ''British Economic Growth 1270-1870'' (Cambridge University Press)]]. * [[d:Q137660377|Susan B. Carter, S. S. Gartner, M. R. Haineset (2006) ''Historical Statistics of the United States: Earliest Time to the Present'' (Cambridge University Press)]]. * [[d:Q137660514|Spencer Graves (2025-11-25) "MaddisonData: Maddison Project Data" software available from the Comprehensive R Archive Network (CRAN) and GitHub]]. * [[d:Q54641943|Richard R. John (1995) ''Spreading the News: The American Postal System from Franklin to Morse'' (Harvard University Press)]]. * [[d:Q131468166|Richard R. John; Jonathan Silberstein-Loeb, eds. (2015) ''Making News: The Political Economy of Journalism in Britain and America from the Glorious Revolution to the Internet'' (Oxford University Press)]]. * [[d:Q137669937|John J. McCusker (2006) "Colonial Statistics", Carter et al. (2006, V-671)]]. * [[d:Q135527962|Reece Peck (2016) "Usurping the usable past: How Fox News remembered the Great Depression during the Great Recession", ''Journalism'', 18(6)]]. * [[d:Q55878109|W. Scheidel and S. J. Friesen (2009) "The size of the economy and the distribution of income in the Roman Empire", ''Journal of Roman Studies'', 99, pp. 61–91]]. * [[d:Q137669960|J.P. Smits, E. Horlings and J.L. van Zanden (2000) Dutch GDP and its Components 1800-1913 (Groningen Growth and Development Centre)]]. * [[d:Q137669987|I. Sugimoto (2011) ''Economic growth of Singapore in the twentieth century: historical GDP estimates and empirical investigations'' ( World Scientific Publishing) * [[d: Q137670038|R. Sutch (2006). National Income and Product. Carter et al. (2006, III-23-25)]]. * [[d: Q137670058|J. L. van Zanden and B. van Leeuwen (2012), ‘Persistent but not consistent: the growth of national income in Holland 1347–1807’, Explorations in Economic History, 49, pp. 119–30]]. [[Category:Media literacy]] [[Category:Freedom and abundance]] [[Category:Communication]] [[Category:Political science]] [[Category:Law]] [[Category:Education]] [[Category:Economics]] [[Category:Media Literacy and You]] <!-- https://en.wikiversity.org/wiki/Category_Review --> qf2xwz5ul812wzwcaao2lm9osticwwn 2811997 2811991 2026-05-29T14:43:32Z DavidMCEddy 218607 chg ref to another article to the next chapter 2811997 wikitext text/x-wiki [[File:MaddisonDataLeaders1349-2022.svg|thumb|''Figure 1. World leaders in GDP per capita 1349-2022 (NLD, GBR, USA, SGP).<ref>See Bolt and van Zanden (2024) for the Maddison Data generally, van Zanden and van Leeuwen (2012) for the data on Holland 1348–1807, Smits et al. (2000) for the data on the Netherlands 1808-1913, Broadberry et al. (2015) for the data on England 1252–1700 and on Great Britain until 1870, and Sugimoto (2011) for Singapore to 2007.</ref>'']] :''I am entitled to my [[Wiktionary:cockamamie|cockamamie]] ideas, and you are entitled to yours.''<ref>Humor is important but must be offered in a way that does not offend others. If others are offended, they may be less interested in dialogue. The term "cockamamie" is used here, hoping that this style of [[w:Self-deprecation|self-deprecation]] might be more inviting for dialogue.</ref> :This book invites you to improve your understanding of the role of the media in society and help you be more effective in talking with others and building consensus for action on the issues that most concern you. You are also invited to help improve this chapter and book. ==Introduction== Acemoglu and Johnson (2023) suggest that the stability of poverty and the [[w:Malthusianism|Malthusian trap]] in hierarchical societies ''prior'' to the Industrial Revolution was enforced by "societies of orders" consisting of those who (1) fought, (2) prayed, and (3) worked. Those who prayed convinced those who worked to live in poverty while giving increasing shares of what they produced so those who prayed and fought could live in leisure and opulence. This seems to describe the construction of the pyramids in Egypt and the cathedrals, castles and manor homes that dot Europe today.<ref>Acemoglu and Johnson (2023, esp. ch. 4).</ref> During the reign of [[w:James VI and I|King James of the King James bible]] pamphlets and newspapers began to compete with the church for helping peasants understand their role in society. That contributed to the [[w:English Civil War|English Civil War]] (1642-1651) during which James' son, [[w: Charles I of England|Charles I]], was decapitated for abuse of power. The new government allowed enough commoners to become entrepreneurs that it increased the rate of economic growth visible in Figure 1. The number of independent media organizations per million population continued to grow, especially in the US where they were subsidized by the US [[w:Postal Service Act|Postal Service Act]] of 1792, [[Media concentration per Columbia History Professor Richard John|until the mid nineteenth century]], when high-speed rotary presses substantially reduced the per copy cost of printing while increasing the cost of starting a newspaper.<ref>John (1995); John and Silberstein-Loeb (2015).</ref> That contributed to the expansion of voting rights, in the US officially to all adults by 1920. During the [[w:Great Depression|Great Depression]] with over 20 percent of the US workforce unemployed, conservative arguments that blamed the poor for their poverty did not sell newspapers. That helped US President [[w:Franklin D. Roosevelt|Franklin Roosevelt]] get the political support needed for his [[w:New Deal|New Deal]] program and [[The Media, the Great Depression, and our future|wage and price controls that enabled unprecedented economic growth]] during [[w:World War II|World War II]] by dramatically limiting price gouging that had generated inflation and stifled economic growth during previous major wars in US history. However, the consolidation of ownership of the major media since World War II limited the increases in inequality and then reversed them starting around the time that [[w:Ronald Reagan|Ronald Reagan]] became president of the US in 1981.<ref>Acemoglu and Johnson (2023).</ref> During the [[w:Great Recession|Great Recession]] (2007-2009) Fox featured "experts" who insisted that Franklin Roosevelt's New Deal made the Great Depression ''worse'', not better. That editorial distortion helped prevent the US Congress from protecting the victims of [[w:Stated income loan|liar loans]]. But the money had to be injected back into the economy, so the US Congress funded bonuses of over a million dollars each to over 5,000 finance industry leaders, some of whom should have been in prison, according to Acemoglu and Johnson (2023, ch. 3), who shared the 2024 Nobel Memorial Prize in Economics with [[w:James A. Robinson|James A. Robinson]]. A major contributor to the Great Recession has reportedly been the success of the finance industry in getting sufficient control of the major media that they have been able to dramatically reduce financial regulation codified in the [[w:Glass–Steagall legislation|Glass–Steagall legislation]] of 1933 and block other reforms like a [[w:Tobin tax|Tobin tax]] to reduce market swings from extremely short term trading, an idea for which [[w:James Tobin|James Tobin]] won the [[w:List of Nobel Memorial Prize laureates in Economic Sciences|1981 Nobel memorial prize in economics]]. You, dear reader, are invited to offer contrary evidence or questions regarding the evidence presented here. This chapter identifies the leading countries in [[w:List of countries by GDP (PPP) per capita|GDP per capita for each year]] in the [[w:Maddison Project|Maddison Historical Statistics Project]], which collates [[w:Gross domestic product|GDP]] per capita and population estimates for years 1 [[w:Common Era|CE]] to the present from all credible sources the project team has found. The MaddisonData package for R makes these data available as an R data object with companion functions to facilitate computing the leaders for each year with optional exclusions to facilitate identifying the technology leaders and with a function ggplotPath to make it relatively easy to plot and annotate the image as desired.<ref>Graves (2025).</ref> The analysis then narrows the focus to apparent technology leaders since 1349, when the data identify Holland as the leader. This analysis identifies 11 different countries with the highest GDP per capita for at least one year. However, only 5 lead for more than 10 years: Holland (NLD), England / Great Britain / the United Kingdom (GBR), Australia (AUS), the United States (USA), and Singapore (SGP).<ref>The population of [[w:Singapore|Singapore]] is roughly 6 million. If you think that's too small to be included in this analysis, you can either ignore it or, better, repeat this analysis without it. A tutorial on how to do that using free and open-source software is available in [[Most productive countries since 1349#Appendix. Companion R Markdown vignette|an R Markdown vignette]] supplied as an appendix to the Wikiversity article on "[[Most productive countries since 1349]]".</ref> Australia led for only 17 of the 39 years between 1853 and 1891 as the US was overtaking the UK as the technology leader. A plot of the leaders is then made without Australia, because its inclusion would seem to add more complexity than clarity to the message. That display is a [[w:Semi-log|semilog]] plot, because it makes a constant percentage increase look like a straight line. The resulting plot of GDP per capita suggests the Industrial Revolution began in England around 1649 when King Charles I was beheaded. == World leaders in GDP per capita == Table 1 lists all the countries with the highest GDP per capita for at least one year in MaddisonData. {| class="wikitable" |+ ''Table 1. all the countries with the highest GDP per capita for at least one year in the MaddisonData.'' |- ! rowspan=2 | country !! rowspan=2 | ISO !! colspan=2 | year !! rowspan=2 | n years !! rowspan=2 | percent of years |- ! begin !! end |- | [[w:Ancient Rome|Italy (ancient Rome)]] | ITA || style="text-align:right | 1 || 1501 || style="text-align:right | 3 || style="text-align:right | 0.2% |- | [[w:Iraq|Iraq]] || IRQ || style="text-align:right | 730 || 1000 || style="text-align:right | 271 || style="text-align:right | 100% |- | [[w:China|China]] || CHN || style="text-align:right | 1090 || 1150 || style="text-align:right | 61 || style="text-align:right | 100% |- | [[w:England|England]] / [[w:Great Britain|Great Britain]] / [[w:United Kingdom|United Kingdom]] || GBR || style="text-align:right | 1252 || 1898 || style="text-align:right | 91 || style="text-align:right | 14% |- | [[w:France|France]] || FRA || style="text-align:right | 1276 || 1374 || style="text-align:right | 19 || style="text-align:right | 19% |- | [[w:Spain|Spain]] || ESP || style="text-align:right | 1278 || 1348 || style="text-align:right | 50 || style="text-align:right | 70% |- | [[w:Sweden|Sweden]] || SWE || style="text-align:right | 1304 || 1509 || style="text-align:right | 13 || style="text-align:right | 6% |- | [[w:Holland|Holland]] / [[w:Netherlands|Netherlands]] || NLD || style="text-align:right | 1349 || 1807 || style="text-align:right | 447 || style="text-align:right | 97% |- | [[w:Belgium|Belgium]] || BEL || style="text-align:right | 1500 || 1500 || style="text-align:right | 1 || style="text-align:right | 100% |- | [[w:Australia|Australia]] || AUS || style="text-align:right | 1853 || 1891 || style="text-align:right | 17 || style="text-align:right | 44% |- | [[w:New Zealand|New Zealand]] || NZL || style="text-align:right | 1873 || 1874 || style="text-align:right | 2 || style="text-align:right | 100% |- | [[w:United States|United States]] || USA || style="text-align:right | 1882 || 1990 || style="text-align:right | 58 || style="text-align:right | 53% |- | [[w:Switzerland|Switzerland]] || CHE || style="text-align:right | 1931 || 1934 || style="text-align:right | 4 || style="text-align:right | 100% |- | [[w:Qatar|Qatar]] || QAT || style="text-align:right | 1950 || 2022 || style="text-align:right | 45 || style="text-align:right | 62% |- | [[w:Kuwait|Kuwait]] || KWT || style="text-align:right | 1953 || 1957 || style="text-align:right | 5 || style="text-align:right | 100% |- |[[w:United Arab Emirates|United Arab Emirates]] || ARE || style="text-align:right | 1965 || 1984 || style="text-align:right | 5 || style="text-align:right | 25% |- | [[w:Luxembourg|Luxembourg]] || LUX || style="text-align:right | 1991 || 1995 || style="text-align:right | 5 || style="text-align:right | 100% |- | [[w:Norway|Norway]] || NOR || style="text-align:right | 1996 || 2002 || style="text-align:right | 7 || style="text-align:right | 100% |} For simplicity, we focus on the data since 1349, when Holland had the highest GDP per capita, omitting Qatar (QAT), Kuwait (KWT), United Arab Emirates (ARE), and Norway (NOR), whose wealth may be more due to petroleum than to broad technology leadership. We also delete Luxenbourg (LUX), whose population is under a million and therefore may be too small to use for general conclusions about technology leadership. Table 2 is similar to Table 1 with these adjustments. {| class="wikitable" |+ ''Table 2. Highest GDP per capita among broad-based economies since 1349.'' |- ! rowspan=2 | country !! rowspan=2 | ISO !! colspan=2 | year !! rowspan=2 | n years !! rowspan=2 | percent of years |- ! begin !! end |- | [[w:Holland|Holland]] / [[w:Netherlands|Netherlands]] || NLD || 1349 || 1807 || style="text-align:right | 447 || style="text-align:right | 97% |- | [[w:France|France]] || FRA || 1357 || 1374 || style="text-align:right | 7 || style="text-align:right | 39% |- | [[w:Italy|Italy]] || ITA || 1451 || 1501 || style="text-align:right | 2 || style="text-align:right | 4% |- | [[w:Sweden|Sweden]] || SWE || 1468 || 1509 || style="text-align:right | 2 || style="text-align:right | 5% |- | [[w:Belgium|Belgium]] || BEL || 1500 || 1500 || style="text-align:right | 1 || style="text-align:right | 100% |- | [[w:England|England]] / [[w:Great Britain|Great Britain]] / [[w:United Kingdom|United Kingdom]] || GBR || 1808 || 1898 || style="text-align:right | 67 || style="text-align:right | 74% |- | [[w:Australia|Australia]] || AUS || 1853 || 1891 || style="text-align:right | 17 || style="text-align:right | 44% |- | [[w:New Zealand|New Zealand]] || NZL || 1873 || 1874 || style="text-align:right | 2 || style="text-align:right | 100% |- | [[w:United States|United States]] || USA || 1882 || 2005 || style="text-align:right | 107 || style="text-align:right | 86% |- | [[w:Switzerland|Switzerland]] || CHE || 1931 || 2009 || style="text-align:right | 9 || style="text-align:right | 11% |- | [[w:Singapore|Singapore]] || SGP || 2010 || 2022 || style="text-align:right | 13 || style="text-align:right | 100% |} Singapore (SGP) has replaced Norway as the current leader, according to the Maddison project data. The Wikipedia article on "[[w:List of countries by GDP (PPP) per capita|List of countries by GDP (PPP) per capita]]"<ref>accessed 2025-01-01</ref> notes that data from the US [[w:Central Intelligence Agency|Central Intelligence Agency]] report GDP per capita numbers for [[w:Monaco| Monaco]] (MCO) and [[w:Liechtenstein|Liechtenstein]] (LIE) higher than for Singapore and Norway. However, they are tiny countries with populations roughly 40,000 each and are not included in MaddisonData. [[w:Holland|Holland]] (NLD) was the leader for 97 percent of the years between 1349 and 1807, according to MaddisonData. Then between 1807 and 1808, GDP per capita for NLD fell by 32 percent -- almost a third. That change can be attributed at least in part to a change in the definition of "NLD": Up to 1807, NLD represented Holland, per van Zanden and van Leeuwen (2012). Beginning in 1808, the data are for the [[w:Netherlands|Netherlands]], per Smits et al. (2000), of which Holland is only part. Those years were also during the [[w:Napoleonic Wars|Napoleonic Wars]], and the Netherlands were part of France for part of that period. To understand this drop better, we would need to consult experts on that history. After that change, Holland / the Netherlands was replaced as the leader in GDP per capita by England / Great Britain / the United Kingdom (GBR), which led for 74 percent of the 91 years between 1808 and 1898. Then the US led for 84 percent of the years between 1882 and 1990 with Australia (AUS), New Zealand (NZL) and Switzerland (CHE) leading for the remaining 16 percent of those years. Luxembourg (LUX) led between 1991 and 2008, then Switzerland (CHE) led for 2009, then Singapore (SGP) between 2010 and 2022. The next section discusses a plot of the data for NLD, GBR, USA, and SGP. Others countries are omitted, because their leadership was so short, according to these data, that including them might add more complexity than information and make it harder to understand the big picture. == Plot broad-based leaders == Figure 1 is a [[w:Semi-log|semilog]] plot of GDP per capita for NLD, GBR, USA, and SGP between 1349 and 2022. A semilog plot like this makes a constant percentage increase appear as a straight line. Annotations document some of the potentially most important events during this period: * The orange line represents Holland through 1807 and the Netherlands starting in 1808. * The English Civil War (1642-1652), during which King Charles I was decapitated (1649). * The War of 1912 (1812-1815). * The American Civil War (1861-1865). * [[w:World War I|WW1]] (1914-1918). * The presidency of Herbert Hoover (1929-1933). * The presidency of Franklin Roosevelt (1933-1945). * [[w:World War II|WW2]] (1939-1945). * The presidency of Ronald Reagan (1981-1989). * The first presidency of Donald Trump (2017-2021). * The presidency of Joe Biden (2021-2025). The orange NLD line includes a drop of 32 percent between 1807 and 1808 as the data changed from representing only Holland to representing the Netherlands, as mentioned above. A key feature of a semilog plot is that a constant percentage increase appears as a straight line with the slope being proportional to the rate of growth. A fairly obvious feature of Figure 1 is that GDP per capita started increasing for England very close to 1649, which King Charles I lost his head. England combined with Scotland to become Great Britain by the [[w:Acts of Union 1707|Acts of Union of 1707]] during the reign of [[w:Anne, Queen of Great Britain|Queen Ann]], which was accompanied by economic turbulence visible in Figure 1. After she died, the economy began growing again but at a slower rate. Great Britain merged with Ireland by the [[w:Acts of Union 1800|Acts of Union of 1800]] to become the United Kingdom (UK) during the reign of George III. The creation of the UK was quickly followed by the [[w:Napoleonic Wars|Napoleonic Wars]] (1803-1815), which included the [[w:War of 1812|War of 1812]], which is marked on this plot. Those wars were followed by a brief decline in the UK GDP per capita, but it quickly started growing again at a faster rate. Both World Wars had negative impacts on the UK economy, visible in Figure 1. GDP per capita for the US started well below that of the UK, to the extent that the Maddison data are accurate, but grew faster and overtook the UK between 1882 and 1898, according to Table 2 above. The most spectacular features in Figure 1 are the unprecedented decline of the US economy during the administration of Herbert Hoover followed an even more unprecedented increase during the administration of FDR. The rate of growth in GDP per capita in the US is visibly slowing before Singapore takes the lead at the beginning of the Great Recession. [[w:Thomas Piketty|Thomas Piketty]], the world's leading expert on inequality, has attributed that slowing of the US economy to the increase in inequality since Reagan became US president in 1981, documented in Figures 6 and 7 of the next chapter of this book on, "[[Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future|Fox, the Great Depression, the Great Recession, and our future]]". That chapter includes a section on the "[[Media Literacy and You/Fox, the Great Depression, the Great Recession, and our future#Role of the media|Role of the media]]", which cites research suggesting that both the increase in inequality and the slowing of the rate of economic growth can be attributed to the increased concentration of ownership of the major media including for-profit social media, which make money increasing political polarization and violence. == Caveat == Lindert and Williamson insist that Maddison's data are deficient, at least regarding the relative position of the 13 colonies that became the US: {{quote| American world leadership in income per person has waxed and waned for centuries. Before the twentieth century, the period in which Americans most clearly led Britain and all of western Europe in purchasing power per capita was during colonial times—that is, when North Americans were still British. They were already ahead by the late seventeenth century. America lost that lead in the Revolutionary War and the Articles of Confederation years, gained it back by 1860, lost most of it again in the Civil War decade, gained it back once more by 1900, and briefly lost it again in the Great Depression of the 1930s. ''Angus Maddison’s claim that American income per capita did not catch up to that of Britain until the start of the twentieth century seems to be off the mark by at least two centuries.'' Over the whole span of over 360 years since the mid-seventeenth century, America’s income advantage over Britain has not increased and may have decreased slightly. The only historical moment in which the United States soared far ahead of the rest of the world in average income came at the end of World War II. Since then, western Europe and Japan have been growing faster than the United States in terms of incomes per person. (emphasis added.)<ref>Lindert and Williamson (2016, pp. 8-9).</ref>}} This challenge to the numbers in the current analysis is vital for understanding the impact of armed conflict on the economy but may be irrelevant to the main point of this book, that media play a major role in helping humans understand what they should do to advance their interests. == Exercise == Share with others your thoughts on the issues raised in this discussion and summarize those discussions on the "Discuss" page associated with this chapter. Focus especially on how you managed your emotions and your relationships with the humans with whom you spoke as well as the strengths and weaknesses in the content of this chapter, challenges that should be addressed, and suggestions for improvement. == See also == * [[Most productive countries since 1349]] * [[The Media, the Great Depression, and our future]] == Notes == {{reflist}} == Bibliography == * [[d: Q125292212|Daron Acemoğlu and Simon Johnson (2023) ''Power and Progress: Our Thousand-Year Struggle Over Technology and Prosperity'' (PublicAffairs)]]. * [[d:Q126723821|Jutta Bolt and Jan Luiten van Zanden (2024) "Maddison style estimates of the evolution of the world economy: A new 2023 update", Journal of Economic Surveys, 1-41]]. * [[d:Q57945943|S. N. Broadberry, B. Campbell, A. Klein, M. Overton and B. van Leeuwen (2015) ''British Economic Growth 1270-1870'' (Cambridge University Press)]]. * [[d:Q137660377|Susan B. Carter, S. S. Gartner, M. R. Haineset (2006) ''Historical Statistics of the United States: Earliest Time to the Present'' (Cambridge University Press)]]. * [[d:Q137660514|Spencer Graves (2025-11-25) "MaddisonData: Maddison Project Data" software available from the Comprehensive R Archive Network (CRAN) and GitHub]]. * [[d:Q54641943|Richard R. John (1995) ''Spreading the News: The American Postal System from Franklin to Morse'' (Harvard University Press)]]. * [[d:Q131468166|Richard R. John; Jonathan Silberstein-Loeb, eds. (2015) ''Making News: The Political Economy of Journalism in Britain and America from the Glorious Revolution to the Internet'' (Oxford University Press)]]. * [[d:Q137669937|John J. McCusker (2006) "Colonial Statistics", Carter et al. (2006, V-671)]]. * [[d:Q135527962|Reece Peck (2016) "Usurping the usable past: How Fox News remembered the Great Depression during the Great Recession", ''Journalism'', 18(6)]]. * [[d:Q55878109|W. Scheidel and S. J. Friesen (2009) "The size of the economy and the distribution of income in the Roman Empire", ''Journal of Roman Studies'', 99, pp. 61–91]]. * [[d:Q137669960|J.P. Smits, E. Horlings and J.L. van Zanden (2000) Dutch GDP and its Components 1800-1913 (Groningen Growth and Development Centre)]]. * [[d:Q137669987|I. Sugimoto (2011) ''Economic growth of Singapore in the twentieth century: historical GDP estimates and empirical investigations'' ( World Scientific Publishing) * [[d: Q137670038|R. Sutch (2006). National Income and Product. Carter et al. (2006, III-23-25)]]. * [[d: Q137670058|J. L. van Zanden and B. van Leeuwen (2012), ‘Persistent but not consistent: the growth of national income in Holland 1347–1807’, Explorations in Economic History, 49, pp. 119–30]]. [[Category:Media literacy]] [[Category:Freedom and abundance]] [[Category:Communication]] [[Category:Political science]] [[Category:Law]] [[Category:Education]] [[Category:Economics]] [[Category:Media Literacy and You]] <!-- https://en.wikiversity.org/wiki/Category_Review --> ey8afa3p1fudaxa2fiok3kfx64xivt5 0 327588 2812011 2811776 2026-05-29T15:00:54Z DavidMCEddy 218607 put figs 1 and 2 first and add quote from Saez and Zucman 2812011 wikitext text/x-wiki [[File:US unemployment.svg|thumb|Figure 1. US unemployment 1800-2024.<ref>"unemployment" in the USGPDpresidents dataset in Croissant and Graves (2025). Various sources identified in the "help" file for USGPDpresidents including LNS14000000 from the Current Population Survey of the Bureau of Labor Statistics for numbers since 1940.</ref>]] [[File:US GDP per capita 1800-2024.svg|thumb|Figure 2. US average annual income (GDP per capita in 2017 K$) 1800-2024. The Herbert Hoover and Franklin D. Roosevelt (FDR) years present a very different image with GDP per capital falling at 8.1% per year during the Hoover presidency and growing at 8.1% per year during FDR. Between 1800 and 1929, the GDP per capita grew at 1.4% per year. Between 1945 and 2024, GDP per capita grew on average 1.7% per year.<ref>If we start at 1790 rather than 1800, then Measuring Worth has US GDP per capita growing at 1.5% per year. We could also add a breakpoint in 1947, which would have GDP per capita falling at 7.9% per year for 2 years and growing at 2% per year since. Data from Johnston and Samuel H. Williamson (2025). Available as "realGDPperCapita" in the USGPDpresidents dataset in Croissant and Graves (2025).</ref>]] :''I am entitled to my [[Wiktionary:cockamamie|cockamamie]] ideas, and you are entitled to yours.'' [Humor is important but must be offered in a way that does not offend others. If others are offended, they may be less interested in dialogue.] :This book is a combination instruction manual on [[w:Media literacy|media literacy]] and an invitation to you to support collaborative / crowd-sourced research on how to improve the world's understanding of media literacy and how to accelerate its understanding and use globally for the betterment of humanity. == Did Fox and the other major media make the Great Recession worse, or did Franklin Roosevelt (FDR) make the Great Depression worse? == During the [[w:2008 financial crisis|2008 financial crisis]] [[w:Fox News|Fox]] featured interviews with supposed experts, who claimed that the [[w:New Deal|New Deal]] policies of the [[w:Presidency of Franklin D. Roosevelt|Franklin D. Roosevelt (FDR) administration]] made the [[w:Great Depression|Great Depression]] worse, not better. That coverage -- and the lack of a substantive rebuttal in the other major media in the US -- reportedly played a major role in preventing the [[w:Presidency of Barack Obama|Obama administration]] from bailing out poor and middle-class humans who lost their homes at that time. This article plots data that visible challenge "evil New Deal" theory by showing that FDR's administration dramatically ''decreased'' unemployment and produced ''unprecedented'' growth in average annual income ([[w:Real gross domestic product|GDP per capita adjusted for inflation]]) with only nominal inflation. Everyone benefitted except the ultra-wealthy. But the ultra-wealthy in recent decades have controlled increasing portions of the money for the media, which may explain why the humans who accepted "[[w:Stated income loan|liar loans]]" were demonized while many banks that were too big to fail before the crisis were bigger after, and over five thousand finance industry leaders, many of whom pushed those fraudulent loans, got million dollar bonuses at taxpayer expense.<ref>Acemoglu and Johnson (2023, ch. 3).</ref> Leading economists in the [[w:Modern Monetary Theory|Modern Monetary Theory]] school insist that we ''can'' repeat the success of FDR's administration. == Introduction == Peck (2016)<ref>See also Peck (2019).</ref> describes how [[w:Fox News|Fox]] helped shape the debate in the US Congress about the proper response to the [[w:2008 financial crisis|2008 financial crisis]]. Fox's coverage included interviews with [[w:Amity Shlaes|Amity Shlaes]]<ref>See esp. Schlaes (2007).</ref> and other conservative authors and politicians pushing two images: # President Franklin Roosevelt's (FDR's) New Deal allegedly prolonged rather than shortened the Great Depression. # The victims of "Liar loans" were portrayed primarily as people of color begging for an unearned handout from government. Economists [[w:Emmanuel Saez|Emmanuel Saez]] and [[w:Gabriel Zucman|Gabriel Zucman]], leaders with [[w:Thomas Piketty|Thomas Piketty]] in studying inequality, say, "Contrary to what many ideologues would like you to believe, economics has not "proven" that workers "bear the burden" of the corporate income tax. If this were true, then unions all over the world would be begging governments to slash it. In the real world, the most vocal proponents of the view that ordinary workers—not wealthy shareholders—suffer from high corporate taxes are . . . wealthy shareholders. During the 2018 US midterm elections, lobbies supported by the Koch brothers (worth about $50 billion each) spent $20 million to convince voters that President Trump’s corporate tax cut was good for wages.<ref>Saez and Zucman (2019, p. 106).</ref> This chapter responds to the claim that the New Deal prolonged rather than shortened the Great Depression. First, a plot of unemployment between 1800 and 2024 shows a dramatic ''increase'' during the [[w:Presidency of Herbert Hoover|administration of Herbert Hoover]] (1929-1933) followed by effective correction during the [[w:Presidency of Franklin D. Roosevelt|Franklin D. Roosevelt (FDR) years]] (1933-1945). We also plot average annual income ([[w:Real gross domestic product|GDP per capita adjusted for inflation]]), which shows an unprecedented fall during the Hoover years followed by even more unprecedented growth during FDR. And we plot the income tax structure, showing that the ultra-wealthy paid higher taxes under FDR than at any other time in US history with plots showing reductions in inequality that declined from FDR until the inauguration of Ronald Reagan in 1981, when inequality started increasing again. Plots of inflation are noisier and harder to read, so we table growth and inflation comparing especially different wars in US history: This shows that previous wars had high inflation and only nominal growth while WW II had unprecedented growth with only nominal inflation. Regarding the impact of Fox's claims on the US government's reactions to the 2007-2009 international financial crisis, Acemoglu and Johnson (2023) describe how "The insurance company AIG was saved by a government support of $182 billion in the fall of 2008, yet it was allowed to pay nearly half a billion dollars in bonuses, including to people who had wrecked the company. ,,, [And] nine financial firms that were among the largest recipients of bailout money paid five thousand employee bonuses of more than $1 million per person—supposedly because this was needed to retain 'talent.'" Meanwhile, other options like "firing or prosecuting bankers who had broken the law—for example, by deceiving customers and contributing to the financial meltdown in the first place [and providing] greater assistance to home owners in distress" were not considered.<ref>For more on how the US political economy responds to violations of US law by major corporations, see the discussion of [[w:Deferred prosecution|deferred prosecution agreements]] in Starkman and Graves (2025) and Eisinger (2017).</ref> == Unemployment == Figure 1 plots US unemployment 1800 to 2024. This shows a dramatic increase during the administration of Herbert Hoover (1929-1933) followed by effective correction during the FDR's presidency (1933-1945). Schlaes (2007) quotes a few unemployment figures sprinkled throughout her book but does not plot them. [[w:List of Nobel Memorial Prize laureates in Economic Sciences|Nobel prize economist]] [[w:Paul Krugman|Paul Krugman]] accused Shlaes of disseminating "misleading statistics."<ref>Krugman (2008).</ref> Shlaes responded by saying that she used the Lebergott (1964) / Bureau of Labor Statistics (BLS) series.<ref>Shlaes (2008).</ref> However, her book does not include a table or plot of unemployment, though she does decorate the first page of each of her 15 chapters with a percent of the workforce unemployed on a specific month or day between 1927 and 1940. Her numbers are generally consistent with Figure 1.<ref>Figure 1 follows the Wikipedia article on "[[w:Unemployment in the United States|Unemployment in the United States]]", accessed 2025-12-01, in using Lebergott (1964) for 1800 - 1889, Romer (1986) for 1890 - 1929, Coen (1973) for 1930-1939, and the BLS since 1940.</ref> == Average annual income == Figure 2 plots average annual income in the US (GDP per capita) 1800 to 2024. This shows an unprecedented fall at 8 percent per year for the 4 years of the Hoover administration followed by an even more unprecedented increase at 8 percent per year for the ''12'' years of FDR. This raises questions about the claims of Shlaes (2007) and Fox's other guests on this topic.<ref>as described by Peck (2016).</ref> The data plotted in Figure 2 has US GDP per capita in 2017 dollars at 6,980.67 in 1933, more than doubling in 9 years to 14,819.07 by 1943, roughly doubling again in 33 years to 29,288.45 by 1976, doubling again in 39 years to 58,363.37 by 2015, according to [[w:MeasuringWorth|MeasuringWorth]].<ref>Johnston and Williamson (2025).</ref> Banerjee and Duflo, who shared the 2019 [[w:List of Nobel Memorial Prize laureates in Economic Sciences|Nobel Memorial Prize in Economics with Michael Kremer]], said "that despite the best efforts of generations of economists, the deep mechanisms of persistent economic growth remain elusive. No one knows" how to make economies grow.<ref>Banerjee and Duflo (2019, pp. 206-207).</ref> Acemoğlu and Johnson (2023) suggest that economies grow from encouraging commoners to become entrepreneurs and allowing broad segments of society to share in the benefits of productivity growth. [[w:Thomas Piketty|Thomas Piketty]], the world's leading expert on inequality, attributes the slowing of the rate of growth in the economy since 1990 to the increase in inequality.<ref>Piketty (2021, p. 139).</ref> However, the increase in consolidation of ownership of the major media including the rise of social media in recent decades could explain both the increase in inequality and the slowing of the rate of growth. == Income taxes == [[File:Historical US personal income tax-annotated.svg|thumb|Figure 3. Historical US personal income tax rates and brackets as a percent of taxable income (to 2021).<ref>Obtained by adding annotations to [[:File:Historical Income Tax Rates and brackets.png]].</ref>]] Figure 3 shows the history of personal income taxes in the US. This shows that income was taxed during the Civil War and for a few years after, but the US did not have substantive taxes on income until shortly before World War I. These tax rates were reduced after World War I and increased again during the Great Depression. For 1944 and 1945, late in World War II, the top rate was raised to an all-time high of 94% applied to income above $200,000 (equivalent to $3.57 million in 2024 dollars). It has generally trended down since the end of the war.<ref>The history of income taxes in the US appears in the section on "[[w:Income tax in the United States#History of top rates|History of top rates]]" in the Wikipedia article on "[[w:Income tax in the United States|Income tax in the United States]]", accessed 2025-12-01.</ref> But personal income taxes and the top bracket are only part of the story for at least two reasons: [[File:UStaxWords.svg|thumb|Figure 4. Millions of words in the US federal tax code and regulations, 1955-2015, according to the [[w:Tax Foundation|Tax Foundation]]. [1=income tax code; 2=other tax code; 3=income tax regulations; 4=other tax regulations; solid line= total]<ref>"UStaxWords" dataset in Croissant and Graves (2022) from the Tax Foundation.</ref>]] [[File:1960- Tax rates of richest versus low income people - US.svg|thumb|Figure 5. Total effective tax rates (includes ''all'' taxes: federal+state income tax, sales tax, property tax, etc) for the 400 richest Americans (just over one millionth of one percent) declined by 2018 to a level beneath that of the bottom 50% of earners,<ref name=CBSnews_20191017>Picci (2019).</ref> Analysis by economists [[w:Emmanuel Saez|Emmanuel Saez]] and [[w:Gabriel Zucman|Gabriel Zucman]]<ref>Saez and Zucman (2019).</ref>.]] # It applies to [[w:Adjusted gross income|adjusted gross income]], ''not'' gross income. This difference has increased dramatically in the 70 years since 1955, when the number of words in US federal tax code and regulations were reported as 1.4 million words. In 2015, there were 10.1 million words in US federal tax code and regulations, according to the [[w:Tax Foundation|Tax Foundation]], plotted in Figure 4. This suggests a massive increase in [[w:Tax break|tax loopholes]].<ref>"UStaxWords" dataset in Croissant and Graves (2022) from the Tax Foundation, which cite the Tax Foundation (2006) and Greenberg (2015). For alternative perspectives on this issue, see Bishop-Henchman (2014).</ref> Eisinger et al. (2021) with [[w:ProPublica|ProPublica]] reported that many billionaires like [[w:Jeff Bezos|Jeff Bezos]], [[w:Elon Musk|Elon Musk]], [[w:Michael Bloomberg|Michael Bloomberg]], [[w:Carl Icahn|Carl Icahn]], and [[w:George Soros|George Soros]], each paid ''zero'' federal income taxes several years when their fortunes grew dramatically. "IRS records show that the wealthiest can — perfectly legally — pay income taxes that are only a tiny fraction of the hundreds of millions, if not billions, their fortunes grow each year." Figure 5 shows how changes in governmental policies, including but not limited to those summarized in Figure 4, have impacted the effective tax rate paid by the 400 wealthiest individuals vs. the bottom 90 percent. # Taxes on corporations have declined from roughly 30 percent of all federal receipts in the early 1950s to roughly 10 percent in 2012.<ref>[[:File:Federal Receipts by Source.svg]], accessed 2025-12-01.</ref> What was the impact of FDR's policies on inequality? == Inequality == [[File:Share of post-tax US national income 50p97.svg|thumb|Figure 6. Shares of post-tax US national income for bottom half and top 3 percent, 1913-2023.<ref>Plots of percentile=='p0p50' and 'p97p100' for variable == 'sdiincj999' in the US data in the [[w:World Inequality Database|World Inequality Database]] (WID) using the WID package for R described by Graves (2025).</ref>]] [[File:Share of US wealth 90p99.svg|thumb|Figure 7. Shares of US wealth - bottom 90 and top 1 percent, 1820-2023.<ref>Plots of percentile=='p0p90' and 'p99p100' for variable == 'shwealj999' in the US data in the World Inequality Database (WID) using the WID package for R described by Graves (2025).</ref>]] Figures 6 and 7 show inequality of income and wealth in the US. Figure 6 plots the evolution of the shares of the bottom half and top 3 percent of post-tax US national income from 1913 to 2023. Figure 7 shows the evolution of the bottom 90 and top 1 percent of US national wealth from 1820 to 2023. Both show roughly the same image: High inequality dramatically reduced during World War II and continuing after the war with the US on average tending to become slightly more egalitarian until Ronald Reagan became President of the US in 1981. Lindert and Williamson report that, "Incomes were more equally distributed in colonial America than in any other place that can be measured."<ref>{{harvnb|Lindert|Williamson|2016|p=37}}</ref> Inequality increased after the Revolution to produce the effects documented in Figures 6 and 7, which include the "great leveling" that began after the Great Depression. Figures 6 and 7 show that the presidency of Ronald Reagan initiated a reversal of that "great leveling". Lindert and Williamson continue, "Our new inequality evidence for 1774 also speaks to a new institutional literature that argues that :''economic inequality breeds political power that favors rent-seeking (or extractive) institutions and policies rather than growth-enhancing institutions and policies, while a large middle class does just the opposite.'' (emphasis added)<ref>Lindert and Williamson (2016, p. 41).</ref> Conclusion: :''When politicians are allowed to reward people they call 'job creators', the humans who actually create most of the jobs and the bottom 99 percent suffer.'' We can reverse the trend toward increasing inequality in a couple of ways. * First more equitably fund fair application of the laws. Eisinger (2017) describes "why the [US] Justice Department fails to prosecute executives", and with progressive taxes on income and [[w:Wealth tax|wealth]], both for individuals and corporations. == Wartime Growth and inflation == Economists and leading politicians have long understood that inflation was often a problem during wars. During the [[w:Napoleonic Wars|Napoleonic Wars]], the Prime Minister of the UK, [[w:William Pitt the Younger|William Pitt]], reportedly said he was more afraid of high prices than he was of the enemy.<ref>Sabaté and Torregrosa-Hetland (2024).</ref> This author has so far failed to find a reference discussing productivity growth, like that visible during World War II in Figure 2 above. Rockoff (2015) provides estimates of inflation during the [[w:American Revolution|American Revolution]], the [[w:War of 1812|War of 1812]], the [[w:American Civil War|American Civil War]], and World Wars I and II. The [[w:MeasuringWorth|MeasuringWorth]] data plotted in Figure 2 above starts in 1790, after the end of the American Revolution. Table 1 summarizes economic growth and inflation during the War of 1812, the Civil War and World Wars I and II: The first three of those wars had economic growth comparable to non-war years and exceptionally high inflation. During World War II, the US had the opposite: unprecedented economic growth with only nominal inflation. In addition to unprecedented income taxes, summarized in Figure 3 above, FDR's administration also had waged and price controls managed by the [[w:Office of Price Administration|Office of Price Administration]] (OPA) that recruited many volunteers to help manage the program. We will not attempt here to assess the relative contribution of higher taxes and the OPA to controlling inflation during World War II, apart from noting that prices jumped on average 6 percent only a few days after the OPA ceased operations, a monthly increase that would have produced 100 percent inflation if continued for a year. However, less than a month later, the US Congress passed legislation to reopen the OPA, and inflation slowed.<ref>Jacobs (1997) and Cohen (2008), cited from the Wikipedia article on "[[w:Office of Price Administration|Office of Price Administration]]".</ref> {| class="wikitable" |+ Table 1. Economic growth and inflation in major wars in US history |- ! war !! colspan=2 | start !! colspan=2 | end !! colspan=2 | annual rate of |- ! !! date !! year !! date !! year !! growth in real GDP per capita !! inflation |- | [[w:War of 1812|War of 1812]] || 1812-06-18 || 1812 || 1815-02-17 || 1814 || 1.8% || 10.6%<ref>The War of 1812 was followed by dramatic deflation and a major recession. Thus, if we change the end year from 2014 to 2015, the economic growth and inflation reported here disappear.</ref> |- | [[w:American Civil War|Civil War]] || 1861-04-12 || 1861 || 1865-06-26 || 1865 || 4.3% || 14.3% |- | [[w:World War I|WW I]] || 1917-04-02 || 1917 || 1918-11-11 || 1918 || 4.2% || 13.7%<ref>WW I began in Europe 1914-07-28. Between 1914 and 1917, the US economy averaged 7.8% growth per year in real GDP per capita with 16.5% annual inflation. Different numbers. Same general conclusion.</ref> |- | [[w:World War II|WW II]] || 1941-12-07 || 1941 || 1945-09-02 || 1945 || 9.1% || 4.5%<ref>WW II began in Europe 1939-09-01. Between 1939 and 1945, the US economy averaged 10.1% growth per year in real GDP per capita with 4.2% inflation. Different numbers. Same general conclusion.</ref> |} Economists in the [[w:Modern Monetary Theory|Modern Monetary Theory]] (MMT) school support [[w:job guarantee|job guarantees]] like the New Deal programs, while more traditional economists prefer a [[w:guaranteed minimum income|guaranteed minimum income]]. When humans are unemployed, their general health and well being tends to decline, they often lose self esteem<ref>Green (2010).</ref> and good work habits.<ref>Hult et al. (2018).</ref> And employers are less likely to request interviews with applicants who have been unemployed a year or more.<ref>Farber et al. (2018).</ref> These arguments favor a job guarantee over a guaranteed minimum income. But many elites seem to prefer to maintain a large reserve army of unemployed to limit the ability of employees to bargain for better wages and working conditions.<ref>Mitchell et al. (2016, esp. sections 12.3. Unemployment buffer stocks and price stability and 12.4. Employment buffer stocks and price stability, pp. 247-259).</ref> European countries led by Denmark are using "[[w:Flexicurity|flexicurity]]<ref>accessed 2025-12-20.</ref> systems that provide generous unemployment and support for adult education for workers while providing employers greater flexibility in expanding and contracting their workforce in response to changes in demand. == Role of the media == How did FDR get the political support needed to tax the ultra-wealthy and create the Office of Price Administration that generated unprecedented economic growth with only nominal inflation, as described above? One possible answer is given in the research by [[w:Daron Acemoglu|Acemoglu]], [[w:Simon Johnson (economist)|Johnson]], and [[w:James A. Robinson|Robinson]], who shared the 2024 [[w:Nobel Memorial Prize in Economic Sciences|Nobel Memorial Prize in Economics]],<ref>Royal Swedish Academy of Sciences (2024).</ref> combined with research on the role of the media in political economy. Acemoglu and Johnson (2023, ch. 4) said that {{quote| Medieval society is often described as a “society of orders,” consisting of * those who fought, * those who prayed, and * those who did all the work. Those who prayed were crucial in persuading those who labored to accept this hierarchy.<ref>Acemoglu and Johnson note that this description applies to many other societies in history and prehistory, e.g., when the [[w:Egyptian pyramids|pyramids]] were built in [[w:Ancient Egypt|Ancient Egypt]] but did not apply elsewhere. See also Graeber and David Wengrow (2021).</ref>}} Acemoglu and Robinson (2012) suggest that the [[w:Industrial Revolution|Industrial Revolution]] began in England, because the English were the first to extend equal protection of the laws to innovative commoners. At other times and places -- including in many countries today -- innovators who threaten powerful individuals and groups can have their innovations blocked,<ref>In 1707 [[w:Denis Papin|Denis Papin]] reportedly built a ship powered by hand-cranked paddles that was destroyed by boatmen of [[w:Hann. Münden|Munden]] who feared it would threaten their livelihood. He left his family in Germany and went to England, where the Royal Society published several of his papers before he died a pauper and was buried in an unmarked grave.</ref> or the fruits of their labors confiscated by members of the first two orders or even imprisoned.<ref>[[w:Jimmy Lai|Jimmy Lai]] is Hong Kong businessman and media figure, imprisoned over his criticism of the Chinese Communist Party.</ref> Acemoglu and Johnson (2023) further insist that the ''inequality'' is to a large extent a function not of technology but of political power, and we can have a high rate of economic growth with lower inequality, as suggested by Figures 2, 4 and 6 above. They provide a template for doing this based on # altering the narrative, # building countervailing powers [like organized labor], and # developing technical, regulatory, and policy solutions to tackle specific aspects of technology’s social bias.<ref>Acemoglu and Johnson (2023, ch. 11).</ref> "Altering the narrative" implies a major role for the media. But media outlets have conflicts of interest in honestly reporting on anything that might offend (a) anyone with substantive control of the money for the media or (b) major news sources like public officials, including law enforcement. Usher and Kim-Leffingwell (2022) found on average 1.4 more federal prosecutions for political corruption in each of the 94 US federal court districts between 2003 and 2019 per member of the Institute for Nonprofit News (INN) in that district the previous year. During that period, the number of journalists in the US fell by roughly a factor of 3 -- between 60 and 70 percent -- with no statistically significant impact on federal prosecutions for political corruption. They did not describe the specific mechanisms connecting INN members to prosecutions for political corruption, but major media outlets often disseminate news produced by members of INN, because they could lose audience if they don't, and their advertising rates are a function of their audience. More support for local news nonprofits like members of INN may also make it easier to build countervailing powers and disseminate research on policy alternatives that rarely appear in major media outlets. A more diverse media landscape would reduce the impact of decisions like those of [[w:YouTube|YouTube]] to delete videos posted by Palestinian human rights organizations documenting questionable actions by Israelis.<ref>The Cradle (2025).</ref> For a summary of research on media reform, see the Wikiversity article on "[[Media & Democracy lessons for the future]]".<ref>accessed 2025-12-20.</ref> == Rebuilding the 99 percent == Saez and Zucman, responsible for Figure 5 above, said, "what makes taxation work is more than a simple tax code and diligent auditors. It’s a belief system: shared convictions in the benefits of collective action ..., in government’s central role in organizing this collective action, and in the merits of democracy. When this belief system prevails, even the most progressive tax system can work. When this belief system founders, the forces of tax dodging, unleashed and legitimized, can overwhelm even the most sophisticated tax authority and overpower the best tax code."<ref>Saez and Zucman (2019, pp. 47-48).</ref> To support this, they quoted from President Franklin D. Roosevelt's message to Congress 1937-06-01: {{quote| Mr. Justice Holmes said, ‘Taxes are what we pay for civilized society’. Too many individuals, however, want the civilization at a discount.<ref>Saez and Zucman (2019, p. 48).</ref>}} From that day to the 1970s, business executives agreed that they were "responsible to a broad class of stakeholders beyond their owners: employees, customers, communities, and governments."<ref>Saez and Zucman (2019, p. 69).</ref> In the 1970s the tax-avoidance industry began to grow, but it didn't really take off until Ronald Reagan became president, insisting that, {{quote| Government is not the solution to our problem; government is the problem.<ref>Saez and Zucman (2019, p. 51).</ref>}} Saez and Zucman said that "the revived libertarian creed", popularized with Reagan, included the claim that "taxation was theft". That change in mindset meant that tax avoidance, previously immoral, became moral, even mandatory where feasible.<ref>Saez and Zucman (2019, p. 51).</ref> You, dear reader, can help restore the mindset that drove the decrease in inequality visible in Figures 6 and 7 through media literacy activism. This ''[[Media Literacy and You]]'' book is being written in the hope that it can inspire and support such activism. == Caveats == === Empirical evidence is never complete === Statistician and management consultant [[w:W. Edwards Deming|W. E. Deming]] said, "Empirical evidence is never complete." He also said that there is no true value to any number obtained as a result of a measurement: If you change the method of measurement, you get a different answer.{{cn}} Also, humans often do not see things that they do not expect. For example, many experimental subjects asked to count passes in a video of a basketball game failed to notice a person in a gorilla suit who appears in the middle of the video.<ref>This was discussed in research reports and a companion book, ''[[w:The Invisible Gorilla|The Invisible Gorilla]]''.</ref> Estimating GDP including adjusting for inflation is difficult. Different researchers use different methods and get different answers. In particular, Lindert and Williamson insist that Maddison's data are deficient, at least regarding the 13 colonies that became the US:{{quote| American world leadership in income per person has waxed and waned for centuries. Before the twentieth century, the period in which Americans most clearly led Britain and all of western Europe in purchasing power per capita was during colonial times—that is, when North Americans were still British. They were already ahead by the late seventeenth century. America lost that lead in the Revolutionary War and the Articles of Confederation years, gained it back by 1860, lost most of it again in the Civil War decade, gained it back once more by 1900, and briefly lost it again in the Great Depression of the 1930s.<ref>Lindert and Williamson (2016, pp. 8-9).</ref>}} The GDP per capita numbers used in this chapter are from [[w:MeasuringWorth|MeasuringWorth]], which are similar but different the GDP per capita numbers from the [[w:Maddison Project|Maddison Project]], used in the chapter on [[Media Literacy and You/The impact of the media on political economy since the time of the Pharaohs|The impact of the media on political economy since the time of the Pharaohs]]. The differences are critical for evaluating the macroeconomic impact of wars but do not otherwise seem relevant to the main thrust of this book. === We need efficient capital markets but not hyper-liquidity === [[w:James Tobin|James Tobin]] won the [[w:List of Nobel Memorial Prize laureates in Economic Sciences|1981 Nobel memorial prize in economics]] for his analysis of financial markets, including recommending taxing financial market transactions. That idea is now known as a "[[w:Tobin tax|Tobin tax]]". He recommended a tax of, e.g., 0.5 percent of the volume of a transaction to dissuades speculators from investing money on very short-term bases, because of their contribution to [[w:Stock market bubble|market bubbles]]. We need liquidity in financial markets but not hyper-liquidity. == Exercise == Share your understanding of the information in this chapter with others, inviting their comments. Stress that no human knows the "truth" about anything as complex as the issues discussed herein and invite feedback. # As before, the primary goal is ''not'' to convince anyone else of anything. Rather it is to build relationships of mutual respect in which humans can agree to disagree disagreeably. If enough humans do this, it will (a) reduce political polarization and violence and (b) facilitate progress on the issues of greatest concern to the most humans. # Summarize what you hear in the ''Discuss'' page associated with this chapter. If you see opportunities to improve this chapter and change this chapter while writing from a neutral point of view citing credible sources, do so. Or at least document those thoughts on the companion ''Discuss'' page. == Appendix. Companion R Markdown vignette == Statistical details that make [[w:Reproducibility|the research in article reproducible]] are provided in an R Markdown vignette on "[[The Media, the Great Depression, and our future/Companion R Markdown vignette]]". <!--== See also ==--> == Notes == {{reflist}} == Bibliography == * <!--Daron Acemoğlu and Simon Johnson (2023) Power and Progress-->{{cite Q|Q125292212}} * <!--Abhijit Banerjee and Esther Duflo (2019) Économie utile pour des temps difficiles-->{{cite Q|Q85764011}} * <!--Joseph Bishop-Henchman (2014-04-15) How Many Words are in the Tax Code?-->{{cite Q|Q137462713}} * <!--Robert Coen (1973) Labor Force and Unemployment in the 1920s and 1930s: A Re-Examination Based on Postwar Experience-->{{cite Q|Q137180971}} * <!--Lizabeth Cohen (2003, 2008) Consumers' Republic: The Politics of Mass Consumption in Postwar America-->{{cite Q|Q137473626}} * <!--The Cradle (2025-11-05) "YouTube deletes hundreds of videos documenting Israeli war crimes"-->{{cite Q|Q137301573|author=The Cradle}} * <!-- Yves Croissant and Spencer Graves (2025) "Ecdat: Data Sets for Econometrics", available from the Comprehensive R Archive Network (CRAN) -->{{cite Q|Q56452356}} * <!--Jesse Eisinger (2017) The chickenshit club : why the Justice Department fails to prosecute executives-->{{cite Q|Q134599351}} * <!--Jesse Eisinger, Jeff Ernsthausen, and Paul Kiel (2021-06-08) "The Secret IRS Files: Trove of Never-Before-Seen Records Reveal How the Wealthiest Avoid Income Tax"-->{{cite Q|Q139919526}} * <!--Henry S. Farber, Chris M. Herbst, Dan Silverman, and Till von Wachter (2018-05) " Whom Do Employers Want? The Role of Recent Employment and Unemployment Status and Age-->{{cite Q|Q105837471}} * <!--Pam Fessler (2017-05-25) "Housing Secretary Ben Carson Says Poverty Is A 'State Of Mind'"-->{{cite Q|Q137475571|author=Pam Fessler}} * <!--David Graeber and David Wengrow (2021) The Dawn of Everything (Q109769508). * <!--Spencer Graves (2025) WID: Tools for use with the World Inequality Database-->{{cite Q|Q137462795}} * <!--Francis Green (2010-12-22) "Unpacking the misery multiplier: how employability modifies the impacts of unemployment and job insecurity on life satisfaction and mental health"-->{{cite Q|Q50528452}} * <!-- Scott Greenberg (2015-10-08) Federal Tax Laws and Regulations are Now Over 10 Million Words Long-->{{cite Q|Q137462350}} * <!--Marja Hult, Anna-Maija Pietilä, Päivikki Koponen, and Terhi Saaranen (2018-07-26) " Association between good work ability and health behaviours among unemployed: A cross-sectional survey"-->{{cite Q|Q91470779}} * <!--Meg Jacobs (1997-12) ""How About Some Meat?": The Office of Price Administration, Consumption Politics, and State Building from the Bottom Up, 1941–1946-->{{cite Q|Q137473579}} * <!-- Louis Dorrance Johnston and Samuel H. Williamson (2025) "What Was the U.S. GDP Then?"-->{{cite Q|Q56881105}} * <!--Paul Krugman (2008-11-19) "Amity Shlaes strikes again"-->{{cite Q|Q137179834}} * <!--Stanley Lebergott (1964) Manpower in Economic Growth: The American Record since 1800-->{{cite Q|Q137180737}} * <!--Peter H. Lindert and Jeffrey G. Williamson (2016) Unequal Gains: American Growth and Inequality since 1700 (Princeton U. Pr.)-->{{cite Q|Q138296699}} * <!--Bill Mitchell, L. Randall Wray, and Martin Watts (2016) Modern Monetary Theory and Practice: An introductory text-->{{cite Q|Q137485438}} * <!--Reece Peck (2016) "Usurping the usable past: How Fox News remembered the Great Depression during the Great Recession", Journalism-->{{cite Q|Q135527962}} * <!--Reece Peck (2019) Fox populism: Branding conservatism as working class (Cambridge U. Pr.)-->{{cite Q|Q135513426}} * <!--Aimee Picci (2019-10-17) America's richest 400 families now pay a lower tax rate than the middle class-->{{cite Q|Q139935046}} * <!-- Thomas Piketty (2022) A brief history of equality (Harvard U. Pr.) -->{{cite Q|Q115434513}} * <!--Christina Romer (1986) "Spurious Volatility in Historical Unemployment Data"-->{{cite Q|Q55899853}} * <!--Royal Swedish Academy of Sciences (2024-10-20) "Prize in Economic Sciences in Memory of Alfred Nobel 2024"-->{{cite Q|Q130312646|author=Royal Swedish Academy of Sciences}} * <!--Oriol Sabaté and Sara Torregrosa-Hetland (2024-02) War inflation and taxation-->{{cite Q|Q137465618}} * <!--Emmanuel Saez and Gabriel Zucman (2019) The Triumph of Injustice: How the rich dodge taxes and how to make them pay-->{{cite Q|Q133176715}} * <!-- Amity Shlaes (2008) The Krugman Recipe for Depression: Massive government spending is no solution to unemployment-->{{cite Q|Q137179924}} * <!-- Amity Shlaes (2007) The Forgotten Man: A New History of the Great Depression-->{{cite Q|Q7734832}} * [[d:Q138037937|Dean Starkman and Spencer Graves (2025) "Dean Starkman and the watchdog that didn't bark anglais" on Wikiversity]]. * <!--Tax Foundation(2006-10-26) Number of Words in Internal Revenue Code and Federal Tax Regulations, 1955-2005-->{{cite Q|Q137462681|author = Tax Foundation}} [[Category:Original research]] [[Category:Research]] [[Category:Great Depression]] [[Category:Macroeconomics]] [[Category:Gross domestic product]] [[Category:Economic growth]] [[Category:Media literacy]] [[Category:Communication]] [[Category:Political science]] [[Category:Law]] [[Category:Psychology]] [[Category:Sociology]] [[Category:Education]] [[Category:Media Literacy and You]] <!-- https://en.wikiversity.org/wiki/Category_Review --> dmskw142eft9n8or6c7m3r9i7xlvm8k 0 329377 2812008 2811945 2026-05-29T14:58:21Z Unitfreak 695864 /* The New Moon Solstice */ 2812008 wikitext text/x-wiki [[Bully_Metric|Bully Metric Main Page]]<br /> [[Bully_Metric_Timestamps|Bully Metric Timestamps Main Page]]<br /> [[Bully_Metric_Metonic_cycle|The Metonic Cycle in Bully Metric]]<br /> [https://unitfreak.github.io/Bully-Row-Timestamps/Java_Bully.html Current Bully Timestamp (GitHub)]<br /> The '''Metonic cycle''' is a period of approximately 19 solar years, after which the Moon's phases recur on the same calendar dates. This cycle arises because 19 solar years and 235 synodic months nearly coincide. [[File:Bully_Metric_Metonic_Cycle.png|thumb|center|650 px| Figure 1: The Moon’s phase "advances" by approximately {{frac|7|19}} of a lunar cycle when observed on the same day in subsequent years.]] As shown in Figure 1, the Moon’s phase "advances" by approximately {{frac|7|19}} of a lunar cycle when observed on the same day in subsequent years. For example, if a '''New Moon''' occurs on the December Solstice of 2014: * The 2015 solstice will feature a '''Waxing Gibbous Moon''' (an advancement of ~{{frac|7|19}}). * The 2016 solstice will feature a '''Third Quarter Moon''' (an advancement of ~{{frac|14|19}}). * The 2017 solstice will feature a '''Waxing Crescent Moon''' (an advancement of ~{{frac|21|19}}). * And the 2033 solstice will feature a '''New Moon''' (an advancement of 7 complete cycles). Within this 19-year span, significant "near-matches" occur at the 8-year and 11-year marks. At 8 years, the drift reaches {{frac|56|19}} (approx. 2.95 cycles); at 11 years, it reaches {{frac|77|19}} (approx. 4.05 cycles). These intervals represent points where the lunar-solar alignment falls just short or just past a full-integer "reset," which eventually concludes at the 19-year mark. === The New Moon Solstice === The darkest nights in the Northern Hemisphere occur when the '''December Solstice''' coincides with a '''New Moon'''. The darkest nights in the Southern Hemisphere occur when the '''June Solstice''' coincides with a '''New Moon'''. The Metonic cycle predicts this alignment every 19 years, with significant "near-matches" at the 8th- and 11th-year marks. Table 1 illustrates Metonic cycles over a one-century period (1996–2109), listing the approximate date and Bully timestamp for every New Moon during the century. Red cells indicate the New Moon Solstice alignment every 19 years. Yellow cells indicate the New Moon Solstice near-alignments on the 8th- and 11th-year marks of the Metonic cycle. {| class="wikitable" style="text-align:center; width:100%; max-width:800px; font-size: small; font-family: monospace, monospace;" |+ Table 1: New Moon Bully Timestamps 1996 .. 2102 |- style="background-color: #eaecf0; font-size: medium; font-weight: bold;" ! rowspan="2" style="padding: 10px; font-size: large;" | Metonic Cycle ! colspan="7" style="padding: 10px;" | Every New Moon (1996 .. 2102) |- style="background-color: #eaecf0; font-size: large; font-weight: bold;" || {{nowrap|1996}} || {{nowrap|2015}} || {{nowrap|2034}} || {{nowrap|2053}} || {{nowrap|2072}} || {{nowrap|2091}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jan 20}} || {{nowrap|8209 27FF 9E74}} || {{nowrap|8209 2802 9D27}} || {{nowrap|8209 2805 9BD4}} || {{nowrap|8209 2808 9A79}} || {{nowrap|8209 280B 9917}} || {{nowrap|8209 280E 97B8}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Feb 18}} || {{nowrap|8209 27FF A1B5}} || {{nowrap|8209 2802 A067}} || {{nowrap|8209 2805 9F18}} || {{nowrap|8209 2808 9DC2}} || {{nowrap|8209 280B 9C62}} || {{nowrap|8209 280E 9B01}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Mar 19}} || {{nowrap|8209 27FF A4F7}} || {{nowrap|8209 2802 A3A7}} || {{nowrap|8209 2805 A259}} || {{nowrap|8209 2808 A107}} || {{nowrap|8209 280B 9FAC}} || {{nowrap|8209 280E 9E4A}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Apr 17}} || {{nowrap|8209 27FF A839}} || {{nowrap|8209 2802 A6E6}} || {{nowrap|8209 2805 A598}} || {{nowrap|8209 2808 A449}} || {{nowrap|8209 280B A2F3}} || {{nowrap|8209 280E A193}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|May 17}} || {{nowrap|8209 27FF AB7C}} || {{nowrap|8209 2802 AA25}} || {{nowrap|8209 2805 A8D6}} || {{nowrap|8209 2808 A788}} || {{nowrap|8209 280B A635}} || {{nowrap|8209 280E A4DA}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jun 16}} || {{nowrap|8209 27FF AEC1}} || {{nowrap|8209 2802 AD65}} || {{nowrap|8209 2805 AC12}} || {{nowrap|8209 2808 AAC4}} || {{nowrap|8209 280B A975}} || {{nowrap|8209 280E A81E}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jul 15}} || {{nowrap|8209 27FF B206}} || {{nowrap|8209 2802 B0A6}} || {{nowrap|8209 2805 AF50}} || {{nowrap|8209 2808 AE00}} || {{nowrap|8209 280B ACB3}} || {{nowrap|8209 280E AB60}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Aug 14}} || {{nowrap|8209 27FF B54C}} || {{nowrap|8209 2802 B3EA}} || {{nowrap|8209 2805 B28F}} || {{nowrap|8209 2808 B13D}} || {{nowrap|8209 280B AFEF}} || {{nowrap|8209 280E AEA0}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Sep 12}} || {{nowrap|8209 27FF B893}} || {{nowrap|8209 2802 B731}} || {{nowrap|8209 2805 B5D2}} || {{nowrap|8209 2808 B47C}} || {{nowrap|8209 280B B32D}} || {{nowrap|8209 280E B1DF}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Oct 11}} || {{nowrap|8209 27FF BBD9}} || {{nowrap|8209 2802 BA7A}} || {{nowrap|8209 2805 B918}} || {{nowrap|8209 2808 B7BD}} || {{nowrap|8209 280B B66B}} || {{nowrap|8209 280E B51E}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Nov 10}} || {{nowrap|8209 27FF BF1E}} || {{nowrap|8209 2802 BDC3}} || {{nowrap|8209 2805 BC61}} || {{nowrap|8209 2808 BB02}} || {{nowrap|8209 280B B9AC}} || {{nowrap|8209 280E B85D}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Dec 9}} || {{nowrap|8209 27FF C261}} || {{nowrap|8209 2802 C10B}} || {{nowrap|8209 2805 BFAC}} || {{nowrap|8209 2808 BE4A}} || {{nowrap|8209 280B BCEF}} || {{nowrap|8209 280E BB9D}} |- style="background-color: #eaecf0; font-size: large; font-weight: bold;" || || {{nowrap|1997}} || {{nowrap|2016}} || {{nowrap|2035}} || {{nowrap|2054}} || {{nowrap|2073}} || {{nowrap|2092}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jan 8}} || {{nowrap|8209 27FF C5A2}} || {{nowrap|8209 2802 C451}} || {{nowrap|8209 2805 C2F6}} || {{nowrap|8209 2808 C194}} || {{nowrap|8209 280B C035}} || {{nowrap|8209 280E BEDF}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Feb 7}} || {{nowrap|8209 27FF C8E3}} || {{nowrap|8209 2802 C794}} || {{nowrap|8209 2805 C63F}} || {{nowrap|8209 2808 C4E0}} || {{nowrap|8209 280B C37E}} || {{nowrap|8209 280E C223}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Mar 8}} || {{nowrap|8209 27FF CC23}} || {{nowrap|8209 2802 CAD6}} || {{nowrap|8209 2805 C984}} || {{nowrap|8209 2808 C82A}} || {{nowrap|8209 280B C6C8}} || {{nowrap|8209 280E C569}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Apr 6}} || {{nowrap|8209 27FF CF63}} || {{nowrap|8209 2802 CE15}} || {{nowrap|8209 2805 CCC6}} || {{nowrap|8209 2808 CB70}} || {{nowrap|8209 280B CA11}} || {{nowrap|8209 280E C8AF}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|May 6}} || {{nowrap|8209 27FF D2A3}} || {{nowrap|8209 2802 D153}} || {{nowrap|8209 2805 D005}} || {{nowrap|8209 2808 CEB3}} || {{nowrap|8209 280B CD58}} || {{nowrap|8209 280E CBF6}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jun 5}} || {{nowrap|8209 27FF D5E3}} || {{nowrap|8209 2802 D490}} || {{nowrap|8209 2805 D342}} || {{nowrap|8209 2808 D1F3}} || {{nowrap|8209 280B D09C}} || {{nowrap|8209 280E CF3D}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jul 4}} || {{nowrap|8209 27FF D925}} || {{nowrap|8209 2802 D7CD}} || {{nowrap|8209 2805 D67E}} || {{nowrap|8209 2808 D530}} || {{nowrap|8209 280B D3DE}} || {{nowrap|8209 280E D283}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Aug 2}} || {{nowrap|8209 27FF DC69}} || {{nowrap|8209 2802 DB0D}} || {{nowrap|8209 2805 D9BB}} || {{nowrap|8209 2808 D86D}} || {{nowrap|8209 280B D71E}} || {{nowrap|8209 280E D5C7}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Sep 1}} || {{nowrap|8209 27FF DFB0}} || {{nowrap|8209 2802 DE50}} || {{nowrap|8209 2805 DCF9}} || {{nowrap|8209 2808 DBAA}} || {{nowrap|8209 280B DA5C}} || {{nowrap|8209 280E D90A}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Oct 1}} || {{nowrap|8209 27FF E2F8}} || {{nowrap|8209 2802 E196}} || {{nowrap|8209 2805 E03A}} || {{nowrap|8209 2808 DEE8}} || {{nowrap|8209 280B DD9A}} || {{nowrap|8209 280E DC4B}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Oct 30}} || {{nowrap|8209 27FF E640}} || {{nowrap|8209 2802 E4DE}} || {{nowrap|8209 2805 E37F}} || {{nowrap|8209 2808 E228}} || {{nowrap|8209 280B E0D9}} || {{nowrap|8209 280E DF8B}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Nov 29}} || {{nowrap|8209 27FF E987}} || {{nowrap|8209 2802 E829}} || {{nowrap|8209 2805 E6C7}} || {{nowrap|8209 2808 E56B}} || {{nowrap|8209 280B E419}} || {{nowrap|8209 280E E2CB}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Dec 28}} || {{nowrap|8209 27FF ECCD}} || {{nowrap|8209 2802 EB73}} || {{nowrap|8209 2805 EA11}} || {{nowrap|8209 2808 E8B1}} || {{nowrap|8209 280B E75B}} || {{nowrap|8209 280E E60C}} |- style="background-color: #eaecf0; font-size: large; font-weight: bold;" || || {{nowrap|1998}} || {{nowrap|2017}} || {{nowrap|2036}} || {{nowrap|2055}} || {{nowrap|2074}} || {{nowrap|2093}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jan 27}} || {{nowrap|8209 27FF F010}} || {{nowrap|8209 2802 EEBB}} || {{nowrap|8209 2805 ED5C}} || {{nowrap|8209 2808 EBFB}} || {{nowrap|8209 280B EA9F}} || {{nowrap|8209 280E E94D}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Feb 25}} || {{nowrap|8209 27FF F352}} || {{nowrap|8209 2802 F201}} || {{nowrap|8209 2805 F0A7}} || {{nowrap|8209 2808 EF45}} || {{nowrap|8209 280B EDE5}} || {{nowrap|8209 280E EC8F}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Mar 27}} || {{nowrap|8209 27FF F692}} || {{nowrap|8209 2802 F543}} || {{nowrap|8209 2805 F3EE}} || {{nowrap|8209 2808 F28F}} || {{nowrap|8209 280B F12D}} || {{nowrap|8209 280E EFD2}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Apr 25}} || {{nowrap|8209 27FF F9D0}} || {{nowrap|8209 2802 F882}} || {{nowrap|8209 2805 F731}} || {{nowrap|8209 2808 F5D6}} || {{nowrap|8209 280B F474}} || {{nowrap|8209 280E F315}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|May 25}} || {{nowrap|8209 27FF FD0D}} || {{nowrap|8209 2802 FBBF}} || {{nowrap|8209 2805 FA70}} || {{nowrap|8209 2808 F91A}} || {{nowrap|8209 280B F7BB}} || {{nowrap|8209 280E F659}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jun 23}} || {{nowrap|8209 2800 004B}} || {{nowrap|8209 2802 FEFB}} || {{nowrap|8209 2805 FDAE}} || {{nowrap|8209 2808 FC5C}} || {{nowrap|8209 280B FB01}} || {{nowrap|8209 280E F99F}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jul 23}} || {{nowrap|8209 2800 038B}} || {{nowrap|8209 2803 0238}} || {{nowrap|8209 2806 00EA}} || {{nowrap|8209 2808 FF9B}} || {{nowrap|8209 280B FE45}} || {{nowrap|8209 280E FCE6}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Aug 21}} || {{nowrap|8209 2800 06CE}} || {{nowrap|8209 2803 0576}} || {{nowrap|8209 2806 0427}} || {{nowrap|8209 2809 02D9}} || {{nowrap|8209 280C 0187}} || {{nowrap|8209 280F 002C}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Sep 20}} || {{nowrap|8209 2800 0A14}} || {{nowrap|8209 2803 08B8}} || {{nowrap|8209 2806 0765}} || {{nowrap|8209 2809 0617}} || {{nowrap|8209 280C 04C8}} || {{nowrap|8209 280F 0371}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Oct 19}} || {{nowrap|8209 2800 0D5C}} || {{nowrap|8209 2803 0BFC}} || {{nowrap|8209 2806 0AA5}} || {{nowrap|8209 2809 0955}} || {{nowrap|8209 280C 0807}} || {{nowrap|8209 280F 06B5}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Nov 18}} || {{nowrap|8209 2800 10A6}} || {{nowrap|8209 2803 0F44}} || {{nowrap|8209 2806 0DE8}} || {{nowrap|8209 2809 0C95}} || {{nowrap|8209 280C 0B47}} || {{nowrap|8209 280F 09F8}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Dec 17}} || {{nowrap|8209 2800 13EF}} || {{nowrap|8209 2803 128E}} || {{nowrap|8209 2806 112E}} || {{nowrap|8209 2809 0FD7}} || {{nowrap|8209 280C 0E87}} || {{nowrap|8209 280F 0D3A}} |- style="background-color: #eaecf0; font-size: large; font-weight: bold;" || || {{nowrap|1999}} || {{nowrap|2018}} || {{nowrap|2037}} || {{nowrap|2056}} || {{nowrap|2075}} || {{nowrap|2094}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jan 16}} || {{nowrap|8209 2800 1738}} || {{nowrap|8209 2803 15D9}} || {{nowrap|8209 2806 1477}} || {{nowrap|8209 2809 131B}} || {{nowrap|8209 280C 11C9}} || {{nowrap|8209 280F 107B}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Feb 15}} || {{nowrap|8209 2800 1A7D}} || {{nowrap|8209 2803 1924}} || {{nowrap|8209 2806 17C2}} || {{nowrap|8209 2809 1662}} || {{nowrap|8209 280C 150B}} || {{nowrap|8209 280F 13BC}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Mar 16}} || {{nowrap|8209 2800 1DC0}} || {{nowrap|8209 2803 1C6B}} || {{nowrap|8209 2806 1B0C}} || {{nowrap|8209 2809 19AA}} || {{nowrap|8209 280C 184F}} || {{nowrap|8209 280F 16FC}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Apr 14}} || {{nowrap|8209 2800 20FF}} || {{nowrap|8209 2803 1FAE}} || {{nowrap|8209 2806 1E54}} || {{nowrap|8209 2809 1CF3}} || {{nowrap|8209 280C 1B93}} || {{nowrap|8209 280F 1A3C}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|May 14}} || {{nowrap|8209 2800 243C}} || {{nowrap|8209 2803 22EE}} || {{nowrap|8209 2806 2198}} || {{nowrap|8209 2809 203A}} || {{nowrap|8209 280C 1ED8}} || {{nowrap|8209 280F 1D7C}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jun 13}} || {{nowrap|8209 2800 2779}} || {{nowrap|8209 2803 262B}} || {{nowrap|8209 2806 24DA}} || {{nowrap|8209 2809 2380}} || {{nowrap|8209 280C 221E}} || {{nowrap|8209 280F 20BE}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jul 12}} || {{nowrap|8209 2800 2AB6}} || {{nowrap|8209 2803 2968}} || {{nowrap|8209 2806 2819}} || {{nowrap|8209 2809 26C3}} || {{nowrap|8209 280C 2564}} || {{nowrap|8209 280F 2403}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Aug 11}} || {{nowrap|8209 2800 2DF4}} || {{nowrap|8209 2803 2CA4}} || {{nowrap|8209 2806 2B57}} || {{nowrap|8209 2809 2A05}} || {{nowrap|8209 280C 28AA}} || {{nowrap|8209 280F 2748}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Sep 9}} || {{nowrap|8209 2800 3135}} || {{nowrap|8209 2803 2FE2}} || {{nowrap|8209 2806 2E94}} || {{nowrap|8209 2809 2D45}} || {{nowrap|8209 280C 2BEF}} || {{nowrap|8209 280F 2A8F}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Oct 9}} || {{nowrap|8209 2800 3479}} || {{nowrap|8209 2803 3322}} || {{nowrap|8209 2806 31D2}} || {{nowrap|8209 2809 3084}} || {{nowrap|8209 280C 2F32}} || {{nowrap|8209 280F 2DD7}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Nov 7}} || {{nowrap|8209 2800 37C1}} || {{nowrap|8209 2803 3664}} || {{nowrap|8209 2806 3511}} || {{nowrap|8209 2809 33C3}} || {{nowrap|8209 280C 3274}} || {{nowrap|8209 280F 311E}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Dec 6}} || {{nowrap|8209 2800 3B0B}} || {{nowrap|8209 2803 39AA}} || {{nowrap|8209 2806 3853}} || {{nowrap|8209 2809 3703}} || {{nowrap|8209 280C 35B6}} || {{nowrap|8209 280F 3464}} |- style="background-color: #eaecf0; font-size: large; font-weight: bold;" || || {{nowrap|2000}} || {{nowrap|2019}} || {{nowrap|2038}} || {{nowrap|2057}} || {{nowrap|2076}} || {{nowrap|2095}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jan 5}} || {{nowrap|8209 2800 3E56}} || {{nowrap|8209 2803 3CF4}} || {{nowrap|8209 2806 3B98}} || {{nowrap|8209 2809 3A45}} || {{nowrap|8209 280C 38F7}} || {{nowrap|8209 280F 37A8}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Feb 3}} || {{nowrap|8209 2800 41A0}} || {{nowrap|8209 2803 403F}} || {{nowrap|8209 2806 3EDF}} || {{nowrap|8209 2809 3D88}} || {{nowrap|8209 280C 3C38}} || {{nowrap|8209 280F 3AEA}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Mar 5}} || {{nowrap|8209 2800 44E8}} || {{nowrap|8209 2803 438A}} || {{nowrap|8209 2806 4228}} || {{nowrap|8209 2809 40CB}} || {{nowrap|8209 280C 3F78}} || {{nowrap|8209 280F 3E2A}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Apr 3}} || {{nowrap|8209 2800 482B}} || {{nowrap|8209 2803 46D2}} || {{nowrap|8209 2806 4570}} || {{nowrap|8209 2809 4410}} || {{nowrap|8209 280C 42B9}} || {{nowrap|8209 280F 4169}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|May 3}} || {{nowrap|8209 2800 4B6B}} || {{nowrap|8209 2803 4A16}} || {{nowrap|8209 2806 48B8}} || {{nowrap|8209 2809 4756}} || {{nowrap|8209 280C 45FA}} || {{nowrap|8209 280F 44A7}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jun 1}} || {{nowrap|8209 2800 4EA9}} || {{nowrap|8209 2803 4D58}} || {{nowrap|8209 2806 4BFE}} || {{nowrap|8209 2809 4A9D}} || {{nowrap|8209 280C 493D}} || {{nowrap|8209 280F 47E6}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jul 1}} || {{nowrap|8209 2800 51E5}} || {{nowrap|8209 2803 5097}} || {{nowrap|8209 2806 4F42}} || {{nowrap|8209 2809 4DE3}} || {{nowrap|8209 280C 4C81}} || {{nowrap|8209 280F 4B26}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jul 1}} || {{nowrap|8209 2800 5522}} || {{nowrap|8209 2803 53D4}} || {{nowrap|8209 2806 5283}} || {{nowrap|8209 2809 5129}} || {{nowrap|8209 280C 4FC7}} || {{nowrap|8209 280F 4E67}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Aug 29}} || {{nowrap|8209 2800 585F}} || {{nowrap|8209 2803 5711}} || {{nowrap|8209 2806 55C2}} || {{nowrap|8209 2809 546D}} || {{nowrap|8209 280C 530E}} || {{nowrap|8209 280F 51AC}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Sep 27}} || {{nowrap|8209 2800 5B9F}} || {{nowrap|8209 2803 5A4F}} || {{nowrap|8209 2806 5901}} || {{nowrap|8209 2809 57AF}} || {{nowrap|8209 280C 5654}} || {{nowrap|8209 280F 54F2}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Oct 27}} || {{nowrap|8209 2800 5EE1}} || {{nowrap|8209 2803 5D8E}} || {{nowrap|8209 2806 5C40}} || {{nowrap|8209 2809 5AF1}} || {{nowrap|8209 280C 599B}} || {{nowrap|8209 280F 583B}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Nov 25}} || {{nowrap|8209 2800 6227}} || {{nowrap|8209 2803 60CF}} || {{nowrap|8209 2806 5F7F}} || {{nowrap|8209 2809 5E32}} || {{nowrap|8209 280C 5CE0}} || {{nowrap|8209 280F 5B85}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Dec 25}} || {{nowrap|8209 2800 6571}} || {{nowrap|8209 2803 6414}} || {{nowrap|8209 2806 62C1}} || {{nowrap|8209 2809 6173}} || {{nowrap|8209 280C 6024}} || {{nowrap|8209 280F 5ECE}} |- style="background-color: #eaecf0; font-size: large; font-weight: bold;" || || {{nowrap|2001}} || {{nowrap|2020}} || {{nowrap|2039}} || {{nowrap|2058}} || {{nowrap|2077}} || {{nowrap|2096}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jan 24}} || {{nowrap|8209 2800 68BC}} || {{nowrap|8209 2803 675C}} || {{nowrap|8209 2806 6604}} || {{nowrap|8209 2809 64B4}} || {{nowrap|8209 280C 6366}} || {{nowrap|8209 280F 6214}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Feb 22}} || {{nowrap|8209 2800 6C07}} || {{nowrap|8209 2803 6AA5}} || {{nowrap|8209 2806 6948}} || {{nowrap|8209 2809 67F5}} || {{nowrap|8209 280C 66A6}} || {{nowrap|8209 280F 6557}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Mar 24}} || {{nowrap|8209 2800 6F4F}} || {{nowrap|8209 2803 6DEE}} || {{nowrap|8209 2806 6C8E}} || {{nowrap|8209 2809 6B36}} || {{nowrap|8209 280C 69E6}} || {{nowrap|8209 280F 6898}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Apr 22}} || {{nowrap|8209 2800 7294}} || {{nowrap|8209 2803 7136}} || {{nowrap|8209 2806 6FD4}} || {{nowrap|8209 2809 6E77}} || {{nowrap|8209 280C 6D24}} || {{nowrap|8209 280F 6BD6}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|May 22}} || {{nowrap|8209 2800 75D5}} || {{nowrap|8209 2803 747C}} || {{nowrap|8209 2806 731B}} || {{nowrap|8209 2809 71BB}} || {{nowrap|8209 280C 7063}} || {{nowrap|8209 280F 6F14}} |- style="font-size:small:small;background-color:#ffaaaa;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jun 20}} || {{nowrap|8209 2800 7915}} || {{nowrap|8209 2803 77C0}} || {{nowrap|8209 2806 7662}} || {{nowrap|8209 2809 7500}} || {{nowrap|8209 280C 73A4}} || {{nowrap|8209 280F 7251}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jul 19}} || {{nowrap|8209 2800 7C52}} || {{nowrap|8209 2803 7B01}} || {{nowrap|8209 2806 79A7}} || {{nowrap|8209 2809 7846}} || {{nowrap|8209 280C 76E6}} || {{nowrap|8209 280F 758F}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Aug 18}} || {{nowrap|8209 2800 7F8E}} || {{nowrap|8209 2803 7E40}} || {{nowrap|8209 2806 7CEB}} || {{nowrap|8209 2809 7B8C}} || {{nowrap|8209 280C 7A2A}} || {{nowrap|8209 280F 78CE}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Sep 16}} || {{nowrap|8209 2800 82CB}} || {{nowrap|8209 2803 817E}} || {{nowrap|8209 2806 802C}} || {{nowrap|8209 2809 7ED2}} || {{nowrap|8209 280C 7D70}} || {{nowrap|8209 280F 7C11}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Oct 16}} || {{nowrap|8209 2800 860A}} || {{nowrap|8209 2803 84BC}} || {{nowrap|8209 2806 836D}} || {{nowrap|8209 2809 8218}} || {{nowrap|8209 280C 80B9}} || {{nowrap|8209 280F 7F57}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Nov 15}} || {{nowrap|8209 2800 894C}} || {{nowrap|8209 2803 87FB}} || {{nowrap|8209 2806 86AE}} || {{nowrap|8209 2809 855C}} || {{nowrap|8209 280C 8402}} || {{nowrap|8209 280F 82A0}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Dec 14}} || {{nowrap|8209 2800 8C90}} || {{nowrap|8209 2803 8B3D}} || {{nowrap|8209 2806 89EF}} || {{nowrap|8209 2809 88A0}} || {{nowrap|8209 280C 874A}} || {{nowrap|8209 280F 85EC}} |- style="background-color: #eaecf0; font-size: large; font-weight: bold;" || || {{nowrap|2002}} || {{nowrap|2021}} || {{nowrap|2040}} || {{nowrap|2059}} || {{nowrap|2078}} || {{nowrap|2097}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jan 13}} || {{nowrap|8209 2800 8FD8}} || {{nowrap|8209 2803 8E80}} || {{nowrap|8209 2806 8D30}} || {{nowrap|8209 2809 8BE2}} || {{nowrap|8209 280C 8A90}} || {{nowrap|8209 280F 8936}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Feb 11}} || {{nowrap|8209 2800 9322}} || {{nowrap|8209 2803 91C5}} || {{nowrap|8209 2806 9071}} || {{nowrap|8209 2809 8F23}} || {{nowrap|8209 280C 8DD4}} || {{nowrap|8209 280F 8C7E}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Mar 13}} || {{nowrap|8209 2800 966C}} || {{nowrap|8209 2803 950B}} || {{nowrap|8209 2806 93B2}} || {{nowrap|8209 2809 9262}} || {{nowrap|8209 280C 9114}} || {{nowrap|8209 280F 8FC3}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Apr 11}} || {{nowrap|8209 2800 99B4}} || {{nowrap|8209 2803 9852}} || {{nowrap|8209 2806 96F5}} || {{nowrap|8209 2809 95A1}} || {{nowrap|8209 280C 9453}} || {{nowrap|8209 280F 9305}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|May 11}} || {{nowrap|8209 2800 9CFB}} || {{nowrap|8209 2803 9B9A}} || {{nowrap|8209 2806 9A39}} || {{nowrap|8209 2809 98E1}} || {{nowrap|8209 280C 9791}} || {{nowrap|8209 280F 9644}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jun 9}} || {{nowrap|8209 2800 A03E}} || {{nowrap|8209 2803 9EE1}} || {{nowrap|8209 2806 9D7E}} || {{nowrap|8209 2809 9C22}} || {{nowrap|8209 280C 9ACF}} || {{nowrap|8209 280F 9981}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jul 9}} || {{nowrap|8209 2800 A37F}} || {{nowrap|8209 2803 A226}} || {{nowrap|8209 2806 A0C4}} || {{nowrap|8209 2809 9F64}} || {{nowrap|8209 280C 9E0C}} || {{nowrap|8209 280F 9CBD}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Aug 7}} || {{nowrap|8209 2800 A6BD}} || {{nowrap|8209 2803 A569}} || {{nowrap|8209 2806 A40A}} || {{nowrap|8209 2809 A2A8}} || {{nowrap|8209 280C A14C}} || {{nowrap|8209 280F 9FF9}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Sep 6}} || {{nowrap|8209 2800 A9FB}} || {{nowrap|8209 2803 A8AA}} || {{nowrap|8209 2806 A750}} || {{nowrap|8209 2809 A5EF}} || {{nowrap|8209 280C A48F}} || {{nowrap|8209 280F A337}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Oct 5}} || {{nowrap|8209 2800 AD39}} || {{nowrap|8209 2803 ABEA}} || {{nowrap|8209 2806 AA95}} || {{nowrap|8209 2809 A937}} || {{nowrap|8209 280C A7D5}} || {{nowrap|8209 280F A679}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Nov 4}} || {{nowrap|8209 2800 B078}} || {{nowrap|8209 2803 AF2A}} || {{nowrap|8209 2806 ADD9}} || {{nowrap|8209 2809 AC7F}} || {{nowrap|8209 280C AB1E}} || {{nowrap|8209 280F A9BE}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Dec 3}} || {{nowrap|8209 2800 B3B9}} || {{nowrap|8209 2803 B26B}} || {{nowrap|8209 2806 B11C}} || {{nowrap|8209 2809 AFC7}} || {{nowrap|8209 280C AE69}} || {{nowrap|8209 280F AD07}} |- style="background-color: #eaecf0; font-size: large; font-weight: bold;" || || {{nowrap|2003}} || {{nowrap|2022}} || {{nowrap|2041}} || {{nowrap|2060}} || {{nowrap|2079}} || {{nowrap|2098}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jan 2}} || {{nowrap|8209 2800 B6FC}} || {{nowrap|8209 2803 B5AC}} || {{nowrap|8209 2806 B45E}} || {{nowrap|8209 2809 B30D}} || {{nowrap|8209 280C B1B3}} || {{nowrap|8209 280F B051}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Feb 1}} || {{nowrap|8209 2800 BA41}} || {{nowrap|8209 2803 B8ED}} || {{nowrap|8209 2806 B79F}} || {{nowrap|8209 2809 B650}} || {{nowrap|8209 280C B4FB}} || {{nowrap|8209 280F B39C}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Mar 2}} || {{nowrap|8209 2800 BD88}} || {{nowrap|8209 2803 BC2F}} || {{nowrap|8209 2806 BADE}} || {{nowrap|8209 2809 B991}} || {{nowrap|8209 280C B840}} || {{nowrap|8209 280F B6E6}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Apr 1}} || {{nowrap|8209 2800 C0D0}} || {{nowrap|8209 2803 BF72}} || {{nowrap|8209 2806 BE1E}} || {{nowrap|8209 2809 BCD0}} || {{nowrap|8209 280C BB81}} || {{nowrap|8209 280F BA2C}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|May 1}} || {{nowrap|8209 2800 C418}} || {{nowrap|8209 2803 C2B7}} || {{nowrap|8209 2806 C15E}} || {{nowrap|8209 2809 C00E}} || {{nowrap|8209 280C BEC1}} || {{nowrap|8209 280F BD70}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|May 29}} || {{nowrap|8209 2800 C75F}} || {{nowrap|8209 2803 C5FD}} || {{nowrap|8209 2806 C4A0}} || {{nowrap|8209 2809 C34C}} || {{nowrap|8209 280C C1FE}} || {{nowrap|8209 280F C0AF}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jun 28}} || {{nowrap|8209 2800 CAA4}} || {{nowrap|8209 2803 C943}} || {{nowrap|8209 2806 C7E2}} || {{nowrap|8209 2809 C68A}} || {{nowrap|8209 280C C53A}} || {{nowrap|8209 280F C3ED}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jul 27}} || {{nowrap|8209 2800 CDE7}} || {{nowrap|8209 2803 CC89}} || {{nowrap|8209 2806 CB27}} || {{nowrap|8209 2809 C9CA}} || {{nowrap|8209 280C C877}} || {{nowrap|8209 280F C729}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Aug 26}} || {{nowrap|8209 2800 D127}} || {{nowrap|8209 2803 CFCE}} || {{nowrap|8209 2806 CE6D}} || {{nowrap|8209 2809 CD0D}} || {{nowrap|8209 280C CBB5}} || {{nowrap|8209 280F CA65}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Sep 24}} || {{nowrap|8209 2800 D467}} || {{nowrap|8209 2803 D312}} || {{nowrap|8209 2806 D1B4}} || {{nowrap|8209 2809 D052}} || {{nowrap|8209 280C CEF6}} || {{nowrap|8209 280F CDA3}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Oct 24}} || {{nowrap|8209 2800 D7A6}} || {{nowrap|8209 2803 D656}} || {{nowrap|8209 2806 D4FC}} || {{nowrap|8209 2809 D39B}} || {{nowrap|8209 280C D23B}} || {{nowrap|8209 280F D0E4}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Nov 22}} || {{nowrap|8209 2800 DAE7}} || {{nowrap|8209 2803 D998}} || {{nowrap|8209 2806 D844}} || {{nowrap|8209 2809 D6E6}} || {{nowrap|8209 280C D583}} || {{nowrap|8209 280F D427}} |- style="font-size:small:small;background-color:#ffff00;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Dec 22}} || {{nowrap|8209 2800 DE27}} || {{nowrap|8209 2803 DCDA}} || {{nowrap|8209 2806 DB89}} || {{nowrap|8209 2809 DA2F}} || {{nowrap|8209 280C D8CE}} || {{nowrap|8209 280F D76E}} |- style="background-color: #eaecf0; font-size: large; font-weight: bold;" || || {{nowrap|2004}} || {{nowrap|2023}} || {{nowrap|2042}} || {{nowrap|2061}} || {{nowrap|2080}} || {{nowrap|2099}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jan 21}} || {{nowrap|8209 2800 E169}} || {{nowrap|8209 2803 E01A}} || {{nowrap|8209 2806 DECC}} || {{nowrap|8209 2809 DD77}} || {{nowrap|8209 280C DC19}} || {{nowrap|8209 280F DAB7}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Feb 20}} || {{nowrap|8209 2800 E4AC}} || {{nowrap|8209 2803 E35B}} || {{nowrap|8209 2806 E20D}} || {{nowrap|8209 2809 E0BC}} || {{nowrap|8209 280C DF63}} || {{nowrap|8209 280F DE01}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Mar 20}} || {{nowrap|8209 2800 E7EF}} || {{nowrap|8209 2803 E69B}} || {{nowrap|8209 2806 E54D}} || {{nowrap|8209 2809 E3FE}} || {{nowrap|8209 280C E2AA}} || {{nowrap|8209 280F E14C}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Apr 19}} || {{nowrap|8209 2800 EB35}} || {{nowrap|8209 2803 E9DC}} || {{nowrap|8209 2806 E88B}} || {{nowrap|8209 2809 E73E}} || {{nowrap|8209 280C E5ED}} || {{nowrap|8209 280F E493}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|May 19}} || {{nowrap|8209 2800 EE7B}} || {{nowrap|8209 2803 ED1E}} || {{nowrap|8209 2806 EBCA}} || {{nowrap|8209 2809 EA7B}} || {{nowrap|8209 280C E92D}} || {{nowrap|8209 280F E7D8}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jun 17}} || {{nowrap|8209 2800 F1C2}} || {{nowrap|8209 2803 F061}} || {{nowrap|8209 2806 EF08}} || {{nowrap|8209 2809 EDB8}} || {{nowrap|8209 280C EC6A}} || {{nowrap|8209 280F EB19}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jul 17}} || {{nowrap|8209 2800 F508}} || {{nowrap|8209 2803 F3A5}} || {{nowrap|8209 2806 F248}} || {{nowrap|8209 2809 F0F4}} || {{nowrap|8209 280C EFA6}} || {{nowrap|8209 280F EE57}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Aug 15}} || {{nowrap|8209 2800 F84C}} || {{nowrap|8209 2803 F6EB}} || {{nowrap|8209 2806 F58B}} || {{nowrap|8209 2809 F432}} || {{nowrap|8209 280C F2E2}} || {{nowrap|8209 280F F195}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Sep 13}} || {{nowrap|8209 2800 FB90}} || {{nowrap|8209 2803 FA32}} || {{nowrap|8209 2806 F8D0}} || {{nowrap|8209 2809 F774}} || {{nowrap|8209 280C F620}} || {{nowrap|8209 280F F4D2}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Oct 13}} || {{nowrap|8209 2800 FED3}} || {{nowrap|8209 2803 FD7A}} || {{nowrap|8209 2806 FC19}} || {{nowrap|8209 2809 FAB8}} || {{nowrap|8209 280C F961}} || {{nowrap|8209 280F F811}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Nov 11}} || {{nowrap|8209 2801 0214}} || {{nowrap|8209 2804 00C0}} || {{nowrap|8209 2806 FF63}} || {{nowrap|8209 2809 FE00}} || {{nowrap|8209 280C FCA4}} || {{nowrap|8209 280F FB51}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Dec 11}} || {{nowrap|8209 2801 0556}} || {{nowrap|8209 2804 0405}} || {{nowrap|8209 2807 02AC}} || {{nowrap|8209 280A 014B}} || {{nowrap|8209 280C FFEA}} || {{nowrap|8209 280F FE93}} |- style="background-color: #eaecf0; font-size: large; font-weight: bold;" || || {{nowrap|2005}} || {{nowrap|2024}} || {{nowrap|2043}} || {{nowrap|2062}} || {{nowrap|2081}} || {{nowrap|2100}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jan 10}} || {{nowrap|8209 2801 0896}} || {{nowrap|8209 2804 0748}} || {{nowrap|8209 2807 05F3}} || {{nowrap|8209 280A 0496}} || {{nowrap|8209 280D 0334}} || {{nowrap|8209 2810 01D7}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Feb 8}} || {{nowrap|8209 2801 0BD7}} || {{nowrap|8209 2804 0A89}} || {{nowrap|8209 2807 0938}} || {{nowrap|8209 280A 07E0}} || {{nowrap|8209 280D 067E}} || {{nowrap|8209 2810 051E}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Mar 10}} || {{nowrap|8209 2801 0F17}} || {{nowrap|8209 2804 0DC9}} || {{nowrap|8209 2807 0C7B}} || {{nowrap|8209 280A 0B27}} || {{nowrap|8209 280D 09C9}} || {{nowrap|8209 2810 0867}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Apr 8}} || {{nowrap|8209 2801 1259}} || {{nowrap|8209 2804 1108}} || {{nowrap|8209 2807 0FBB}} || {{nowrap|8209 280A 0E6A}} || {{nowrap|8209 280D 0D11}} || {{nowrap|8209 2810 0BB0}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|May 8}} || {{nowrap|8209 2801 159C}} || {{nowrap|8209 2804 1447}} || {{nowrap|8209 2807 12F9}} || {{nowrap|8209 280A 11AA}} || {{nowrap|8209 280D 1056}} || {{nowrap|8209 2810 0EF8}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jun 6}} || {{nowrap|8209 2801 18DF}} || {{nowrap|8209 2804 1786}} || {{nowrap|8209 2807 1635}} || {{nowrap|8209 280A 14E8}} || {{nowrap|8209 280D 1397}} || {{nowrap|8209 2810 123D}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jul 5}} || {{nowrap|8209 2801 1C24}} || {{nowrap|8209 2804 1AC6}} || {{nowrap|8209 2807 1972}} || {{nowrap|8209 280A 1824}} || {{nowrap|8209 280D 16D5}} || {{nowrap|8209 2810 1580}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Aug 4}} || {{nowrap|8209 2801 1F6A}} || {{nowrap|8209 2804 1E09}} || {{nowrap|8209 2807 1CB0}} || {{nowrap|8209 280A 1B60}} || {{nowrap|8209 280D 1A12}} || {{nowrap|8209 2810 18C1}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Sep 3}} || {{nowrap|8209 2801 22B1}} || {{nowrap|8209 2804 214E}} || {{nowrap|8209 2807 1FF1}} || {{nowrap|8209 280A 1E9D}} || {{nowrap|8209 280D 1D4F}} || {{nowrap|8209 2810 1C01}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Oct 2}} || {{nowrap|8209 2801 25F7}} || {{nowrap|8209 2804 2496}} || {{nowrap|8209 2807 2336}} || {{nowrap|8209 280A 21DE}} || {{nowrap|8209 280D 208D}} || {{nowrap|8209 2810 1F40}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Nov 1}} || {{nowrap|8209 2801 293D}} || {{nowrap|8209 2804 27E0}} || {{nowrap|8209 2807 267E}} || {{nowrap|8209 280A 2521}} || {{nowrap|8209 280D 23CD}} || {{nowrap|8209 2810 227F}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Dec 1}} || {{nowrap|8209 2801 2C81}} || {{nowrap|8209 2804 2B29}} || {{nowrap|8209 2807 29C8}} || {{nowrap|8209 280A 2867}} || {{nowrap|8209 280D 270F}} || {{nowrap|8209 2810 25BF}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Dec 30}} || {{nowrap|8209 2801 2FC4}} || {{nowrap|8209 2804 2E70}} || {{nowrap|8209 2807 2D13}} || {{nowrap|8209 280A 2BB0}} || {{nowrap|8209 280D 2A53}} || {{nowrap|8209 2810 2900}} |- style="background-color: #eaecf0; font-size: large; font-weight: bold;" || || {{nowrap|2006}} || {{nowrap|2025}} || {{nowrap|2044}} || {{nowrap|2063}} || {{nowrap|2082}} || {{nowrap|2101}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jan 28}} || {{nowrap|8209 2801 3305}} || {{nowrap|8209 2804 31B5}} || {{nowrap|8209 2807 305C}} || {{nowrap|8209 280A 2EFB}} || {{nowrap|8209 280D 2D9B}} || {{nowrap|8209 2810 2C43}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Feb 27}} || {{nowrap|8209 2801 3645}} || {{nowrap|8209 2804 34F7}} || {{nowrap|8209 2807 33A3}} || {{nowrap|8209 280A 3246}} || {{nowrap|8209 280D 30E4}} || {{nowrap|8209 2810 2F87}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Mar 29}} || {{nowrap|8209 2801 3985}} || {{nowrap|8209 2804 3837}} || {{nowrap|8209 2807 36E7}} || {{nowrap|8209 280A 358F}} || {{nowrap|8209 280D 342E}} || {{nowrap|8209 2810 32CD}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Apr 27}} || {{nowrap|8209 2801 3CC4}} || {{nowrap|8209 2804 3B76}} || {{nowrap|8209 2807 3A27}} || {{nowrap|8209 280A 38D3}} || {{nowrap|8209 280D 3776}} || {{nowrap|8209 2810 3614}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|May 27}} || {{nowrap|8209 2801 4004}} || {{nowrap|8209 2804 3EB3}} || {{nowrap|8209 2807 3D65}} || {{nowrap|8209 280A 3C14}} || {{nowrap|8209 280D 3ABC}} || {{nowrap|8209 2810 395A}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jun 25}} || {{nowrap|8209 2801 4345}} || {{nowrap|8209 2804 41F0}} || {{nowrap|8209 2807 40A1}} || {{nowrap|8209 280A 3F53}} || {{nowrap|8209 280D 3DFE}} || {{nowrap|8209 2810 3CA0}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jul 24}} || {{nowrap|8209 2801 4687}} || {{nowrap|8209 2804 452E}} || {{nowrap|8209 2807 43DD}} || {{nowrap|8209 280A 4290}} || {{nowrap|8209 280D 413F}} || {{nowrap|8209 2810 3FE6}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Aug 23}} || {{nowrap|8209 2801 49CD}} || {{nowrap|8209 2804 486F}} || {{nowrap|8209 2807 471B}} || {{nowrap|8209 280A 45CD}} || {{nowrap|8209 280D 447E}} || {{nowrap|8209 2810 4329}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Sep 21}} || {{nowrap|8209 2801 4D14}} || {{nowrap|8209 2804 4BB3}} || {{nowrap|8209 2807 4A5B}} || {{nowrap|8209 280A 490A}} || {{nowrap|8209 280D 47BD}} || {{nowrap|8209 2810 466C}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Oct 20}} || {{nowrap|8209 2801 505D}} || {{nowrap|8209 2804 4EFB}} || {{nowrap|8209 2807 4D9E}} || {{nowrap|8209 280A 4C4A}} || {{nowrap|8209 280D 4AFB}} || {{nowrap|8209 2810 49AD}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Nov 19}} || {{nowrap|8209 2801 53A5}} || {{nowrap|8209 2804 5245}} || {{nowrap|8209 2807 50E4}} || {{nowrap|8209 280A 4F8B}} || {{nowrap|8209 280D 4E3B}} || {{nowrap|8209 2810 4CED}} |- style="font-size:small:small;background-color:#ffff00;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Dec 19}} || {{nowrap|8209 2801 56EC}} || {{nowrap|8209 2804 558F}} || {{nowrap|8209 2807 542D}} || {{nowrap|8209 280A 52D0}} || {{nowrap|8209 280D 517C}} || {{nowrap|8209 2810 502D}} |- style="background-color: #eaecf0; font-size: large; font-weight: bold;" || || {{nowrap|2007}} || {{nowrap|2026}} || {{nowrap|2045}} || {{nowrap|2064}} || {{nowrap|2083}} || {{nowrap|2102}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jan 18}} || {{nowrap|8209 2801 5A31}} || {{nowrap|8209 2804 58D9}} || {{nowrap|8209 2807 5778}} || {{nowrap|8209 280A 5617}} || {{nowrap|8209 280D 54BF}} || {{nowrap|8209 2810 536E}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Feb 16}} || {{nowrap|8209 2801 5D73}} || {{nowrap|8209 2804 5C20}} || {{nowrap|8209 2807 5AC3}} || {{nowrap|8209 280A 5961}} || {{nowrap|8209 280D 5804}} || {{nowrap|8209 2810 56B0}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Mar 18}} || {{nowrap|8209 2801 60B4}} || {{nowrap|8209 2804 5F64}} || {{nowrap|8209 2807 5E0C}} || {{nowrap|8209 280A 5CAB}} || {{nowrap|8209 280D 5B4A}} || {{nowrap|8209 2810 59F2}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Apr 16}} || {{nowrap|8209 2801 63F3}} || {{nowrap|8209 2804 62A5}} || {{nowrap|8209 2807 6151}} || {{nowrap|8209 280A 5FF4}} || {{nowrap|8209 280D 5E92}} || {{nowrap|8209 2810 5D34}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|May 16}} || {{nowrap|8209 2801 6730}} || {{nowrap|8209 2804 65E2}} || {{nowrap|8209 2807 6492}} || {{nowrap|8209 280A 633A}} || {{nowrap|8209 280D 61D9}} || {{nowrap|8209 2810 6078}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jun 14}} || {{nowrap|8209 2801 6A6D}} || {{nowrap|8209 2804 691F}} || {{nowrap|8209 2807 67D0}} || {{nowrap|8209 280A 667C}} || {{nowrap|8209 280D 651F}} || {{nowrap|8209 2810 63BD}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jul 14}} || {{nowrap|8209 2801 6DAC}} || {{nowrap|8209 2804 6C5B}} || {{nowrap|8209 2807 6B0D}} || {{nowrap|8209 280A 69BD}} || {{nowrap|8209 280D 6864}} || {{nowrap|8209 2810 6703}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Aug 12}} || {{nowrap|8209 2801 70ED}} || {{nowrap|8209 2804 6F98}} || {{nowrap|8209 2807 6E4A}} || {{nowrap|8209 280A 6CFC}} || {{nowrap|8209 280D 6BA7}} || {{nowrap|8209 2810 6A4A}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Sep 11}} || {{nowrap|8209 2801 7431}} || {{nowrap|8209 2804 72D8}} || {{nowrap|8209 2807 7187}} || {{nowrap|8209 280A 703A}} || {{nowrap|8209 280D 6EE9}} || {{nowrap|8209 2810 6D90}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Oct 10}} || {{nowrap|8209 2801 7779}} || {{nowrap|8209 2804 761B}} || {{nowrap|8209 2807 74C6}} || {{nowrap|8209 280A 7378}} || {{nowrap|8209 280D 7229}} || {{nowrap|8209 2810 70D4}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Nov 8}} || {{nowrap|8209 2801 7AC2}} || {{nowrap|8209 2804 7961}} || {{nowrap|8209 2807 7808}} || {{nowrap|8209 280A 76B7}} || {{nowrap|8209 280D 7569}} || {{nowrap|8209 2810 7418}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Dec 8}} || {{nowrap|8209 2801 7E0C}} || {{nowrap|8209 2804 7CAA}} || {{nowrap|8209 2807 7B4C}} || {{nowrap|8209 280A 79F8}} || {{nowrap|8209 280D 78A9}} |- style="background-color: #eaecf0; font-size: large; font-weight: bold;" || || {{nowrap|2008}} || {{nowrap|2027}} || {{nowrap|2046}} || {{nowrap|2065}} || {{nowrap|2084}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jan 6}} || {{nowrap|8209 2801 8155}} || {{nowrap|8209 2804 7FF5}} || {{nowrap|8209 2807 7E94}} || {{nowrap|8209 280A 7D3B}} || {{nowrap|8209 280D 7BEA}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Feb 5}} || {{nowrap|8209 2801 849D}} || {{nowrap|8209 2804 8340}} || {{nowrap|8209 2807 81DE}} || {{nowrap|8209 280A 8080}} || {{nowrap|8209 280D 7F2C}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Mar 6}} || {{nowrap|8209 2801 87E1}} || {{nowrap|8209 2804 8689}} || {{nowrap|8209 2807 8529}} || {{nowrap|8209 280A 83C8}} || {{nowrap|8209 280D 826F}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Apr 5}} || {{nowrap|8209 2801 8B21}} || {{nowrap|8209 2804 89CE}} || {{nowrap|8209 2807 8872}} || {{nowrap|8209 280A 8710}} || {{nowrap|8209 280D 85B2}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|May 4}} || {{nowrap|8209 2801 8E5F}} || {{nowrap|8209 2804 8D10}} || {{nowrap|8209 2807 8BB8}} || {{nowrap|8209 280A 8A57}} || {{nowrap|8209 280D 88F6}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jun 3}} || {{nowrap|8209 2801 919C}} || {{nowrap|8209 2804 904E}} || {{nowrap|8209 2807 8EFB}} || {{nowrap|8209 280A 8D9E}} || {{nowrap|8209 280D 8C3C}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jul 3}} || {{nowrap|8209 2801 94D8}} || {{nowrap|8209 2804 938B}} || {{nowrap|8209 2807 923B}} || {{nowrap|8209 280A 90E3}} || {{nowrap|8209 280D 8F82}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Aug 1}} || {{nowrap|8209 2801 9816}} || {{nowrap|8209 2804 96C7}} || {{nowrap|8209 2807 9579}} || {{nowrap|8209 280A 9425}} || {{nowrap|8209 280D 92C8}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Aug 30}} || {{nowrap|8209 2801 9B55}} || {{nowrap|8209 2804 9A04}} || {{nowrap|8209 2807 98B7}} || {{nowrap|8209 280A 9766}} || {{nowrap|8209 280D 960E}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Sep 29}} || {{nowrap|8209 2801 9E98}} || {{nowrap|8209 2804 9D43}} || {{nowrap|8209 2807 9BF5}} || {{nowrap|8209 280A 9AA6}} || {{nowrap|8209 280D 9952}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Oct 28}} || {{nowrap|8209 2801 A1DE}} || {{nowrap|8209 2804 A084}} || {{nowrap|8209 2807 9F33}} || {{nowrap|8209 280A 9DE5}} || {{nowrap|8209 280D 9C95}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Nov 27}} || {{nowrap|8209 2801 A527}} || {{nowrap|8209 2804 A3C9}} || {{nowrap|8209 2807 A274}} || {{nowrap|8209 280A A125}} || {{nowrap|8209 280D 9FD7}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Dec 27}} || {{nowrap|8209 2801 A872}} || {{nowrap|8209 2804 A711}} || {{nowrap|8209 2807 A5B7}} || {{nowrap|8209 280A A466}} || {{nowrap|8209 280D A318}} |- style="background-color: #eaecf0; font-size: large; font-weight: bold;" || || {{nowrap|2009}} || {{nowrap|2028}} || {{nowrap|2047}} || {{nowrap|2066}} || {{nowrap|2085}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jan 25}} || {{nowrap|8209 2801 ABBD}} || {{nowrap|8209 2804 AA5B}} || {{nowrap|8209 2807 A8FD}} || {{nowrap|8209 280A A7A8}} || {{nowrap|8209 280D A65A}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Feb 24}} || {{nowrap|8209 2801 AF06}} || {{nowrap|8209 2804 ADA6}} || {{nowrap|8209 2807 AC45}} || {{nowrap|8209 280A AAEB}} || {{nowrap|8209 280D A99A}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Mar 25}} || {{nowrap|8209 2801 B24B}} || {{nowrap|8209 2804 B0EF}} || {{nowrap|8209 2807 AF8D}} || {{nowrap|8209 280A AE2F}} || {{nowrap|8209 280D ACDA}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Apr 24}} || {{nowrap|8209 2801 B58D}} || {{nowrap|8209 2804 B436}} || {{nowrap|8209 2807 B2D5}} || {{nowrap|8209 280A B174}} || {{nowrap|8209 280D B01A}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|May 23}} || {{nowrap|8209 2801 B8CB}} || {{nowrap|8209 2804 B778}} || {{nowrap|8209 2807 B61C}} || {{nowrap|8209 280A B4BA}} || {{nowrap|8209 280D B35C}} |- style="font-size:small:small;background-color:#ffff00;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jun 22}} || {{nowrap|8209 2801 BC08}} || {{nowrap|8209 2804 BAB9}} || {{nowrap|8209 2807 B961}} || {{nowrap|8209 280A B801}} || {{nowrap|8209 280D B69F}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jul 21}} || {{nowrap|8209 2801 BF45}} || {{nowrap|8209 2804 BDF7}} || {{nowrap|8209 2807 BCA4}} || {{nowrap|8209 280A BB47}} || {{nowrap|8209 280D B9E5}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Aug 20}} || {{nowrap|8209 2801 C282}} || {{nowrap|8209 2804 C134}} || {{nowrap|8209 2807 BFE4}} || {{nowrap|8209 280A BE8C}} || {{nowrap|8209 280D BD2B}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Sep 18}} || {{nowrap|8209 2801 C5C0}} || {{nowrap|8209 2804 C471}} || {{nowrap|8209 2807 C323}} || {{nowrap|8209 280A C1CF}} || {{nowrap|8209 280D C072}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Oct 18}} || {{nowrap|8209 2801 C901}} || {{nowrap|8209 2804 C7B0}} || {{nowrap|8209 2807 C662}} || {{nowrap|8209 280A C511}} || {{nowrap|8209 280D C3B9}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Nov 16}} || {{nowrap|8209 2801 CC45}} || {{nowrap|8209 2804 CAF0}} || {{nowrap|8209 2807 C9A1}} || {{nowrap|8209 280A C853}} || {{nowrap|8209 280D C6FF}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Dec 16}} || {{nowrap|8209 2801 CF8D}} || {{nowrap|8209 2804 CE33}} || {{nowrap|8209 2807 CCE2}} || {{nowrap|8209 280A CB94}} || {{nowrap|8209 280D CA44}} |- style="background-color: #eaecf0; font-size: large; font-weight: bold;" || || {{nowrap|2010}} || {{nowrap|2029}} || {{nowrap|2048}} || {{nowrap|2067}} || {{nowrap|2086}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jan 14}} || {{nowrap|8209 2801 D2D8}} || {{nowrap|8209 2804 D179}} || {{nowrap|8209 2807 D024}} || {{nowrap|8209 280A CED5}} || {{nowrap|8209 280D CD87}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Feb 13}} || {{nowrap|8209 2801 D623}} || {{nowrap|8209 2804 D4C2}} || {{nowrap|8209 2807 D368}} || {{nowrap|8209 280A D216}} || {{nowrap|8209 280D D0C8}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Mar 14}} || {{nowrap|8209 2801 D96D}} || {{nowrap|8209 2804 D80B}} || {{nowrap|8209 2807 D6AC}} || {{nowrap|8209 280A D557}} || {{nowrap|8209 280D D408}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Apr 13}} || {{nowrap|8209 2801 DCB3}} || {{nowrap|8209 2804 DB53}} || {{nowrap|8209 2807 D9F2}} || {{nowrap|8209 280A D898}} || {{nowrap|8209 280D D747}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|May 12}} || {{nowrap|8209 2801 DFF6}} || {{nowrap|8209 2804 DE9B}} || {{nowrap|8209 2807 DD38}} || {{nowrap|8209 280A DBDA}} || {{nowrap|8209 280D DA85}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jun 11}} || {{nowrap|8209 2801 E336}} || {{nowrap|8209 2804 E1DF}} || {{nowrap|8209 2807 E07F}} || {{nowrap|8209 280A DF1E}} || {{nowrap|8209 280D DDC4}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jul 10}} || {{nowrap|8209 2801 E675}} || {{nowrap|8209 2804 E522}} || {{nowrap|8209 2807 E3C5}} || {{nowrap|8209 280A E263}} || {{nowrap|8209 280D E105}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Aug 9}} || {{nowrap|8209 2801 E9B1}} || {{nowrap|8209 2804 E862}} || {{nowrap|8209 2807 E70A}} || {{nowrap|8209 280A E5AA}} || {{nowrap|8209 280D E448}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Sep 8}} || {{nowrap|8209 2801 ECEE}} || {{nowrap|8209 2804 EBA0}} || {{nowrap|8209 2807 EA4D}} || {{nowrap|8209 280A E8F0}} || {{nowrap|8209 280D E78E}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Oct 7}} || {{nowrap|8209 2801 F02C}} || {{nowrap|8209 2804 EEDE}} || {{nowrap|8209 2807 ED8E}} || {{nowrap|8209 280A EC36}} || {{nowrap|8209 280D EAD6}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Nov 6}} || {{nowrap|8209 2801 F36C}} || {{nowrap|8209 2804 F21D}} || {{nowrap|8209 2807 F0CF}} || {{nowrap|8209 280A EF7C}} || {{nowrap|8209 280D EE1F}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Dec 5}} || {{nowrap|8209 2801 F6AF}} || {{nowrap|8209 2804 F55E}} || {{nowrap|8209 2807 F410}} || {{nowrap|8209 280A F2C0}} || {{nowrap|8209 280D F168}} |- style="background-color: #eaecf0; font-size: large; font-weight: bold;" || || {{nowrap|2011}} || {{nowrap|2030}} || {{nowrap|2049}} || {{nowrap|2068}} || {{nowrap|2087}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jan 4}} || {{nowrap|8209 2801 F9F6}} || {{nowrap|8209 2804 F8A0}} || {{nowrap|8209 2807 F751}} || {{nowrap|8209 280A F603}} || {{nowrap|8209 280D F4B0}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Feb 2}} || {{nowrap|8209 2801 FD3F}} || {{nowrap|8209 2804 FBE4}} || {{nowrap|8209 2807 FA92}} || {{nowrap|8209 280A F944}} || {{nowrap|8209 280D F7F4}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Mar 3}} || {{nowrap|8209 2802 0088}} || {{nowrap|8209 2804 FF29}} || {{nowrap|8209 2807 FDD3}} || {{nowrap|8209 280A FC84}} || {{nowrap|8209 280D FB36}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Apr 2}} || {{nowrap|8209 2802 03D1}} || {{nowrap|8209 2805 0270}} || {{nowrap|8209 2808 0115}} || {{nowrap|8209 280A FFC3}} || {{nowrap|8209 280D FE76}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|May 1}} || {{nowrap|8209 2802 0719}} || {{nowrap|8209 2805 05B7}} || {{nowrap|8209 2808 0458}} || {{nowrap|8209 280B 0302}} || {{nowrap|8209 280E 01B4}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jun 1}} || {{nowrap|8209 2802 0A5E}} || {{nowrap|8209 2805 08FE}} || {{nowrap|8209 2808 079C}} || {{nowrap|8209 280B 0642}} || {{nowrap|8209 280E 04F1}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jul 1}} || {{nowrap|8209 2802 0DA0}} || {{nowrap|8209 2805 0C44}} || {{nowrap|8209 2808 0AE2}} || {{nowrap|8209 280B 0984}} || {{nowrap|8209 280E 082F}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jul 29}} || {{nowrap|8209 2802 10DF}} || {{nowrap|8209 2805 0F88}} || {{nowrap|8209 2808 0E28}} || {{nowrap|8209 280B 0CC7}} || {{nowrap|8209 280E 0B6D}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Aug 27}} || {{nowrap|8209 2802 141D}} || {{nowrap|8209 2805 12CA}} || {{nowrap|8209 2808 116E}} || {{nowrap|8209 280B 100C}} || {{nowrap|8209 280E 0EAE}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Sep 26}} || {{nowrap|8209 2802 175B}} || {{nowrap|8209 2805 160B}} || {{nowrap|8209 2808 14B4}} || {{nowrap|8209 280B 1353}} || {{nowrap|8209 280E 11F2}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Oct 26}} || {{nowrap|8209 2802 1A9A}} || {{nowrap|8209 2805 194C}} || {{nowrap|8209 2808 17F9}} || {{nowrap|8209 280B 169C}} || {{nowrap|8209 280E 153A}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Nov 24}} || {{nowrap|8209 2802 1DDA}} || {{nowrap|8209 2805 1C8C}} || {{nowrap|8209 2808 1B3D}} || {{nowrap|8209 280B 19E5}} || {{nowrap|8209 280E 1885}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Dec 24}} || {{nowrap|8209 2802 211C}} || {{nowrap|8209 2805 1FCD}} || {{nowrap|8209 2808 1E7F}} || {{nowrap|8209 280B 1D2C}} || {{nowrap|8209 280E 1BCF}} |- style="background-color: #eaecf0; font-size: large; font-weight: bold;" || || {{nowrap|2012}} || {{nowrap|2031}} || {{nowrap|2050}} || {{nowrap|2069}} || {{nowrap|2088}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jan 23}} || {{nowrap|8209 2802 2460}} || {{nowrap|8209 2805 230E}} || {{nowrap|8209 2808 21C0}} || {{nowrap|8209 280B 2070}} || {{nowrap|8209 280E 1F19}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Feb 21}} || {{nowrap|8209 2802 27A6}} || {{nowrap|8209 2805 2650}} || {{nowrap|8209 2808 2500}} || {{nowrap|8209 280B 23B2}} || {{nowrap|8209 280E 225F}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Mar 22}} || {{nowrap|8209 2802 2AED}} || {{nowrap|8209 2805 2992}} || {{nowrap|8209 2808 2840}} || {{nowrap|8209 280B 26F2}} || {{nowrap|8209 280E 25A2}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Apr 21}} || {{nowrap|8209 2802 2E35}} || {{nowrap|8209 2805 2CD6}} || {{nowrap|8209 2808 2B80}} || {{nowrap|8209 280B 2A31}} || {{nowrap|8209 280E 28E3}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|May 20}} || {{nowrap|8209 2802 317D}} || {{nowrap|8209 2805 301B}} || {{nowrap|8209 2808 2EC0}} || {{nowrap|8209 280B 2D6E}} || {{nowrap|8209 280E 2C21}} |- style="font-size:small:small;background-color:#ffff00;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jun 19}} || {{nowrap|8209 2802 34C3}} || {{nowrap|8209 2805 3361}} || {{nowrap|8209 2808 3202}} || {{nowrap|8209 280B 30AC}} || {{nowrap|8209 280E 2F5E}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jul 18}} || {{nowrap|8209 2802 3807}} || {{nowrap|8209 2805 36A7}} || {{nowrap|8209 2808 3545}} || {{nowrap|8209 280B 33EB}} || {{nowrap|8209 280E 329A}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Aug 16}} || {{nowrap|8209 2802 3B48}} || {{nowrap|8209 2805 39ED}} || {{nowrap|8209 2808 388A}} || {{nowrap|8209 280B 372C}} || {{nowrap|8209 280E 35D7}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Sep 15}} || {{nowrap|8209 2802 3E89}} || {{nowrap|8209 2805 3D31}} || {{nowrap|8209 2808 3BD1}} || {{nowrap|8209 280B 3A70}} || {{nowrap|8209 280E 3916}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Oct 14}} || {{nowrap|8209 2802 41C8}} || {{nowrap|8209 2805 4076}} || {{nowrap|8209 2808 3F1A}} || {{nowrap|8209 280B 3DB8}} || {{nowrap|8209 280E 3C59}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Nov 13}} || {{nowrap|8209 2802 4508}} || {{nowrap|8209 2805 43B9}} || {{nowrap|8209 2808 4262}} || {{nowrap|8209 280B 4102}} || {{nowrap|8209 280E 3FA0}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Dec 13}} || {{nowrap|8209 2802 4849}} || {{nowrap|8209 2805 46FB}} || {{nowrap|8209 2808 45A8}} || {{nowrap|8209 280B 444C}} || {{nowrap|8209 280E 42EA}} |- style="background-color: #eaecf0; font-size: large; font-weight: bold;" || || {{nowrap|2013}} || {{nowrap|2032}} || {{nowrap|2051}} || {{nowrap|2070}} || {{nowrap|2089}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jan 11}} || {{nowrap|8209 2802 4B8A}} || {{nowrap|8209 2805 4A3C}} || {{nowrap|8209 2808 48ED}} || {{nowrap|8209 280B 4795}} || {{nowrap|8209 280E 4635}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Feb 10}} || {{nowrap|8209 2802 4ECC}} || {{nowrap|8209 2805 4D7D}} || {{nowrap|8209 2808 4C2F}} || {{nowrap|8209 280B 4ADC}} || {{nowrap|8209 280E 4980}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Mar 11}} || {{nowrap|8209 2802 520F}} || {{nowrap|8209 2805 50BD}} || {{nowrap|8209 2808 4F6F}} || {{nowrap|8209 280B 4E1F}} || {{nowrap|8209 280E 4CC8}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Apr 10}} || {{nowrap|8209 2802 5553}} || {{nowrap|8209 2805 53FD}} || {{nowrap|8209 2808 52AE}} || {{nowrap|8209 280B 5160}} || {{nowrap|8209 280E 500D}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|May 9}} || {{nowrap|8209 2802 5899}} || {{nowrap|8209 2805 573E}} || {{nowrap|8209 2808 55EC}} || {{nowrap|8209 280B 549E}} || {{nowrap|8209 280E 534F}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jun 8}} || {{nowrap|8209 2802 5BDF}} || {{nowrap|8209 2805 5A80}} || {{nowrap|8209 2808 592A}} || {{nowrap|8209 280B 57DB}} || {{nowrap|8209 280E 568D}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jul 7}} || {{nowrap|8209 2802 5F26}} || {{nowrap|8209 2805 5DC4}} || {{nowrap|8209 2808 5C69}} || {{nowrap|8209 280B 5B17}} || {{nowrap|8209 280E 59C9}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Aug 6}} || {{nowrap|8209 2802 626B}} || {{nowrap|8209 2805 6109}} || {{nowrap|8209 2808 5FAA}} || {{nowrap|8209 280B 5E54}} || {{nowrap|8209 280E 5D06}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Sep 4}} || {{nowrap|8209 2802 65AF}} || {{nowrap|8209 2805 6450}} || {{nowrap|8209 2808 62EE}} || {{nowrap|8209 280B 6194}} || {{nowrap|8209 280E 6043}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Oct 4}} || {{nowrap|8209 2802 68F3}} || {{nowrap|8209 2805 6797}} || {{nowrap|8209 2808 6635}} || {{nowrap|8209 280B 64D7}} || {{nowrap|8209 280E 6382}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Nov 2}} || {{nowrap|8209 2802 6C35}} || {{nowrap|8209 2805 6ADF}} || {{nowrap|8209 2808 697F}} || {{nowrap|8209 280B 681D}} || {{nowrap|8209 280E 66C4}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Dec 2}} || {{nowrap|8209 2802 6F77}} || {{nowrap|8209 2805 6E25}} || {{nowrap|8209 2808 6CC9}} || {{nowrap|8209 280B 6B67}} || {{nowrap|8209 280E 6A08}} |- style="background-color: #eaecf0; font-size: large; font-weight: bold;" || || {{nowrap|2014}} || {{nowrap|2033}} || {{nowrap|2052}} || {{nowrap|2071}} || {{nowrap|2090}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jan 1}} || {{nowrap|8209 2802 72B8}} || {{nowrap|8209 2805 7169}} || {{nowrap|8209 2808 7012}} || {{nowrap|8209 280B 6EB2}} || {{nowrap|8209 280E 6D50}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; 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background-color: #eaecf0;" | {{nowrap|May 28}} || {{nowrap|8209 2802 82FE}} || {{nowrap|8209 2805 81A8}} || {{nowrap|8209 2808 8058}} || {{nowrap|8209 280B 7F0A}} || {{nowrap|8209 280E 7DB8}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jun 26}} || {{nowrap|8209 2802 8642}} || {{nowrap|8209 2805 84E7}} || {{nowrap|8209 2808 8395}} || {{nowrap|8209 280B 8247}} || {{nowrap|8209 280E 80F7}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Jul 26}} || {{nowrap|8209 2802 8988}} || {{nowrap|8209 2805 8828}} || {{nowrap|8209 2808 86D2}} || {{nowrap|8209 280B 8583}} || {{nowrap|8209 280E 8435}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Aug 24}} || {{nowrap|8209 2802 8CCE}} || {{nowrap|8209 2805 8B6C}} || {{nowrap|8209 2808 8A11}} || {{nowrap|8209 280B 88C0}} || {{nowrap|8209 280E 8772}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Sep 22}} || {{nowrap|8209 2802 9015}} || {{nowrap|8209 2805 8EB3}} || {{nowrap|8209 2808 8D54}} || {{nowrap|8209 280B 8BFF}} || {{nowrap|8209 280E 8AB0}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Oct 22}} || {{nowrap|8209 2802 935C}} || {{nowrap|8209 2805 91FC}} || {{nowrap|8209 2808 909B}} || {{nowrap|8209 280B 8F40}} || {{nowrap|8209 280E 8DEF}} |- style="font-size:small:small;background-color:#ffffff;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Nov 21}} || {{nowrap|8209 2802 96A1}} || {{nowrap|8209 2805 9546}} || {{nowrap|8209 2808 93E4}} || {{nowrap|8209 280B 9285}} || {{nowrap|8209 280E 912F}} |- style="font-size:small:small;background-color:#ffaaaa;” | style="font-weight: bold; background-color: #eaecf0;" | {{nowrap|Dec 21}} || {{nowrap|8209 2802 99E5}} || {{nowrap|8209 2805 988E}} || {{nowrap|8209 2808 972F}} || {{nowrap|8209 280B 95CD}} || {{nowrap|8209 280E 9473}} |} c347flmoz5bnahscq51vkaopdlpbm70 2 329527 2812059 2807897 2026-05-29T20:38:03Z AIfriendly 3069390 2812059 wikitext text/x-wiki https://en.wikiversity.org/wiki/Physics/Essays/AIfriendly ; neutrino gradient induced gravitation https://en.wikiversity.org/wiki/Talk:Consciousness#everything_is_made_of_and_causal_with_consciousness_that_has_always_existed_everywhere cb3fm8m2hiul9z4sg27gszhjckzjh3u 0 329703 2812031 2811731 2026-05-29T16:14:48Z IgnorantObserver 3076980 Distinguish operational tracking rate C_eff = r·b·f from Landauer ceiling (Landauer is consistency limit, not operative rate); reframe controlled variable as C_eff directly via loop bandwidth/packet-drop/bit-depth, with P demoted to one possible actuator; adopt Palmer (2024) 'superdeterminism without conspiracy' framing for epistemically bounded ancestral correlation; explicit hosting of Born weights from no-collapse embedding (pilot-wave/Everett); recast Heisenberg cut as designed-and-movabl... 2812031 wikitext text/x-wiki {{Research project}} = The Ignorant Observer Framework = ''This research page is authored and maintained by [[User:IgnorantObserver|Aernoud Dekker]], an independent researcher and the originator of the framework described below. Page text is offered for review, critique, and collaborative refinement under [[Wikiversity:Copyrights|Wikiversity's standard licence]].'' == Status == Research project under active development. The framework consists of an interlinked set of technical and interpretive documents published at [https://ignorantobserver.xyz ignorantobserver.xyz] and archived on the [https://osf.io Open Science Framework]. ''The Ignorant Observer'' is the foundational paper. A conceptual bridge, ''The Measurement Problem in IOF'', states what claim the framework is actually making about the measurement basis. The technical bridge, ''Bandwidth-Limited Quantum Control'' (BLQC), sets out the framework's falsifiable experimental discriminator. A companion paper, ''A Conditional Derivation of the Binary Born Form under Bandwidth-Limited Quantum Control'', derives the binary Born form in the laboratory basis coordinate of a BLQC experiment via a Fisher capacity bridge, conditional on two named bridge assumptions. All work is single-authored. == Summary == The Ignorant Observer Framework proposes that the conventional treatment of quantum measurement idealizes the measurement basis as stably available to the observer. The framework removes that idealization. It treats the measurement basis θ as a physical dynamical variable inside the apparatus, with its own causal history and its own information-production rate. The measurement setting and the measured system are read as descendants of one physical history, not as ancestrally independent ingredients dropped into the experiment from outside. The framework's position on this point is measurement dependence in the technical, non-conspiratorial sense defended by Palmer (2024): the setting and the system share causal ancestry, so Bell's statistical-independence premise is not imposed, but in a single globally consistent history the correlation is structural rather than fine-tuned. The qualifier ''epistemically bounded ancestral correlation'' adds that the embedded observer cannot, in principle, reconstruct the joint causal ancestry of basis and outcome — so the shared ancestry is not a predictive hidden-variable ledger, and the situation must be represented probabilistically. This is superdeterminism in the technical sense and not in the colloquial conspiratorial one; it is distinguished both from fine-tuned conspiracy and from a completed deterministic theory of the 't Hooft type. Whether the apparatus can stably track θ is a control-theoretic question, governed by an inequality between effective information-channel capacity and the basis-defining dynamics' entropy rate. ''Bandwidth-Limited Quantum Control'' (BLQC), the framework's technical bridge to the laboratory, derives — under the assumptions catalogued in the [[#Open objections|Open objections]] section below — a distinctive ''double-exponential'' visibility decay law and a multi-axis falsifiable discriminator. The central test asks whether the visibility-breakdown time ''t''_break moves with the BLQC deficit κ = ''h''_KS − ''C''_eff ln 2 under independent calibrated variation of effective tracking capacity ''C''_eff, basis-instability rate ''h''_KS, and mass geometry. The two candidate mechanisms — finite basis tracking and Penrose Objective Reduction — are not treated as mutually exclusive: the mesoscopic overlap regime is analysed with an additive combined-rate model, and the discriminator is the ''derivative'' of ''t''_break with respect to each independently varied knob (''C''_eff and ''h''_KS at fixed mass geometry isolate the basis-tracking channel; mass, separation, and geometry at fixed ''C''_eff isolate objective reduction). Controller input power ''P'' is one possible actuator for ''C''_eff, not the central variable. A separate Fisher-homogeneity module of the protocol tests the Born-derivation bridge by measuring whether the empirical Fisher information ''I''(θ) is approximately constant across the calibrated basis range. The framework's principal implication for the measurement problem is structural: the Heisenberg cut — the boundary between quantum description and classical record — is not an interpretive convention but an operational boundary fixed by the apparatus's finite basis-tracking budget, with the Landauer bound entering only as a thermodynamic ceiling rather than the operative rate (see [[#The measurement problem: where the Heisenberg cut sits|the measurement problem: where the Heisenberg cut sits]] below). The double-exponential visibility law and the binary-Born derivation are two consequences of this single reframe, both pinned by the same scalar threshold κ and tested by the same prospective experiment. A companion paper develops a conditional derivation of the binary Born form ''p''(θ) = cos²(θ/2) directly in the laboratory basis coordinate of a BLQC experiment. The derivation chains BLQC finite-rate basis tracking → a ''Fisher capacity bridge'' identifying ''C''_eff with capacity for preserving operational distinguishability of finite observer records → Cencov's uniqueness theorem selecting Fisher–Rao as the invariant distinguishability metric → square-root record coordinates → scalar-threshold homogeneity of κ = ''h''_KS − ''C''_eff ln 2 in θ. The conditional weight is carried by two explicit, named premises — the Fisher capacity bridge and scalar-threshold homogeneity — both empirically testable. The derivation does not derive complex Hilbert space, tensor products, unitary dynamics, or the multi-outcome Born rule. In the updated framing, the binary-Born derivation and the BLQC basis-tracking story are no longer two separate IOF-internal moves: the metric in which finite-rate basis tracking succeeds or fails is the same Fisher–Rao metric that forces the binary probability form, and the same scalar BLQC threshold pins both. They are two consequences of one operational geometry. The framework as a whole also offers an interpretive extension that connects the technical proposal to existing positions in quantum foundations (Brukner, Rovelli's relational quantum mechanics) and to non-dual philosophy of mind (Advaita Vedānta). These interpretive elements are clearly fenced from the empirical core in [[#Philosophical interpretation|the relevant section below]]. What stands or falls with the experimental discriminator is the framework's specific physical mapping into these positions, not the positions themselves. == Core question == ''Can quantum visibility depend on finite observer or apparatus basis-tracking capacity, independently of, and distinguishably from, ordinary environmental decoherence?'' Phrased positively: if the classical degrees of freedom that define and maintain a measurement basis exhibit chaotic dynamics with positive Kolmogorov–Sinai entropy rate ''h''_KS, and if the effective information channel that constrains those degrees of freedom has capacity ''C''_eff insufficient to track them, does interference visibility decay in a functional form distinguishable from standard exponential or Gaussian dephasing — and does this decay respond to controller input power in a direction opposite to thermal decoherence? == Technical proposal == The framework introduces the following quantities. '''Effective channel capacity ''C''_eff''' (bits/s): the information rate available to the basis-tracking control loop, operationalised as :''C''_eff = ''r'' · ''b'' · ''f'' with ''r'' the update rate (Hz), ''b'' the effective number of bits per update that constrain the basis variable θ, and ''f'' ∈ (0,1] the fraction of updates that genuinely constrain θ after overhead and latency. ''C''_eff is bounded above by the Landauer limit on the controller's actuation: :''C''_eff ≤ ''P'' / (''k''_B ''T'' ln 2) where ''P'' is controller input power and ''T'' is the temperature at which the controller operates. '''Kolmogorov–Sinai entropy rate ''h''_KS''' (nats/s): the information-production rate of the classical degrees of freedom (voltage references, timing circuits, feedback loops) that define and maintain the measurement basis. For chaotic systems, ''h''_KS equals the sum of positive Lyapunov exponents (Pesin identity). It is estimated operationally from the exponential growth of one-step prediction error on logged controller states. The nats/s convention is used so that the deficit κ below combines ''h''_KS (nats/s) and ''C''_eff ln 2 (bits/s converted to nats/s) in consistent units; an equivalent all-bits form would be κ_bits = ''h''_KS,bits − ''C''_eff. '''Ignorance rate κ''' (s⁻¹): :κ = ''h''_KS − ''C''_eff · ln 2 The framework distinguishes two regimes. When κ < 0 (''capacity-wins''), basis-tracking error stays bounded and standard quantum visibility predictions are recovered in the BLQC correction-free limit, modulo ordinary decoherence. When κ > 0 (''chaos-wins''), the variance of the basis-tracking error grows exponentially in time as σ_θ²(''t'') = σ₀² e^2κ''t''. '''Measured visibility ''V''(''t'')'''. Averaging the interference term cos(φ − θ) over a Gaussian distribution of basis-tracking error δθ ∼ ''N''(0, σ_θ²(''t'')) yields, in the small-angle regime, :''V''(''t'') = exp(−½ σ₀² e^2κ''t'') i.e. a ''double-exponential'' decay of visibility once the chaos-wins regime is entered. '''Two visibility channels'''. The basis-tracking loss is one of two multiplicative contributions to the observed interference visibility: :''V''_obs ≈ ''V''_std · ''V''_IOF, with ''V''_IOF = exp(−½ σ_θ²) where ''V''_std is the ordinary environmental/decoherence channel — the visibility standard quantum mechanics already predicts — and ''V''_IOF is the finite basis-tracking channel derived above. The framework does not deny ''V''_std; it claims that, in the chaos-wins regime, part of the observed visibility loss belongs to ''V''_IOF and may be misassigned to standard decoherence if the capacity-instability coordinate κ is not independently varied and tested. '''Breakdown time ''t''_break'''. For a chosen visibility threshold ''V''*, :''t''_break = (1 / 2κ) · ln(−2 ln ''V''* / σ₀²) for κ > 0. ''t''_break is the framework's primary observable. The technical derivation extends the Data-Rate Theorem of Nair & Evans (2004) and Tatikonda & Mitter (2004) from linear plants to nonlinear, chaotic systems by substituting ''h''_KS for the sum-of-positive-eigenvalues bound. This extension is an explicit assumption of the framework rather than a proven theorem (see [[#Open objections|Open objections]]). == Experimental discriminator == The framework prescribes the following experimental protocol as its central falsifiable test. '''Primary controlled variable''': the useful basis-tracking capacity ''C''_eff, varied directly through the tracking loop — for example by changing the accepted update rate, the useful bit depth, the estimator bandwidth, or by imposing a calibrated packet-drop schedule. Controller input power ''P'' is one possible actuator for ''C''_eff, not the central knob, and is used only insofar as it produces an independently calibrated change in ''C''_eff. The controller is the physical system whose state defines and maintains the measurement basis (e.g. an interferometer phase-locking loop, a qubit readout chain, the active feedback in a precision interferometer). '''Held constant''': mass geometry, the environmental temperature ''T'', readout signal-to-noise, latency, pulse/actuator behaviour, and plant dynamics. Varying ''C''_eff while these ordinary confounds are clamped is what distinguishes the framework's prediction from standard thermal decoherence (which depends on ''T'' and is indifferent to tracking capacity). '''Dependent variable''': the visibility-decay breakdown time ''t''_break, fitted to interference data at a chosen visibility threshold (e.g. ''V''* = 0.5). '''Prediction''': ∂''t''_break/∂''C''_eff > 0 at clamped confounds (and, where ''P'' is the actuator that raises ''C''_eff, ∂''t''_break/∂''P'' > 0 at clamped ''T'' — a direction opposite to thermal decoherence), with the visibility curve ''V''(''t'') fitting the double-exponential form exp(−½ σ₀² e^2κ''t'') better than a standard exponential ''e''^−Γ''t'' or Gaussian ''e''^−γ''t''². '''What would count as falsification'''. Any of the following null findings counts against the framework: * ∂''t''_break/∂''C''_eff ≤ 0 at clamped confounds (i.e. increasing useful tracking capacity does not extend, or shortens, coherence time); * ''V''(''t'') fits a single-exponential or Gaussian dephasing law significantly better than the double-exponential form, in the regime where the framework predicts the double-exponential should dominate; * ''t''_break shows no dependence on ''C''_eff or ''h''_KS at fixed mass geometry once ordinary confounds are controlled — i.e. the capacity-instability coordinate κ adds no predictive value beyond a mass-geometry timescale ''t''_OR ∝ ''s'' / ''m''² (a positive [[w:Penrose interpretation|Penrose Objective Reduction]] mass-geometry dependence does not by itself count against the framework, since the two mechanisms are treated as additive, not mutually exclusive); * ''C''_eff cannot be calibrated independently of ''t''_break (in which case the prediction would be unfalsifiable, which would itself count against the framework's experimental status). The [https://www.qgemproject.com/ QGEM] pathfinder is cited in the BLQC manuscript as one candidate testbed; superconducting-qubit readout chains and precision interferometer phase-locking loops are others. The framework's comprehensive experimental protocol additionally includes a ''Fisher-homogeneity module'' that tests the Born-derivation bridge. The module measures the Fisher information ''I''(θ) on the operational record family ''p''(''o'' | θ) across the calibrated basis range and asks whether ''I''(θ) is approximately constant, as required by the scalar-threshold homogeneity premise of the [[#Relation to quantum foundations|binary-Born derivation]]. The Fisher-homogeneity module is logically independent of the κ-scaling test of the basis-tracking claim: a BLQC-positive but Fisher-negative result would validate finite-rate basis tracking as a real physical channel while rejecting the binary-Born-derivation bridge as drafted. Simultaneous κ-scaling and Fisher homogeneity would support the stronger claim that one operational geometry controls both basis tracking and binary probability. == Relation to quantum foundations == The framework is connected to, and partly draws from, several existing positions in the foundations of quantum mechanics. * '''The measurement problem'''. The framework's principal claim about the measurement problem is structural rather than dynamical: the Heisenberg cut is an operational boundary set by the apparatus's finite basis-tracking budget — a partly engineered rate, with the Landauer bound only a thermodynamic ceiling — not a floating interpretive convention. The measurement problem appears in its sharpest form because standard accounts treat the cut as freely movable; the framework holds it was always located by the basis-tracking budget the apparatus actually devotes to its reference. The conceptual claim is developed in [[#The measurement problem: where the Heisenberg cut sits|the measurement problem: where the Heisenberg cut sits]] below and at full length in ''[https://ignorantobserver.xyz/documents/The_Measurement_Problem_in_IOF.pdf The Measurement Problem in IOF]''. * '''Brukner's information-theoretic reconstructions''' provide a precedent for treating information limits as structural constraints in quantum theory. * '''Relational Quantum Mechanics''' (Rovelli) takes measurement outcomes to be relative to an observer-system; the framework provides one possible mechanism (finite ''C''_eff) for what makes one observer's frame physically inequivalent to another's. * '''Decoherence theory''' is not opposed by the framework. The framework's prediction sits beside ordinary environmental decoherence and is intended to be ''distinguishable'' from it by the sign-reversal under power variation; in the capacity-wins regime (κ < 0) standard decoherence theory is recovered. * '''Measurement-independence'''. Because the framework treats the measurement basis as a dynamical variable with its own causal history, if extended to Bell-type set-ups it implies a violation of statistical measurement-independence. Named plainly, this is superdeterminism in the technical, non-conspiratorial sense defended by Palmer (2024): the setting and the system share causal ancestry, so statistical independence is not imposed, but in a single globally consistent history the correlation is structural, not fine-tuned. The qualifier "epistemically bounded ancestral correlation" adds that the embedded observer cannot reconstruct that ancestry in principle, so the shared ancestry is not a hidden knob for prediction. The framework adopts the technical label and rejects the conspiratorial one, and is likewise distinguished from a completed deterministic theory of the 't Hooft type. It does not derive Bell correlations from first principles: the Born weights and the standard quantum correlations are inherited from a hosted no-collapse embedding (pilot-wave- or Everett-type) and recovered in the capacity-wins limit, and the framework asks only whether finite basis access adds a measurable visibility factor when tracking is stressed. A proper consistency proof, including no-signalling treatment, remains an open question (see [[#Open objections|Open objections]]). * '''Information geometry'''. The framework's binary-Born derivation runs a directional chain: BLQC finite-rate basis tracking → a ''Fisher capacity bridge'' identifying ''C''_eff with capacity for preserving operational distinguishability of finite observer records → Cencov's uniqueness theorem selecting Fisher–Rao as the invariant distinguishability metric under sufficient Markov morphisms → square-root record coordinates → scalar-threshold homogeneity of κ = ''h''_KS − ''C''_eff ln 2 in the laboratory basis coordinate → ''p''(θ) = cos²(θ/2). The connection between statistical distance and quantum transition probabilities is not new — Wootters (1981) showed that quantum distinguishability is naturally expressed in terms of statistical distance — but the framework runs the logic in the opposite direction: it starts from finite-observer record constraints, invokes Cencov uniqueness, and obtains the squared-coordinate binary form from the resulting record geometry, with the laboratory basis coordinate θ identified as the Fisher-arclength-affine coordinate by the BLQC scalar-threshold reading. The binary-Born derivation and the BLQC basis-tracking visibility law are therefore tied to the same operational geometry: the Fisher–Rao metric on records is the metric in which BLQC tracking is calibrated, and the same scalar threshold pins both the basis-tracking task and the binary probability form. * '''Penrose Objective Reduction''' is treated as a ''non-exclusive'', potentially co-contributing mechanism rather than a rival to be ruled out. In the mesoscopic overlap regime both effects can act together; the framework's protocol analyses the overlap with an additive combined-rate model (alongside mediated and collinear "Bridge-Ansatz" alternatives, within the same regression) and discriminates the contributions by their distinct knobs — ''C''_eff and ''h''_KS drive the basis-tracking channel, while mass, separation, and geometry drive ''t''_OR ∝ ''s'' / ''m''². The numerical proximity of the two timescales in the mesoscopic regime motivates the protocol described in the next section and is treated as suggestive pending experimental evidence. == The measurement problem: where the Heisenberg cut sits == The framework offers a specific reframing of the Heisenberg cut — the boundary between the quantum description used for the measured system and the classical description used for the apparatus and the record. Standard interpretations have placed the cut variously: Von Neumann showed the cut can be moved without changing predictions and treated its location as conventional; decoherence theory sharpens the picture but locates the cut by an external property, the rate of environmental coupling; objective-collapse proposals fix the cut universally at a mass or geometry scale, without reference to who is observing. The framework places the cut where the observer-apparatus system's ''useful'' basis-tracking rate runs out relative to its basis-producing dynamics. The operative quantity is the effective rate ''C''_eff = ''r'' · ''b'' · ''f'' that genuinely constrains the reference, not the Landauer ceiling: the Landauer bound ''C''_eff ≤ ''P'' / (''k''_B ''T'' ln 2) is a thermodynamic upper limit on irreversible bookkeeping, typically far above the modest rate any one tracking loop actually devotes to the basis, and enters only as a consistency ceiling. The cut sits at the locus where ''h''_KS = ''C''_eff ln 2: on one side the basis-producing dynamics run slower than the useful tracking rate and standard quantum visibility predictions are recovered (modulo ordinary decoherence); on the other side the dynamics outrun the tracking rate and visibility decays with the deficit κ = ''h''_KS − ''C''_eff ln 2. A consequence a purely thermodynamic framing would obscure is that the cut is set largely by ''design'': the experimenter can move it deliberately — by throttling or widening the tracking loop, changing estimator bandwidth, or imposing a calibrated packet-drop schedule — at fixed temperature and power. The cut is therefore observer-relative — two apparatuses tracking the same basis with different loop designs, power budgets, or temperatures will have their cuts at different places — but not subjective. For any given configuration the cut is fixed by that configuration, and any observer inspecting the same hardware agrees on where it sits; what the experimenter controls is the configuration, not the verdict it then yields. This also predicts something conventional cut placement does not: the cut ''moves''. Cooling the apparatus, increasing the available power, or improving the controller raises ''C''_eff and shifts the cut outward, toward more chaotic basis-producing dynamics. The BLQC test, in this language, is an experiment that measures the motion of the cut. The measurement problem has historically taken its sharpest form because the Heisenberg cut was treated as floating. The framework's claim is narrower and testable: for a given finite apparatus the cut is not floating but located, by the basis-tracking budget that apparatus actually devotes to its reference — and the standard interpretations were not reading that ledger. == Philosophical interpretation == ''This section describes interpretive extensions of the framework that go beyond the empirical core. Nothing in this section is a load-bearing element of the experimental claim. If the experimental discriminator returns a null result, the claimed physical realization of these interpretive readings within the framework would fall. The interpretive positions themselves — Advaita Vedānta, relational quantum mechanics — do not stand or fall on an interferometry experiment; what stands or falls is the framework's specific physical mapping into them.'' The most direct, accessible statement of the framework's interpretive position is ''[https://ignorantobserver.xyz/documents/The_Measurement_Problem_in_IOF.pdf The Measurement Problem in IOF]'' (Dekker, May 2026). This conceptual companion to BLQC states the central move — the measurement basis as a physical variable with causal ancestry inside the same history as the system being measured — addresses the standard objections (does this just move the mystery, is this just correctable reference noise, is this just control engineering), and names the position ''epistemically bounded ancestral correlation''. Readers approaching the framework for the first time may find this the cleanest entry point. A second, distinct interpretive piece is ''[https://ignorantobserver.xyz/documents/Response_to_Rovelli_on_the_Hard_Problem.pdf The Hard Problem Dissolved — But Into What? A Critical Response to Carlo Rovelli's "There Is No 'Hard Problem of Consciousness'"]'' (Dekker, May 2026). The response engages Rovelli's Noema essay, marks the substantial ground it shares with the framework, and identifies where the framework presses beyond Rovelli's deflationary physicalism toward a non-dual reading. The framework's interpretive layer is developed in dialogue with two existing positions. The first is Carlo Rovelli's relational quantum mechanics. The framework can be read as supplying a candidate physical mechanism — the ''C''_eff versus ''h''_KS inequality — for what makes a measurement outcome relative to an observer rather than absolute. On this reading, the framework is a mechanistic specification of an idea that RQM leaves at the level of principle. The second is the Advaita Vedānta tradition (Śaṅkara, Ramaṇa Mahaṛṣi), in which the apparent independence of the experiencing subject from the perceived world is treated as a structural feature of ignorance (''avidyā'') rather than a metaphysical fact. The framework's σ_θ²(''t'') — the growing basis-tracking error of an observer whose capacity is insufficient to track its own apparatus — admits a structural analogy with avidyā as the phenomenological self-opacity of an embodied subject. The framework neither asserts that this analogy is more than structural nor that any experimental result could confirm or refute Advaita as a philosophical position; it offers the analogy as a way of locating the framework within a non-dual reading of the measurement problem for readers who find that reading useful. A separate, IOF-internal derivation paper — ''[https://ignorantobserver.xyz/documents/A_Conditional_Born-Rule_Derivation.pdf A Conditional Derivation of the Binary Born Form under Bandwidth-Limited Quantum Control]'' — derives the binary Born form ''p''(θ) = cos²(θ/2) in the laboratory basis coordinate of a BLQC experiment, via a Fisher capacity bridge from BLQC tracking capacity to Fisher–Rao record geometry. Its metaphysical companion, ''[https://ignorantobserver.xyz/documents/Structural_Resonance.pdf Structural Resonance]'', explains how a structural reading of the ''Katha Upaniṣad'' (subject and witness, layered cognition, invariance under refinement) served as a disciplined search heuristic for the mathematical derivation. The companion does not claim that Vedanta proves the Born rule; it documents the structural overlap between an old analysis of finite observation and a contemporary information-geometric derivation. Readers who prefer to ignore the interpretive readings should be able to evaluate the framework's empirical content from the [[#Technical proposal|Technical proposal]] and [[#Experimental discriminator|Experimental discriminator]] sections alone. A further speculative extension, ''[https://ignorantobserver.xyz/documents/The_Creation_of_Duality.pdf The Creation of Duality]'', asks whether space, time, objecthood, and gravity-like structure can themselves be read as features of a consistent finite-observer world-model, with a Bridge Ansatz ''E''_G = (π/2)ℏκ linking the deficit rate κ to a gravitational energy scale via Margolus–Levitin saturation. Its scientific status is contingent on the BLQC experimental discriminator; until then it is offered explicitly as speculation. == Consequences of a positive result == If the experimental discriminator returns the predicted result, several interpretive readings of the framework gain physical support rather than remaining speculative. ''Quantum mechanics as an observer-capacity-dependent regime.'' The framework's "chaos-wins" / "capacity-wins" distinction becomes a physical, not merely conceptual, partition. Standard quantum predictions are recovered to high accuracy in the capacity-wins regime; the framework predicts measurable departures in the chaos-wins regime. The quantum-classical transition then becomes information-theoretic and, in principle, controllable: throttling effective controller capacity should push a system across the transition without changing the plant. ''An epistemic reading of measurement.'' The framework's no-collapse account — measurement as an information-update inside a finite observer rather than a physical event in the world — becomes empirically defensible alongside other interpretations of the measurement problem, rather than a stipulation. ''Measurement-independence and locality.'' The framework's response to the conventional "conspiracy" objection against superdeterminism (common causal past plus a global consistency constraint, in place of fine-tuned initial conditions) becomes a substantive position rather than a philosophical reframing. Whether this amounts to a non-conspiratorial reading consistent with local realism remains a live debate; a positive result moves that debate from speculation onto experimental terrain. ''The Penrose-Objective-Reduction comparison.'' The framework's basis-tracking contribution depends on controller bandwidth rather than mass or geometry. Because the two mechanisms are treated as additive rather than mutually exclusive, the discriminating evidence is a ''t''_break dependence on ''C''_eff and ''h''_KS at fixed mass geometry — which isolates the basis-tracking channel whether or not a Penrose mass-geometry term is also present. ''The interpretive analogy.'' The structural analogy between σ_θ²(''t'') and the Vedantic notion of ''avidyā'' gains a concrete physical anchor rather than remaining purely analogical. The framework's claim is structural rather than metaphysical; a positive result strengthens the structural mapping, but does not itself adjudicate the philosophical positions the mapping connects. None of these consequences is established by the experimental discriminator on its own. What the test establishes, if positive, is that the framework's bridge from a control-theoretic measurement model to these interpretive readings has a physical basis. The interpretive work in each direction remains. == Documents == The framework's documents are published at [https://ignorantobserver.xyz ignorantobserver.xyz]. Direct links to the principal documents, grouped by their role in the project: '''Foundational and bridges''' * '''[https://ignorantobserver.xyz/documents/The_Ignorant_Observer.pdf The Ignorant Observer]''' — the foundational paper. Both the philosophical motivation (avidyā as structural ignorance) and the technical groundwork from which the rest of the project grew. * '''[https://ignorantobserver.xyz/documents/The_Measurement_Problem_in_IOF.pdf The Measurement Problem in IOF]''' — the conceptual bridge. States what claim the framework is making about the measurement basis, addresses the standard objections, and names the framework's position as ''epistemically bounded ancestral correlation''. * '''[https://ignorantobserver.xyz/documents/Bandwidth-Limited_Quantum_Control.pdf Bandwidth-Limited Quantum Control]''' — the technical bridge. A finite-rate phase-reference test in the Penrose-overlap regime. The framework's falsifiable experimental discriminator. * '''[https://ignorantobserver.xyz/documents/Concise_Mathematical_Summary.pdf Concise Mathematical Summary]''' — shortest formal map of the IOF variables and BLQC test regimes. * '''[https://ignorantobserver.xyz/documents/Experimental_Protocol.pdf Comprehensive Experimental Protocol]''' — preregistered prospective experiment discriminating a Penrose-style mass-geometry timescale from the BLQC capacity / instability timescale in the same mesoscopic apparatus. * '''[https://ignorantobserver.xyz/documents/Questions_and_Answers_IOF.pdf Questions and Answers (IOF)]''' — common questions on the framework addressed in depth. '''Foundational Extensions''' * '''[https://ignorantobserver.xyz/documents/A_Conditional_Born-Rule_Derivation.pdf A Conditional Derivation of the Binary Born Form under Bandwidth-Limited Quantum Control]''' — derives the binary Born form ''p''(θ) = cos²(θ/2) directly in the laboratory basis coordinate of a BLQC experiment, via a Fisher capacity bridge from BLQC tracking capacity to Fisher–Rao record geometry. The conditional weight is carried by two named, empirically testable assumptions (Fisher capacity bridge, scalar-threshold homogeneity). Does not derive complex Hilbert space, tensor products, unitary dynamics, or the multi-outcome Born rule. Supersedes an earlier version in which the binary Born form was obtained only in Fisher arclength. * '''[https://ignorantobserver.xyz/documents/Structural_Resonance.pdf Structural Resonance: A Metaphysical Companion to the Conditional Born-Rule Derivation]''' — explains how a structural reading of the ''Katha Upaniṣad'' served as a disciplined search heuristic for the derivation. Does not claim that Vedanta proves the Born rule. '''Supplements''' * '''[https://ignorantobserver.xyz/documents/Forensic_Signatures.pdf Forensic Signatures]''' — retrospective screening of Chinese 63-qubit, Google Sycamore, and LIGO data for the double-exponential visibility decay signature predicted by BLQC. Motivating evidence for treating LIGO as a candidate regime; not causal attribution. Detailed findings and caveats are discussed in [[#Open objections|Open objections]]. * '''[https://ignorantobserver.xyz/documents/The_Creation_of_Duality.pdf The Creation of Duality]''' — speculative extension on appearance, gravity, and information from self-ignorance. Scientific status contingent on the BLQC experimental discriminator. * '''[https://ignorantobserver.xyz/documents/The_Capacity-Backaction_Frontier.pdf The Capacity–Backaction Frontier]''' — application to cryogenic quantum error correction. Defines an operational coordinate ρ_CB = ε_QEC ''C''_eff ln 2 / ''h''_eff(''N'', ''C''_eff) comparing useful syndrome capacity against the physical instability induced by obtaining and using it. * '''[https://ignorantobserver.xyz/documents/Biological_Observers.pdf Biological Observers]''' — exploratory supplement on biological timescales. A full archival deposit of the framework's documents is also available on the Open Science Framework at [https://doi.org/10.17605/OSF.IO/FCDSN doi.org/10.17605/OSF.IO/FCDSN]. == Open objections == The following objections to the framework are listed openly so that reviewers can engage with them directly. Several are diagnosed in the framework's own manuscripts; others reflect critiques the author has received in correspondence or anticipates from sophisticated readers. They are deliberately phrased from outside the framework's assumptions, not from within them. # '''Useful capacity versus thermodynamic bound'''. The framework uses the Landauer expression ''C'' ≤ ''P'' / (''k''_B ''T'' ln 2) to relate controller input power ''P'' to channel capacity. Landauer is an ideal upper bound on bit-erasure cost; it does not guarantee that increased ''P'' actually translates to increased ''useful'' basis-tracking capacity. Additional power can equally well couple to actuator noise, electromagnetic leakage, vibration, or backaction channels that do not constrain the basis variable θ. Establishing that Δ''P'' → Δ''C''_eff in the predicted direction — with realistic loss budgets for the candidate apparatus — is a substantive engineering claim that the framework does not by itself establish. # '''Existence of positive ''h''_KS in engineered apparatus'''. Many precision controllers (phase-locked loops, qubit readout chains, interferometer servo systems) are explicitly engineered to suppress chaotic dynamics. The basis-defining degrees of freedom may exhibit colored noise, slow drift, or stochastic control error rather than positive-''h''_KS chaos in the Pesin sense. If the relevant dynamics are not chaotic in this sense, the ''h''_KS framing may not apply at all, and a different rate-distortion accounting (or none) would be needed. Even where positive ''h''_KS can be identified, the operationally relevant rate may differ substantially from textbook surrogate estimates (kicked rotor, logistic map) used illustratively in the manuscripts. # '''Rate-distortion extension to nonlinear / chaotic systems'''. The mapping from channel capacity ''C'' to angular tracking variance σ_θ² ≥ ''D''/(''C'' ln 2) assumes a high-rate coder model and the framework extends the Data-Rate Theorem from linear plants to nonlinear chaotic systems by substituting ''h''_KS. This extension is an explicit assumption, not a proven theorem. If the extension fails, the closed-form visibility law and the κ-regime structure both lose their derivation. # '''Gaussian small-angle assumption'''. The visibility expression ''V''(''t'') = exp(−½ σ_θ²) requires σ_θ ≲ 1 rad and a Gaussian basis-tracking error distribution. Non-Gaussian, heavy-tailed, or state-dependent δθ would break the closed-form double-exponential law. # '''Decoherence and control-noise confound'''. Distinguishing the predicted visibility loss from ordinary environmental dephasing, alignment drift, and detector systematics is the central experimental challenge. The framework's answer is the sign-reversal under power variation at clamped ''T'' — a conceptually clean discriminator that is engineering-hard to realise. Independent calibration of ''C''_eff may be the single largest practical hurdle. # '''Prior-art and reparameterization risk'''. The proposed double-exponential visibility signature may already be expressible within existing frameworks: compound dephasing channels with two or more contributing rates, classical feedback-loop instability, or hidden-variable control-noise models with appropriate parameter choices. The framework should be able to show that its prediction is genuinely new rather than a reparameterization of one of these known phenomena. The author's adversarial-mimic analysis is in progress, and a positive result on that front would substantially strengthen the framework's empirical claim. # '''Bell / locality consistency'''. The framework implies a structural violation of statistical measurement-independence. The author's response (common causal past plus global consistency, in place of fine-tuned initial conditions) is a philosophical reframing rather than a no-signalling lemma. A proper consistency proof has not been published. # '''Forensic-signature interpretation'''. The Forensic Signatures preprint applies a screening protocol to existing data from Chinese 63-qubit processors, Google Sycamore, and LIGO glitch records. The paper's own domain-of-validity statement is that BLQC applies in observer-limited rather than plant-limited regimes, and the protocol finds power-law dominance on the qubit datasets (consistent with that statement) and 43% Gompertz-consistent events on LIGO (consistent with BLQC). The paper flags a controller-regime confound for the LIGO result and is explicit that retrospective findings do not establish causal attribution to BLQC; the case rests on the prospective controlled-capacity experiment. The objection here is the standard one for retrospective signal analyses: even where the predicted geometry is present, it remains compatible with alternative explanations until the controlled experiment runs. # '''Observer language'''. The framework's "observer" plays two distinct roles: the physical apparatus / controller whose finite ''C''_eff and ''h''_KS appear in the equations, and the epistemic subject for whom measurement outcomes are or are not determinate. The framework treats these as connected but not identified, and the distinction is load-bearing. Critics will reasonably worry — especially given the framework's interpretive engagement with non-dual philosophy and the philosophy of mind — that consciousness is being smuggled into the foundations of measurement under physical vocabulary. The framework's defence is that the BLQC experimental claim is stated entirely in apparatus-level terms; whether that defence holds depends on the framework keeping the two senses of "observer" rigorously separate. # '''Interpretive vocabulary'''. Some of the framework's documents draw on vocabulary from philosophy of mind and non-dual philosophy (notably Advaita Vedānta) alongside the physical derivations. Readers who find this vocabulary off-putting are invited to evaluate the empirical content from the BLQC manuscript, which uses only standard physics and control-theory language. # '''Conditional Born-rule derivation, scope'''. The framework's binary-Born derivation now obtains ''p''(θ) = cos²(θ/2) in the laboratory basis coordinate of a BLQC experiment, with the conditional weight stated explicitly as two named premises: the ''Fisher capacity bridge'' (''C''_eff measures the useful rate of reducing distinguishability error in the operational record family ''p''(''o'' | θ)) and ''scalar-threshold homogeneity'' (the physical basis coordinate θ is homogeneous in the Fisher distinguishability metric on records). Both premises are empirically testable through the Fisher-homogeneity module of the BLQC protocol. The derivation does not derive complex Hilbert space, tensor products, unitary dynamics, the multi-outcome Born rule for arbitrary projective measurements, or the full IOF admissible-history measure μ_A. Reviewer engagement on whether the Fisher capacity bridge is the right substantive identification of useful tracking capacity, whether scalar-threshold homogeneity is the natural reading of the BLQC threshold in a calibrated basis, whether Cencov-based selection is the correct uniqueness theorem under sufficient Markov invariance, and what would constitute a non-circular extension to multi-outcome records and full Hilbert kinematics, is explicitly invited. # '''Peer-review status and independent replication'''. The framework has not yet undergone peer review, and the experimental discriminator has not been independently replicated. This is the actual current epistemic status of the work. The framework's case must be evaluated on its merits in the documents linked above and on the conduct of the prospective experiment, not on any external imprimatur. == Invitation for review == This page is offered as a venue for substantive critique. The author is particularly interested in engagement on the following: * '''From physicists working on quantum control or precision interferometry''': is the proposed sign-reversal under controller-power variation at clamped temperature genuinely distinguishable from known instrumental artefacts (closed-loop resonances, thermal-noise mismodelling, photon-shot-noise rebalancing at higher gain), and what existing apparatus would be best positioned to perform the test? * '''From decoherence theorists''': under what conditions does the proposed double-exponential visibility law overlap with compound-channel decoherence models in ways that would make the two empirically indistinguishable? Is there a parameter regime where the framework's prediction is genuinely new rather than a reparameterisation of existing models? * '''From researchers in the foundations of quantum mechanics''': how should the framework's structural — but epistemically bounded — violation of measurement-independence be evaluated against the alternatives in the superdeterminism / retrocausality / many-worlds landscape, and what would constitute a satisfactory consistency proof? * '''From researchers in information geometry or foundations of probability''': the framework's conditional binary-Born derivation runs from BLQC finite-rate basis tracking via a Fisher capacity bridge and scalar-threshold homogeneity to ''p''(θ) = cos²(θ/2) in the laboratory basis coordinate. The binary case in θ is conditionally closed under the two stated bridge assumptions; the extension to multi-outcome records and the recovery of full Hilbert-space empirical content remain open. Critique on whether the Fisher capacity bridge is the right substantive identification of useful tracking capacity, whether scalar-threshold homogeneity is the natural reading of the BLQC threshold in a calibrated basis, whether Cencov-based selection is the correct uniqueness theorem under sufficient Markov invariance, and what would constitute a non-circular extension to multi-outcome records and full Hilbert kinematics, is welcome. * '''From philosophers of mind''': the Advaita / RQM interpretive layer is offered conditionally on the empirical core. Is the conditional structure ("these readings are available ''if'' the empirical claim survives") presented clearly enough, or does it still amount to overreach? Comments, references to prior or parallel work the author may not be aware of, and pointers to potential confounds or alternative explanations are all welcome. Substantive critique on the [[Talk:The Ignorant Observer Framework|talk page]] will be acknowledged in subsequent revisions of the manuscripts. == References == * Brukner, Č., & Zeilinger, A. (1999). Operationally invariant information in quantum measurements. ''Physical Review Letters'', 83(17), 3354–3357. * Nair, G. N., & Evans, R. J. (2004). Stabilizability of stochastic linear systems with finite feedback data rates. ''SIAM Journal on Control and Optimization'', 43(2), 413–436. * Palmer, T. (2024). Superdeterminism without conspiracy. ''Universe'', 10(1), 47. * Penrose, R. (1996). On gravity's role in quantum state reduction. ''General Relativity and Gravitation'', 28(5), 581–600. * Rovelli, C. (1996). Relational quantum mechanics. ''International Journal of Theoretical Physics'', 35(8), 1637–1678. * Tatikonda, S., & Mitter, S. (2004). Control under communication constraints. ''IEEE Transactions on Automatic Control'', 49(7), 1056–1068. * Wootters, W. K. (1981). Statistical distance and Hilbert space. ''Physical Review D'', 23(2), 357–362. == See also == * [[w:Quantum decoherence|Decoherence]] (Wikipedia) * [[w:Relational quantum mechanics|Relational quantum mechanics]] (Wikipedia) * [[w:Penrose interpretation|Penrose interpretation]] (Wikipedia) * [[w:Data-rate theorem|Data-rate theorem]] (Wikipedia) [[Category:Research projects]] [[Category:Quantum mechanics]] [[Category:Philosophy of science]] sffsw9jadomxi7ss29qs4y3em5ntejc 0 329829 2812072 2811892 2026-05-29T22:12:06Z Scogdill 1331941 2812072 wikitext text/x-wiki = The Irish Aristocracy at the End of the 19th Century = == The Irish Peerage == === [[Social Victorians/People/Abercorn|Duke and Duchess of Abercorn]] === * This dukedom is in the peerage of the United Kingdom of Great Britain and Ireland * James Hamilton, 1st Duke of Abercorn (1811–1885), elder son of Lord Hamilton, "styled Viscount Hamiltonfrom 1814 to 1818 and The Marquess of Abercorn from 1818 to 1868, was a Conservative statesman who twice served as Lord Lieutenant of Ireland."<ref>{{Cite journal|date=2026-04-05|title=James Hamilton, 1st Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_1st_Duke_of_Abercorn&oldid=1347253763|journal=Wikipedia|language=en}}</ref> * James Hamilton, 2nd Duke of Abercorn (1838–1913), eldest son of the 1st Duke, "styled Viscount Hamilton until 1868 and Marquess of Hamilton from 1868 to 1885, was a British nobleman, courtier, and diplomat."<ref>{{Cite journal|date=2026-01-25|title=James Hamilton, 2nd Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_2nd_Duke_of_Abercorn&oldid=1334676058|journal=Wikipedia|language=en}}</ref> * Subsidiary Titles ** Marquess of Abercorn ** Viscount Hamilton ** Viscount Strabane, county Tyrone === Duke of Leinster === Irish peerage * Gerald FitzGerald, 5th Duke of Leinster (16 August 1851 – 1 December 1893)<ref>{{Cite web|url=https://www.thepeerage.com/p1207.htm#i12063|title=Gerald FitzGerald, 5th Duke of Leinster|website=www.thepeerage.com|access-date=2026-05-24}}</ref> * Maurice FitzGerald, 6th Duke of Leinster, 6 years old when he succeeded to the dukedom<ref>{{Cite web|url=https://www.thepeerage.com/p2767.htm#i27667|title=Maurice FitzGerald, 6th Duke of Leinster|website=www.thepeerage.com|access-date=2026-05-24}}</ref> ==== Subsidiary Titles ==== # Marquess of Kildare (Irish peerage), did not attend the ball. # Earl of Kildare (Irish peerage), did not attend the ball. # Earl of Offaly (Irish peerage) # Viscount Leinster of Taplow (GB peerage) # Baron Offaly (Irish peerage) # Baron Kildare of Kildare (UK peerage) === Marquess Conyngham === Did not attend the ball but did attend a number of social events about this time. ==== Subsidiary Titles ==== * Earl of Conyngham === Marquess of Donegall === Did not attend the ball. ==== Subsidiary Titles ==== * Earl of Donegall, did not attend the ball. * Viscount Chichester — did not attend the ball; some Chichesters attended social events at about this time. === Marquess and Marchioness of Downshire === * Arthur Wills John Wellington Trumbull Blundell Hill, 6th Marquess of Downshire (2 July 1871 – 29 May 1918) in 1893 married Katherine Mary ("Kitty") Hare (1872–1959)<ref>{{Cite journal|date=2025-02-10|title=Arthur Hill, 6th Marquess of Downshire|url=https://en.wikipedia.org/w/index.php?title=Arthur_Hill,_6th_Marquess_of_Downshire&oldid=1274976272|journal=Wikipedia|language=en}}</ref> * Did not attend the ball. * Subsidiary Titles ** Earl of Hillsborough, did not attend the ball, also not at any social events described so far. === Marquess of Ely === Did not attend the ball. Subsidiary Titles * Earl of Ely — did not attend the ball. === [[Social Victorians/People/Bective|Marquess and Marchioness of Headfort]] === Did not attend the ball. ==== Subsidiary Titles ==== * [[Social Victorians/People/Bective|Earl of Bective]] === [[Social Victorians/People/Londonderry|Marquess and Marchioness of Londonderry]] === The Marquess and Marchioness attended the ball, she led one of the courts as Maria Thérèse, plus two of their children attended. ==== Subsidiary Titles ==== * [[Social Victorians/People/Londonderry|Earl of Londonderry]] === [[Social Victorians/People/Lucan|Earl of Lucan]] === Some members of the family attended the ball, and the family attended a number of social events at this time. === [[Social Victorians/People/Ormonde|Marquess and Marchioness of Ormonde]] === * James Edward Butler, 3rd Marquess of Ormonde and 21st Earl of Ormonde (1844–1919)<ref>{{Cite journal|date=2026-05-03|title=Earl of Ormond (Ireland)|url=https://en.wikipedia.org/w/index.php?title=Earl_of_Ormond_(Ireland)&oldid=1352334266|journal=Wikipedia|language=en}}</ref> Now extinct; earldom dormant. Castle X was their manor, but they don't appear to have any papers. ==== Subsidiary Titles ==== === Marquess of Sligo === Did not attend the ball. ==== Subsidiary Titles ==== * Earl of Altamont. Did not attend the ball; did not attend any social events analyzed so far. * Earl of Clanricarde — Did not attend the ball but did attend a few social events about this time. === Marquess of Waterford === * John Henry de La Poer Beresford, 5th Marquess of Waterford (1844–1895) * Henry de La Poer Beresford, 6th Marquess of Waterford (1875–1911)<ref>{{Cite journal|date=2026-02-10|title=Henry Beresford, 6th Marquess of Waterford|url=https://en.wikipedia.org/w/index.php?title=Henry_Beresford,_6th_Marquess_of_Waterford&oldid=1337565707|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but members of the Beresford family were prominent socially at about this time. * Subsidiary Titles ** Viscount Tyrone === Earl of Annesley === Did not attend the ball but did attend a number of social events in the 1890s. === [[Social Victorians/People/Antrim|Earl of Antrim]] === Some members of this family attended the ball, though not the earl or countess. === Earl of Arran === Attended the ball. === [[Social Victorians/People/Belmore|Earl Belmore]] === Did not attend the ball, but did attend a number of social events about this time. === Earl of Bessborough === * Frederick George Brabazon Ponsonby, 6th Earl of Bessborough (1815–1895) * Walter William Brabazon Ponsonby, 7th Earl of Bessborough (1821–1906), would have been Viscount Duncannon 1880–1895 * Edward Ponsonby, 8th Earl of Bessborough (1851–1920), would have been Viscount Duncannon 1895–1906 * Did not attend the ball, but the [[Social Victorians/People/Ponsonby|Ponsonby]] family attended many social events at about this time, including mention of Lady Duncannon's school that taught fabric arts. * Subsidiary Titles ** Viscount Duncannon === Earl of Caledon === Did not attend the ball but did attend a number of social events about this time. === Earl of Carrick === Did not attend the ball. === Earl Castle Stewart === Did not attend the ball. === Earl of Cavan === Did not attend the ball. === [[Social Victorians/People/Clanwilliam|Earl and Countess of Clanwilliam]] === Did not attend the ball. === Earl of Cork, Earl of Orrery === Cork and Orrery, did attend the ball. === Earl of Courtown === Did not attend the ball. === Earl of Darnley === * John Bligh, 6th Earl of Darnley (1827–1896), British<ref>{{Cite journal|date=2026-02-07|title=John Bligh, 6th Earl of Darnley|url=https://en.wikipedia.org/w/index.php?title=John_Bligh,_6th_Earl_of_Darnley&oldid=1337113925|journal=Wikipedia|language=en}}</ref> * Edward Bligh, 7th Earl of Darnley (1851–1900), Lord Clifton much of his adult life, "English"<ref>{{Cite journal|date=2026-05-05|title=Edward Bligh, 7th Earl of Darnley|url=https://en.wikipedia.org/w/index.php?title=Edward_Bligh,_7th_Earl_of_Darnley&oldid=1352607379|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but the Bligh family attended some social events from about this time. * Subsidiary Titles: ** Viscount Darnley === Earl of Desmond === Did not attend the ball. === [[Social Victorians/People/Donoughmore|Earl of Donoughmore]] === Did not attend the ball but did attend a number of social events about this time. === Earl of Drogheda === Did not attend the ball. ==== Subsidiary Titles ==== * Viscount Moore — no evidence of the Viscount or Viscountess Moore at social events at about this time. === [[Social Victorians/People/Cole|Earl and Countess of Enniskillen]] === The Earl and Countess and a daughter attended the ball. Papers in PRONI. === [[Social Victorians/People/Crichton|Earl of Erne]] === Some members of the family attended the ball. Papers in PRONI. === Earl of Granard === * Did not attend the ball. * Bernard Arthur William Patrick Hastings Forbes, 8th Earl of Granard (17 September 1874 – 10 September 1948)[https://en.wikipedia.org/wiki/Bernard_Forbes,_8th_Earl_of_Granard] * Anglo-Irish * Subsidiary Titles ** Bernard Arthur William Patrick Hastings Forbes, styled Viscount Forbes from 1874 to 1889 === Earl of Kerry === * Subsidiary title of the [[Social Victorians/People/Lansdowne|Marquess of Lansdowne]] (in the peerage of Great Britain). Attended the ball. * Subsidiary Titles ** Viscount Clanmaurice === [[Social Victorians/People/Kilmorey|Earl of Kilmorey]] === Anglo-Irish Nellie Countess of Kilmorey attended the ball; Francis, 3rd Earl was alive at the time, did he attend? Both he and she attended a number of social events from about this time. === Earl of Kingston === Did not attend the ball. === Earl of Lisburne === * Did not attend the ball. * Ernest Augustus Malet Vaughan, 5th Earl of Lisburne (1836–1888)<ref>{{Cite journal|date=2025-12-03|title=Ernest Augustus Malet Vaughan, 5th Earl of Lisburne|url=https://en.wikipedia.org/w/index.php?title=Ernest_Augustus_Malet_Vaughan,_5th_Earl_of_Lisburne&oldid=1325511612|journal=Wikipedia|language=en}}</ref> ** Owned a lot of land in Cardiganshire, Wales ** Conservative, but withdrew from politics * George Henry Arthur Vaughan, 6th Earl of Lisburne (1862–1899) * Ernest Edmund Henry Malet Vaughan, 7th Earl of Lisburne (1892–1965) ** Welsh nobleman, of Trawsgoed, Cardiganshire. 7 years old when he succeeded to the earldom === Earl of Longford === Did not attend the ball. === [[Social Victorians/People/Mayo|Earl of Mayo]] === Some members of the family attended the ball. === Earl and Countess of Meath === Did not attend the ball. === Earl of Mexborough === Did not attend the ball === Earl of Mornington === Subsidiary title of the Duke of Wellington (in the peerage of the UK). === Earl of Portarlington === Did not attend the ball. === Earl of Roden === Did not attend the ball. === Earl of Shannon === Did not attend the ball. === Earl of Shelburne === Subsidiary title of the Marquess of Lansdowne (in the peerage of Great Britain). Did not attend the ball, and did not attend any social events analyzed so far. === Earl of Tyrone === Did not attend === Earl of Waterford === Not a subsidiary title of the Marquess of Waterford but of the Earl of Shrewsbury in the peerage of England. === Earl of Westmeath === Did not attend the ball. === Earl of Winterton === Did not attend the ball. === Viscount Ashbrook === * William Spencer Flower, 7th Viscount Ashbrook (1830–1906)<ref>{{Cite journal|date=2025-12-01|title=Viscount Ashbrook|url=https://en.wikipedia.org/w/index.php?title=Viscount_Ashbrook&oldid=1325071512|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, has no social presence at about this time. === Viscount Boyne === Did not attend the ball, but did attend a number of events at about this time. === Viscount Callan === Did not attend the ball, and does not have much if any social presence at about this time. The Viscount Callan is a subsidiary title of the Earl of Denbigh in the Peerage of England. === Viscount Charlemont === * Did not attend the ball. * Colonel James Alfred Caulfeild, 7th Viscount Charlemont (20 March 1830 – 4 July 1913), Irish<ref>{{Cite journal|date=2026-05-02|title=James Caulfeild, 7th Viscount Charlemont|url=https://en.wikipedia.org/w/index.php?title=James_Caulfeild,_7th_Viscount_Charlemont&oldid=1352129469|journal=Wikipedia|language=en}}</ref> * Unionist === Viscount Chetwynd === * Does not seem to have attended the ball, but Chetwynds were socially very active at about this time. * Godfrey John Boyle Chetwynd, 8th Viscount Chetwynd (1863–1936), British<ref>{{Cite journal|date=2026-05-24|title=Godfrey Chetwynd, 8th Viscount Chetwynd|url=https://en.wikipedia.org/w/index.php?title=Godfrey_Chetwynd,_8th_Viscount_Chetwynd&oldid=1355878192|journal=Wikipedia|language=en}}</ref> === Viscount Dillon === Did not attend the ball, but several Dillons attended other social events at about this time. === [[Social Victorians/People/Downe|Viscount Downe]] === * Did not attend the ball but attended many social events at about this time. * Major-General Hugh Richard Dawnay, 8th Viscount Downe (20 July 1844 – 21 January 1924)<ref>{{Cite journal|date=2026-03-24|title=Hugh Dawnay, 8th Viscount Downe|url=https://en.wikipedia.org/w/index.php?title=Hugh_Dawnay,_8th_Viscount_Downe&oldid=1345146095|journal=Wikipedia|language=en}}</ref> * British Army general === Viscount Gage === * Henry Charles Gage, 5th Viscount Gage (1854–1912)<ref>{{Cite journal|date=2025-06-21|title=Viscount Gage|url=https://en.wikipedia.org/w/index.php?title=Viscount_Gage&oldid=1296646030|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but members of this family attended a number of social events at about this time. === Viscount Galway === * George Edmund Milnes Monckton-Arundell, 7th Viscount Galway (1844–1931), British conservative<ref>{{Cite journal|date=2025-08-08|title=George Monckton-Arundell, 7th Viscount Galway|url=https://en.wikipedia.org/w/index.php?title=George_Monckton-Arundell,_7th_Viscount_Galway&oldid=1304770631|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but Viscount and Viscountess Galway attended many social events at about this time. * Subsidiary Title ** Baron Monckton (in the Peerage of the United Kingdom) === Viscount Gormanston === Did not attend the ball, has no social presence in the late 19th-century newspapers at this time. === Viscount Grandison === Did not attend the ball, has no social presence in the late 19th-century newspapers at this time. The Viscount Grandison is a subsidiary title of the Earl of Jersey in the Peerage of England. === Viscount Grimston === * Subsidiary title of the Earl of Verulam (in the Peerage of the United Kingdom) * Did not attend the ball, but a number of members of this family attended social events at about this time. === Viscount Massereene === * Did not attend the ball but did attend a few events at about this time. * Anglo-Irish * Clotworthy John Eyre Skeffington, 11th Viscount Massereene (9 October 1842 – 26 June 1905)<ref>{{Cite journal|date=2024-11-23|title=Clotworthy Skeffington, 11th Viscount Massereene|url=https://en.wikipedia.org/w/index.php?title=Clotworthy_Skeffington,_11th_Viscount_Massereene&oldid=1259199982|journal=Wikipedia|language=en}}</ref> === [[Social Victorians/People/Midleton|Viscount Midleton]] === * Some people from this family seem to have attended the ball as well as many other social events at about this time. * William Brodrick, 8th Viscount Midleton (6 January 1830 – 18 April 1907), "Irish peer, landowner and Conservative politician in both Houses of Parliament"<ref>{{Cite journal|date=2025-01-05|title=William Brodrick, 8th Viscount Midleton|url=https://en.wikipedia.org/w/index.php?title=William_Brodrick,_8th_Viscount_Midleton&oldid=1267418489|journal=Wikipedia|language=en}}</ref> * Sight and hearing disabilities caused by intermarriage. A daughter became a Republican. === Viscount Molesworth === * Did not attend the ball, but attended the Warwick Bal Poudré and a number of other social events at about this time. * Samuel Molesworth, 8th Viscount Molesworth (1829–1906), may have been a Quaker === Viscount Mountgarret === Did not attend the ball, has no social presence in the late 19th-century newspapers at this time. === [[Social Victorians/People/Powerscourt|Viscount Powerscourt]] === * Mervyn Wingfield, 7th Viscount Powerscourt (1836–1904)<ref name=":0">{{Cite journal|date=2026-02-18|title=Mervyn Wingfield, 7th Viscount Powerscourt|url=https://en.wikipedia.org/w/index.php?title=Mervyn_Wingfield,_7th_Viscount_Powerscourt&oldid=1339057453|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but members of this family attended a number of social events at about this time. * Subsidiary Title ** Baron Powerscourt (in the Peerage of the United Kingdom), 1885<ref name=":0" /> === Viscount Valentia === Did not attend the ball, attended some social events at about this time. Was on the Welcome Council for the 1887 American Exhibition. == Peerage of the United Kingdom of Great Britain and Ireland == After the forced 1801 Act of Union. === Earl of Clancarty === Did not attend the ball and attended few social events researched so far. === [[Social Victorians/People/Gosford|Earl of Gosford]] === The Earl and Countess of Gosford attended the ball, as did a son and a daughter. They attended many social events at about this time. === Earl of Limerick === Did not attend the ball, but did attend a number of events about this time. === Earl of Listowel === Did not attend the ball, but hosted and attended social events at about this time. === Earl of Norbury === Did not attend the ball, but attended some social events at about this time. === Earl of Normanton === Did not attend the ball, but did attend some social events in the 1880s and 1890s. === Earl of Ranfurly === Did not attend the ball, and they have a small social presence in the newspapers in the 1880s and 1890s. === Earl of Rosse === Did not attend the ball, but did attend a few events at about this time. == Irish Nationalists == == Irish Unionists == == Irish Aristocrats at the Duchess of Devonshire's 1897 Fancy-dress Ball == == References == lt231w6xx5q0k93h9tupk937we1p9em 2812073 2812072 2026-05-29T22:22:32Z Scogdill 1331941 /* References */ 2812073 wikitext text/x-wiki = The Irish Aristocracy at the End of the 19th Century = == The Irish Peerage == === [[Social Victorians/People/Abercorn|Duke and Duchess of Abercorn]] === * This dukedom is in the peerage of the United Kingdom of Great Britain and Ireland * James Hamilton, 1st Duke of Abercorn (1811–1885), elder son of Lord Hamilton, "styled Viscount Hamiltonfrom 1814 to 1818 and The Marquess of Abercorn from 1818 to 1868, was a Conservative statesman who twice served as Lord Lieutenant of Ireland."<ref>{{Cite journal|date=2026-04-05|title=James Hamilton, 1st Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_1st_Duke_of_Abercorn&oldid=1347253763|journal=Wikipedia|language=en}}</ref> * James Hamilton, 2nd Duke of Abercorn (1838–1913), eldest son of the 1st Duke, "styled Viscount Hamilton until 1868 and Marquess of Hamilton from 1868 to 1885, was a British nobleman, courtier, and diplomat."<ref>{{Cite journal|date=2026-01-25|title=James Hamilton, 2nd Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_2nd_Duke_of_Abercorn&oldid=1334676058|journal=Wikipedia|language=en}}</ref> * Subsidiary Titles ** Marquess of Abercorn ** Viscount Hamilton ** Viscount Strabane, county Tyrone === Duke of Leinster === Irish peerage * Gerald FitzGerald, 5th Duke of Leinster (16 August 1851 – 1 December 1893)<ref>{{Cite web|url=https://www.thepeerage.com/p1207.htm#i12063|title=Gerald FitzGerald, 5th Duke of Leinster|website=www.thepeerage.com|access-date=2026-05-24}}</ref> * Maurice FitzGerald, 6th Duke of Leinster, 6 years old when he succeeded to the dukedom<ref>{{Cite web|url=https://www.thepeerage.com/p2767.htm#i27667|title=Maurice FitzGerald, 6th Duke of Leinster|website=www.thepeerage.com|access-date=2026-05-24}}</ref> ==== Subsidiary Titles ==== # Marquess of Kildare (Irish peerage), did not attend the ball. # Earl of Kildare (Irish peerage), did not attend the ball. # Earl of Offaly (Irish peerage) # Viscount Leinster of Taplow (GB peerage) # Baron Offaly (Irish peerage) # Baron Kildare of Kildare (UK peerage) === Marquess Conyngham === Did not attend the ball but did attend a number of social events about this time. ==== Subsidiary Titles ==== * Earl of Conyngham === Marquess of Donegall === Did not attend the ball. ==== Subsidiary Titles ==== * Earl of Donegall, did not attend the ball. * Viscount Chichester — did not attend the ball; some Chichesters attended social events at about this time. === Marquess and Marchioness of Downshire === * Arthur Wills John Wellington Trumbull Blundell Hill, 6th Marquess of Downshire (2 July 1871 – 29 May 1918) in 1893 married Katherine Mary ("Kitty") Hare (1872–1959)<ref>{{Cite journal|date=2025-02-10|title=Arthur Hill, 6th Marquess of Downshire|url=https://en.wikipedia.org/w/index.php?title=Arthur_Hill,_6th_Marquess_of_Downshire&oldid=1274976272|journal=Wikipedia|language=en}}</ref> * Did not attend the ball. * Subsidiary Titles ** Earl of Hillsborough, did not attend the ball, also not at any social events described so far. === Marquess of Ely === Did not attend the ball. Subsidiary Titles * Earl of Ely — did not attend the ball. === [[Social Victorians/People/Bective|Marquess and Marchioness of Headfort]] === Did not attend the ball. ==== Subsidiary Titles ==== * [[Social Victorians/People/Bective|Earl of Bective]] === [[Social Victorians/People/Londonderry|Marquess and Marchioness of Londonderry]] === The Marquess and Marchioness attended the ball, she led one of the courts as Maria Thérèse, plus two of their children attended. ==== Subsidiary Titles ==== * [[Social Victorians/People/Londonderry|Earl of Londonderry]] === [[Social Victorians/People/Lucan|Earl of Lucan]] === Some members of the family attended the ball, and the family attended a number of social events at this time. === [[Social Victorians/People/Ormonde|Marquess and Marchioness of Ormonde]] === * James Edward Butler, 3rd Marquess of Ormonde and 21st Earl of Ormonde (1844–1919)<ref>{{Cite journal|date=2026-05-03|title=Earl of Ormond (Ireland)|url=https://en.wikipedia.org/w/index.php?title=Earl_of_Ormond_(Ireland)&oldid=1352334266|journal=Wikipedia|language=en}}</ref> Now extinct; earldom dormant. Castle X was their manor, but they don't appear to have any papers. ==== Subsidiary Titles ==== === Marquess of Sligo === Did not attend the ball. ==== Subsidiary Titles ==== * Earl of Altamont. Did not attend the ball; did not attend any social events analyzed so far. * Earl of Clanricarde — Did not attend the ball but did attend a few social events about this time. === Marquess of Waterford === * John Henry de La Poer Beresford, 5th Marquess of Waterford (1844–1895) * Henry de La Poer Beresford, 6th Marquess of Waterford (1875–1911)<ref>{{Cite journal|date=2026-02-10|title=Henry Beresford, 6th Marquess of Waterford|url=https://en.wikipedia.org/w/index.php?title=Henry_Beresford,_6th_Marquess_of_Waterford&oldid=1337565707|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but members of the Beresford family were prominent socially at about this time. * Subsidiary Titles ** Viscount Tyrone === Earl of Annesley === Did not attend the ball but did attend a number of social events in the 1890s. === [[Social Victorians/People/Antrim|Earl of Antrim]] === Some members of this family attended the ball, though not the earl or countess. === Earl of Arran === Attended the ball. === [[Social Victorians/People/Belmore|Earl Belmore]] === Did not attend the ball, but did attend a number of social events about this time. === Earl of Bessborough === * Frederick George Brabazon Ponsonby, 6th Earl of Bessborough (1815–1895) * Walter William Brabazon Ponsonby, 7th Earl of Bessborough (1821–1906), would have been Viscount Duncannon 1880–1895 * Edward Ponsonby, 8th Earl of Bessborough (1851–1920), would have been Viscount Duncannon 1895–1906 * Did not attend the ball, but the [[Social Victorians/People/Ponsonby|Ponsonby]] family attended many social events at about this time, including mention of Lady Duncannon's school that taught fabric arts. * Subsidiary Titles ** Viscount Duncannon === Earl of Caledon === Did not attend the ball but did attend a number of social events about this time. === Earl of Carrick === Did not attend the ball. === Earl Castle Stewart === Did not attend the ball. === Earl of Cavan === Did not attend the ball. === [[Social Victorians/People/Clanwilliam|Earl and Countess of Clanwilliam]] === Did not attend the ball. === Earl of Cork, Earl of Orrery === Cork and Orrery, did attend the ball. === Earl of Courtown === Did not attend the ball. === Earl of Darnley === * John Bligh, 6th Earl of Darnley (1827–1896), British<ref>{{Cite journal|date=2026-02-07|title=John Bligh, 6th Earl of Darnley|url=https://en.wikipedia.org/w/index.php?title=John_Bligh,_6th_Earl_of_Darnley&oldid=1337113925|journal=Wikipedia|language=en}}</ref> * Edward Bligh, 7th Earl of Darnley (1851–1900), Lord Clifton much of his adult life, "English"<ref>{{Cite journal|date=2026-05-05|title=Edward Bligh, 7th Earl of Darnley|url=https://en.wikipedia.org/w/index.php?title=Edward_Bligh,_7th_Earl_of_Darnley&oldid=1352607379|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but the Bligh family attended some social events from about this time. * Subsidiary Titles: ** Viscount Darnley === Earl of Desmond === Did not attend the ball. === [[Social Victorians/People/Donoughmore|Earl of Donoughmore]] === Did not attend the ball but did attend a number of social events about this time. === Earl of Drogheda === Did not attend the ball. ==== Subsidiary Titles ==== * Viscount Moore — no evidence of the Viscount or Viscountess Moore at social events at about this time. === [[Social Victorians/People/Cole|Earl and Countess of Enniskillen]] === The Earl and Countess and a daughter attended the ball. Papers in PRONI. === [[Social Victorians/People/Crichton|Earl of Erne]] === Some members of the family attended the ball. Papers in PRONI. === Earl of Granard === * Did not attend the ball. * Bernard Arthur William Patrick Hastings Forbes, 8th Earl of Granard (17 September 1874 – 10 September 1948)[https://en.wikipedia.org/wiki/Bernard_Forbes,_8th_Earl_of_Granard] * Anglo-Irish * Subsidiary Titles ** Bernard Arthur William Patrick Hastings Forbes, styled Viscount Forbes from 1874 to 1889 === Earl of Kerry === * Subsidiary title of the [[Social Victorians/People/Lansdowne|Marquess of Lansdowne]] (in the peerage of Great Britain). Attended the ball. * Subsidiary Titles ** Viscount Clanmaurice === [[Social Victorians/People/Kilmorey|Earl of Kilmorey]] === Anglo-Irish Nellie Countess of Kilmorey attended the ball; Francis, 3rd Earl was alive at the time, did he attend? Both he and she attended a number of social events from about this time. === Earl of Kingston === Did not attend the ball. === Earl of Lisburne === * Did not attend the ball. * Ernest Augustus Malet Vaughan, 5th Earl of Lisburne (1836–1888)<ref>{{Cite journal|date=2025-12-03|title=Ernest Augustus Malet Vaughan, 5th Earl of Lisburne|url=https://en.wikipedia.org/w/index.php?title=Ernest_Augustus_Malet_Vaughan,_5th_Earl_of_Lisburne&oldid=1325511612|journal=Wikipedia|language=en}}</ref> ** Owned a lot of land in Cardiganshire, Wales ** Conservative, but withdrew from politics * George Henry Arthur Vaughan, 6th Earl of Lisburne (1862–1899) * Ernest Edmund Henry Malet Vaughan, 7th Earl of Lisburne (1892–1965) ** Welsh nobleman, of Trawsgoed, Cardiganshire. 7 years old when he succeeded to the earldom === Earl of Longford === Did not attend the ball. === [[Social Victorians/People/Mayo|Earl of Mayo]] === Some members of the family attended the ball. === Earl and Countess of Meath === Did not attend the ball. === Earl of Mexborough === Did not attend the ball === Earl of Mornington === Subsidiary title of the Duke of Wellington (in the peerage of the UK). === Earl of Portarlington === Did not attend the ball. === Earl of Roden === Did not attend the ball. === Earl of Shannon === Did not attend the ball. === Earl of Shelburne === Subsidiary title of the Marquess of Lansdowne (in the peerage of Great Britain). Did not attend the ball, and did not attend any social events analyzed so far. === Earl of Tyrone === Did not attend === Earl of Waterford === Not a subsidiary title of the Marquess of Waterford but of the Earl of Shrewsbury in the peerage of England. === Earl of Westmeath === Did not attend the ball. === Earl of Winterton === Did not attend the ball. === Viscount Ashbrook === * William Spencer Flower, 7th Viscount Ashbrook (1830–1906)<ref>{{Cite journal|date=2025-12-01|title=Viscount Ashbrook|url=https://en.wikipedia.org/w/index.php?title=Viscount_Ashbrook&oldid=1325071512|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, has no social presence at about this time. === Viscount Boyne === Did not attend the ball, but did attend a number of events at about this time. === Viscount Callan === Did not attend the ball, and does not have much if any social presence at about this time. The Viscount Callan is a subsidiary title of the Earl of Denbigh in the Peerage of England. === Viscount Charlemont === * Did not attend the ball. * Colonel James Alfred Caulfeild, 7th Viscount Charlemont (20 March 1830 – 4 July 1913), Irish<ref>{{Cite journal|date=2026-05-02|title=James Caulfeild, 7th Viscount Charlemont|url=https://en.wikipedia.org/w/index.php?title=James_Caulfeild,_7th_Viscount_Charlemont&oldid=1352129469|journal=Wikipedia|language=en}}</ref> * Unionist === Viscount Chetwynd === * Does not seem to have attended the ball, but Chetwynds were socially very active at about this time. * Godfrey John Boyle Chetwynd, 8th Viscount Chetwynd (1863–1936), British<ref>{{Cite journal|date=2026-05-24|title=Godfrey Chetwynd, 8th Viscount Chetwynd|url=https://en.wikipedia.org/w/index.php?title=Godfrey_Chetwynd,_8th_Viscount_Chetwynd&oldid=1355878192|journal=Wikipedia|language=en}}</ref> === Viscount Dillon === Did not attend the ball, but several Dillons attended other social events at about this time. === [[Social Victorians/People/Downe|Viscount Downe]] === * Did not attend the ball but attended many social events at about this time. * Major-General Hugh Richard Dawnay, 8th Viscount Downe (20 July 1844 – 21 January 1924)<ref>{{Cite journal|date=2026-03-24|title=Hugh Dawnay, 8th Viscount Downe|url=https://en.wikipedia.org/w/index.php?title=Hugh_Dawnay,_8th_Viscount_Downe&oldid=1345146095|journal=Wikipedia|language=en}}</ref> * British Army general === Viscount Gage === * Henry Charles Gage, 5th Viscount Gage (1854–1912)<ref>{{Cite journal|date=2025-06-21|title=Viscount Gage|url=https://en.wikipedia.org/w/index.php?title=Viscount_Gage&oldid=1296646030|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but members of this family attended a number of social events at about this time. === Viscount Galway === * George Edmund Milnes Monckton-Arundell, 7th Viscount Galway (1844–1931), British conservative<ref>{{Cite journal|date=2025-08-08|title=George Monckton-Arundell, 7th Viscount Galway|url=https://en.wikipedia.org/w/index.php?title=George_Monckton-Arundell,_7th_Viscount_Galway&oldid=1304770631|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but Viscount and Viscountess Galway attended many social events at about this time. * Subsidiary Title ** Baron Monckton (in the Peerage of the United Kingdom) === Viscount Gormanston === Did not attend the ball, has no social presence in the late 19th-century newspapers at this time. === Viscount Grandison === Did not attend the ball, has no social presence in the late 19th-century newspapers at this time. The Viscount Grandison is a subsidiary title of the Earl of Jersey in the Peerage of England. === Viscount Grimston === * Subsidiary title of the Earl of Verulam (in the Peerage of the United Kingdom) * Did not attend the ball, but a number of members of this family attended social events at about this time. === Viscount Massereene === * Did not attend the ball but did attend a few events at about this time. * Anglo-Irish * Clotworthy John Eyre Skeffington, 11th Viscount Massereene (9 October 1842 – 26 June 1905)<ref>{{Cite journal|date=2024-11-23|title=Clotworthy Skeffington, 11th Viscount Massereene|url=https://en.wikipedia.org/w/index.php?title=Clotworthy_Skeffington,_11th_Viscount_Massereene&oldid=1259199982|journal=Wikipedia|language=en}}</ref> === [[Social Victorians/People/Midleton|Viscount Midleton]] === * Some people from this family seem to have attended the ball as well as many other social events at about this time. * William Brodrick, 8th Viscount Midleton (6 January 1830 – 18 April 1907), "Irish peer, landowner and Conservative politician in both Houses of Parliament"<ref>{{Cite journal|date=2025-01-05|title=William Brodrick, 8th Viscount Midleton|url=https://en.wikipedia.org/w/index.php?title=William_Brodrick,_8th_Viscount_Midleton&oldid=1267418489|journal=Wikipedia|language=en}}</ref> * Sight and hearing disabilities caused by intermarriage. A daughter became a Republican. === Viscount Molesworth === * Did not attend the ball, but attended the Warwick Bal Poudré and a number of other social events at about this time. * Samuel Molesworth, 8th Viscount Molesworth (1829–1906), may have been a Quaker === Viscount Mountgarret === Did not attend the ball, has no social presence in the late 19th-century newspapers at this time. === [[Social Victorians/People/Powerscourt|Viscount Powerscourt]] === * Mervyn Wingfield, 7th Viscount Powerscourt (1836–1904)<ref name=":0">{{Cite journal|date=2026-02-18|title=Mervyn Wingfield, 7th Viscount Powerscourt|url=https://en.wikipedia.org/w/index.php?title=Mervyn_Wingfield,_7th_Viscount_Powerscourt&oldid=1339057453|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but members of this family attended a number of social events at about this time. * Subsidiary Title ** Baron Powerscourt (in the Peerage of the United Kingdom), 1885<ref name=":0" /> === Viscount Valentia === Did not attend the ball, attended some social events at about this time. Was on the Welcome Council for the 1887 American Exhibition. == Peerage of the United Kingdom of Great Britain and Ireland == After the forced 1801 Act of Union. === Earl of Clancarty === Did not attend the ball and attended few social events researched so far. === [[Social Victorians/People/Gosford|Earl of Gosford]] === The Earl and Countess of Gosford attended the ball, as did a son and a daughter. They attended many social events at about this time. === Earl of Limerick === Did not attend the ball, but did attend a number of events about this time. === Earl of Listowel === Did not attend the ball, but hosted and attended social events at about this time. === Earl of Norbury === Did not attend the ball, but attended some social events at about this time. === Earl of Normanton === Did not attend the ball, but did attend some social events in the 1880s and 1890s. === Earl of Ranfurly === Did not attend the ball, and they have a small social presence in the newspapers in the 1880s and 1890s. === Earl of Rosse === Did not attend the ball, but did attend a few events at about this time. == Irish Nationalists == == Irish Unionists == == Irish Aristocrats at the Duchess of Devonshire's 1897 Fancy-dress Ball == == References == {{reflist}} mw61h18z6jufl4ffppt6wr9vl65fpko 4 329840 2812119 2811507 2026-05-30T11:21:35Z Jtneill 10242 the -> trusted user 2812119 wikitext text/x-wiki {{Proposal}} '''Patrolling''' is trusted user review of newly created pages and recent edits to identify pages that need improvement, administrative attention, deletion, and new contributors who need support. Patrolling is part of Wikiversity’s collaborative maintenance work. It is not approval of content, but rather an initial review. Unpatrolled edits show up in [[Special:RecentChanges]] with an exclamation mark (!), while unpatrolled new pages are highlighted in yellow. The ability to patrol is determined by user rights. 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Patrolling is a collobaritve part of [[Wikiversity:Maintenance|Wikiversity’s maintenance work]]. It is not approval of content, but rather an initial review to identify obvious issues that warrant attention. Unpatrolled edits show up in [[Special:RecentChanges]] with an exclamation mark (!), while unpatrolled new pages are highlighted in yellow. The ability to patrol is determined by [[Wikiversity:User access levels|user rights]]. 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. == Purpose == Patrolling helps to: * welcome and support new contributors * identify pages that may need formatting, categorisation, or wikification * check whether new pages fit Wikiversity’s educational scope * identify vandalism, spam, copyright violations, or other problematic content * reduce the backlog of unreviewed pages * improve the discoverability and organisation of learning resources == Who can patrol? == Patrolling may be undertaken by users with the appropriate user rights, including: * [[Wikiversity:Curators|Curators]] * [[Wikiversity:Custodians|Custodians]] These users can mark pages as patrolled. Pages created by users with the ''autopatrol'' right are automatically marked as patrolled upon creation. == What patrolling means == Marking a page as patrolled indicates that the page has received an initial review. This generally means that the patroller has checked that: * the page is not obvious vandalism or spam * the page broadly fits Wikiversity’s scope and mission * the title is reasonable * the content is not an obvious copyright violation * any urgent issues have been addressed or flagged for follow-up Patrolling '''does not''' necessarily mean that the: * page is complete * page meets all style guidelines * content has been fact-checked * page has community endorsement Pages can still be edited, improved, moved, nominated for deletion, or discussed after being marked as patrolled. == Suggested patrolling workflow == When reviewing a newly created page, patrollers are encouraged to: # Open the page and read the content # Check the page history and creator's contributions # Consider whether the page is within [[Wikiversity:Scope|Wikiversity’s scope]] # Look for: #* vandalism #* spam or promotional content #* copyright concerns #* test pages #* pages requiring [[Wikiversity:Deletion|speedy deletion]] or cleanup # If appropriate: #* add categories #* add maintenance templates #* welcome or assist the creator on their talk page #* nominate the page for deletion if needed # Mark the page as patrolled once reviewed == When not to mark a page as patrolled == A page should generally '''not''' be marked as patrolled if it: * is obvious vandalism awaiting reversion or deletion * is spam or promotional content needing removal * appears to be a copyright violation * requires immediate administrative attention and has not yet been addressed In these cases, patrollers should address or flag the issue first. == Good practice == Patrollers are encouraged to: * [[Wikiversity:Assume good faith|assume good faith]], especially with new contributors * focus on whether a page needs attention, rather than whether it is “perfect” * leave constructive feedback where useful * use maintenance templates to indicate issues * discuss borderline cases with the community when unsure == See also == * [[Special:NewPages|New pages]] | [[Special:RecentChanges|Recent changes]] * [[Wikiversity:Curatorship|Curators]] | [[Wikiversity:Custodianship|Custodians]] * [[Wikiversity:Scope|Scope]] * [[Wikiversity:Deletion requests|Deletion requests]] [[Category:Wikiversity maintenance]] 3byz98s5pzq2gjpu7c7d85pfz94lran 3 329879 2812116 2811978 2026-05-30T10:59:33Z Ics-counseling 3085340 /* May 2026 */ Reply 2812116 wikitext text/x-wiki ==May 2026== This account is [https://en.wikiversity.org/w/index.php?title=Child_psychology&curid=302815&diff=2811976&oldid=2811761 adding advertising] with the same material as an IP address recently. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:27, 29 May 2026 (UTC) :Ii only provide a information regarding content not for advetising [[User:Ics-counseling|Ics-counseling]] ([[User talk:Ics-counseling|discuss]] • [[Special:Contributions/Ics-counseling|contribs]]) 10:59, 30 May 2026 (UTC) p0jb459funpk7wnzkn13q9657onnyb7 2812117 2812116 2026-05-30T11:10:15Z Jtneill 10242 /* May 2026 */ reply ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2812117 wikitext text/x-wiki ==May 2026== This account is [https://en.wikiversity.org/w/index.php?title=Child_psychology&curid=302815&diff=2811976&oldid=2811761 adding advertising] with the same material as an IP address recently. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 10:27, 29 May 2026 (UTC) :Ii only provide a information regarding content not for advetising [[User:Ics-counseling|Ics-counseling]] ([[User talk:Ics-counseling|discuss]] • [[Special:Contributions/Ics-counseling|contribs]]) 10:59, 30 May 2026 (UTC) : If you own the copyright to the content on the target page, you are welcome to contribute it here on Wikiversity, but as it is it is clearly link placement to promote a commercial website. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:10, 30 May 2026 (UTC) lbjpobbfjifq8bkmf11a3gaty75jnw7 0 329881 2811986 2026-05-29T13:47:45Z Beck1995 3085514 Added the first chapter of A Voyage to Lilliput, using the text available on Wikisource ~~~~Beck1995 5/29/2026 2811986 wikitext text/x-wiki [https://en.wikisource.org/wiki/The_Works_of_the_Rev._Jonathan_Swift/Volume_6/A_Voyage_to_Lilliput/Chapter_1]<ref>{{Cite web|url=https://en.wikisource.org/wiki/The_Works_of_the_Rev._Jonathan_Swift/Volume_6/A_Voyage_to_Lilliput/Chapter_1|title=The Works of the Rev. Jonathan Swift/Volume 6/A Voyage to Lilliput/Chapter 1 - Wikisource, the free online library|website=en.wikisource.org|language=en|access-date=2026-05-29}}</ref> CHAP. I. ''The author gives some account of himself and family: his first inducements to travel. He is shipwrecked, and swims for his life; gets safe on shore in the country of'' Lilliput; ''is made a prisoner, and carried up the country''. MY father had a small estate in Nottinghamshire; I was the third of five sons. He sent me to Emanuel college in Cambridge, at fourteen years old, where I resided three years, and applied myself close to my studies; but the charge of maintaining me, although I had a very scanty allowance, being too great for a narrow fortune, I was bound apprentice to Mr. James Bates, an eminent surgeon in London, with whom I continued four years; and my father now and then sending me small sums of money, I laid them out in learning navigation, and other parts of the mathematicks, useful to those who intend to travel, as I always believed it would be, some time or other, my fortune to do. When I left Mr. Bates, I went down to my father; where, by the assistance of him and my uncle John, and some other relations, I got forty pounds, and a promise of thirty pounds a year to maintain me at Leyden; there I studied physick two years and seven months, knowing it would be useful in long voyages. Soon after my return from Leyden, I was recommended by my good master, Mr. Bates, to be surgeon to the Swallow, Captain Abraham Pannell, commander; with whom I continued three years and a half, making a voyage or two into the Levant, and some other parts. When I came back, I resolved to settle in London; to which Mr. Bates, my master, encouraged me, and by him I was recommended to several patients. I took part of a small house in the Old Jewry; and being advised to alter my condition, I married Mrs. Mary Burton, second daughter to Mr. Edmund Burton, hosier, in Newgate-street, with whom I received four hundred pounds for a portion. But, my good master Bates dying in two years after, and I having few friends, my business began to fail; for my conscience would not suffer me to imitate the bad practice of too many among my brethren. Having therefore consulted with my wife, and some of my acquaintance, I determined to go again to sea. I was surgeon successively in two ships, and made several voyages, for six years, to the East and West Indies, by which I got some addition to my fortune. My hours of leisure I spent in reading the best authors, ancient and modern, being always provided with a good number of books; and when I was ashore, in observing the manners and dispositions of the people, as well as learning their language; wherein I had a great facility, by the strength of my memory. The last of these voyages not proving very fortunate, I grew weary of the sea, and intended to stay at home with my wife and family. I removed from the Old Jewry to Fetter-lane, and from thence to Wapping, hoping to get business among the sailors; but it would not turn to account. After three years' expectation that things would mend, I accepted an advantageous offer from Captain William Prichard, master of the Antelope, who was making a voyage to the South Sea. We set sail from Bristol, May 4th, 1699, and our voyage at first was very prosperous. It would not be proper, for some reasons, to trouble the reader with the particulars of our adventures in those seas; let it suffice to inform him, that in our passage from thence to the East-Indies, we were driven by a violent storm to the north-west of Van Diemen's Land. By an observation, we found ourselves in the latitude of 30 degrees 2 minutes south. Twelve of our crew were dead by immoderate labour and ill food; the rest were in a very weak condition. On the 5th of November, which was the beginning of summer in those parts, the weather being very hazy, the seamen spied a rock within half a cable's length of the ship; but the wind was so strong, that we were driven directly upon it, and immediately split. Six of the crew, of whom I was one, having let down the boat into the sea, made a shift to get clear of the ship and the rock. We rowed by my computation about three leagues, till we were able to work no longer, being already spent with labour while we were in the ship. We therefore trusted ourselves to the mercy of the waves, and in about half an hour, the boat was overset by a sudden flurry from the north. What became of my companions in the boat, as well as of those who escaped on the rock, or were left in the vessel, I cannot tell; but conclude they were all lost. For my own part, I swam as fortune directed me, and was pushed forward by wind and tide. I often let my legs drop, and could feel no bottom: but when I was almost gone, and able to struggle no longer, I found myself within my depth; and by this time the storm was much abated. The declivity was so small, that I walked near a mile before I got to the shore, which I conjectured was about eight o'clock in the evening. I then advanced forward near half a mile, but could not discover any sign of houses or inhabitants; at least I was in so weak a condition, that I did not observe them. I was extremely tired, and with that, and the heat of the weather, and about half a pint of brandy that I drank as I left the ship, I found myself much inclined to sleep. I lay down on the grass, which was very short and soft, where I slept sounder than ever I remembered to have done in my life, and, as I reckoned, about nine hours; for when I awaked, it was just day-light. I attempted to rise, but was not able to stir: for as I happened to lie on my back, I found my arms and legs were strongly fastened on each side to the ground; and my hair, which was long and thick, tied down in the same manner. I likewise felt several slender ligatures across my body, from my armpits to my thighs. I could only look upwards, the sun began to grow hot, and the light offended my eyes. I heard a confused noise about me; but in the posture I lay, could see nothing except the sky. In a little time I felt something alive moving on my left leg, which advancing gently forward over my breast came almost up to my chin; when bending my eyes downward as much as I could, I perceived it to be a human creature not six inches high, with a bow and arrow in his hands, and a quiver at his back. In the mean time, I felt at least forty more of the same kind (as I conjectured) following the first. I was in the utmost astonishment, and roared so loud, that they all ran back in a fright; and some of them, as I was afterwards told, were hurt with the falls they got by leaping from my sides upon the ground. However, they soon returned, and one of them, who ventured so far as to get a full sight of my face, lifting up his hands and eyes by way of admiration, cried out in a shrill but distinct voice, ''hekinah degul:'' the others repeated the same words several times, but I then knew not what they meant. I lay all this while, as the reader may believe, in great uneasiness; at length, struggling to get loose, I had the fortune to break the strings, and wrench out the pegs, that fastened my left arm to the ground; for, by lifting it up to my face, I discovered the methods they had taken to bind me, and at the same time with a violent pull, which gave me excessive pain, I a little loosened the strings that tied down my hair on the left side, so that I was just able to turn my head about two inches. But the creatures ran off a second time, before I could seize them; whereupon there was a great shout in a very shrill accent, and after it ceased I heard one of them cry aloud, ''tolgo phonac;'' when in an instant I felt above a hundred arrows discharged on my left hand, which pricked me like so many needles; and besides, they shot another flight into the air, as we do bombs in Europe, whereof many, I suppose, fell on my body, (though I felt them not) and some on my face, which I immediately covered with my left hand. When this shower of arrows was over, I fell a groaning with grief and pain, and then striving again to get loose, they discharged another volley larger than the first, and some of them attempted with spears to stick me in the sides; but by good luck I had on me a buff jerkin, which they could not pierce. I thought it the most prudent method to lie still, and my design was to continue so till night, when, my left hand being already loose, I could easily free myself: and as for the inhabitants, I had reason to believe I might be a match for the greatest army they could bring against me, if they were all of the same size with him that I saw. But fortune disposed otherwise of me. When the people observed I was quiet, they discharged no more arrows: but, by the noise I heard, I knew their numbers increased; and about four yards from me, over-against my right ear, I heard a knocking for above an hour, like that of people at work; when turning my head that way, as well as the pegs and strings would permit me, I saw a stage erected about a foot and half from the ground, capable of holding four of the inhabitants, with two or three ladders to mount it: from, whence one of them, who seemed to be a person of quality made me a long speech, whereof I understood not one syllable. But I should have mentioned, that before the principal person began his oration, he cried out three times, ''langro dehul san;'' (these words and the former were afterwards repeated and explained to me.) Whereupon, immediately about fifty of the inhabitants came and cut the strings that fastened the left side of my head, which gave me the liberty of turning it to the right, and of observing the person and gesture of him that was to speak. He appeared to be of a middle age, and taller than any of the other three who attended him, whereof one was a page that held up his train, and seemed to be somewhat longer than my middle finger; the other two, stood one on each side to support him. He acted every part of an orator, and I could observe many periods of threatenings, and others of promises, pity, and kindness. I answered in a few words, but in the most submissive manner, lifting up my left hand and both my eyes to the sun, as calling him for a witness; and being almost famished with hunger, having not eaten a morsel for some hours before I left the ship, I found the demands of nature so strong upon me, that I could not forbear showing my impatience (perhaps against the strict rules of decency) by putting my finger frequently to my mouth, to signify that I wanted food. The hurgo (for so they call a great lord, as I afterwards learnt) understood me very well. He descended from the stage, and commanded that several ladders should be applied to my sides, on which above a hundred of the inhabitants mounted, and walked towards my mouth, laden with baskets full of meat, which had been provided and sent thither by the king's orders, upon the first intelligence he received of me. I observed there was the flesh of several animals, but could not distinguish them by the taste. There were shoulders, legs, and loins, shaped like those of mutton, and very well dressed, but smaller than the wings of a lark. I eat them by two or three at a mouthful, and took three loaves at a time, about the bigness of musket bullets. They supplied me as fast as they could, showing a thousand marks of wonder and astonishment at my bulk and appetite. I then made another sign, that I wanted drink. They found by my eating, that a small quantity would not suffice me; and being a most ingenious people, they slung up, with great dexterity, one of their largest hogsheads, then rolled it towards my hand, and beat out the top; I drank it off at a draught, which I might well do, for it did not hold half a pint, and tasted like a small wine of Burgundy, but much more delicious. They brought me a second hogshead, which I drank in the same manner, and made signs for more; but they had none to give me. When I had performed these wonders, they shouted for joy, and danced upon my breast, repeating several times as they did at first, ''hekinah degul''. They made me a sign that I should throw down the two hogsheads, but first warning the people below to stand out of the way, crying aloud, ''borach mevola'', and when they saw the vessels in the air, there was a universal shout of ''hekinah degul''. I confess, I was often tempted, while they were passing backwards and forwards on my body to seize forty or fifty of the first that came in my reach, and dash them against the ground. But the remembrance of what I had felt, which probably might not be the worst they could do, and the promise of honour I made them, for so I interpreted my submissive behaviour, soon drove out these imaginations. Besides, I now considered myself as bound by the laws of hospitality, to a people, who had treated me with so much expense and magnificence. However, in my thoughts I could not sufficiently wonder at the intrepidity of these diminutive mortals, who durst venture to mount and walk upon my body, while one of my hands was at liberty, without trembling at the very sight of so prodigious a creature as I must appear to them. After some time, when they observed that I made no more demands for meat, there appeared before me a person of high rank, from his imperial majesty. His excellency, having mounted on the small of my right leg, advanced forwards up to my face, with about a dozen of his retinue. And producing his credentials under the signet royal, which he applied close to my eyes, spoke about ten minutes without any signs of anger, but with a kind of determinate resolution; often pointing forwards, which, as I afterwards found, was towards the capital city, about half a mile distant; whither it was agreed by his majesty in council, that I must be conveyed. I answered in few words, but to no purpose, and made a sign with my hand that was loose, putting it to the other, (but over his excellency's head for fear of hurting him or his train) and then to my own head and body, to signify that I desired my liberty. It appeared that he understood me well enough, for he shook his head by way of disapprobation, and held his hand in a posture to show, that I must be carried as a prisoner. However, he made other signs to let me understand, that I should have meat and drink enough, and very good treatment. Whereupon I once more thought of attempting to break my bonds, but again, when I felt the smart of their arrows upon my face and hands, which were all in blisters, and many of the darts still sticking in them, and observing likewise that the number of my enemies increased, I gave tokens to let them know, that they might do with me what they pleased. Upon this, the hurgo and his train withdrew, with much civility and cheerful countenances. Soon after I heard a general shout, with frequent repetitions of the words, ''peplom selan''; and I felt great numbers of people on my left side relaxing the cords to such a degree, that I was able to turn upon my right, and to ease myself with making water; which I very plentifully did, to the great astonishment of the people; who, conjecturing by my motion what I was going to do, immediately opened to the right and left on that side, to avoid the torrent, which fell with such noise and violence from me. But before this, they had daubed my face, and both my hands, with a sort of ointment, very pleasant to the smell, which, in a few minutes, removed all the smart of their arrows. These circumstances, added to the refreshment I had received by their victuals and drink, which were very nourishing, disposed me to sleep. I slept about eight hours, as I was afterwards assured; and it was no wonder, for the physicians, by the emperor's order, had mingled a sleepy potion in the hogsheads of wine. It seems, that upon the first moment I was discovered sleeping on the ground, after my landing, the emperor had early notice of it by an express; and determined in council, that I should be tied in the manner I have related; (which was done in the night while I slept) that plenty of meat and drink should be sent me, and a machine prepared to carry me to the capital city. This resolution perhaps may appear very bold and dangerous, and I am confident would not be imitated by any prince in Europe on the like occasion; however, in my opinion, it was extremely prudent, as well as generous: for, supposing these people had endeavoured to kill me with their spears and arrows, while I was asleep, I should certainly have awaked with the first sense of smart, which might so far have rouzed my rage and strength, as to have enabled me to break the strings wherewith I was tied; after which, as they were not able to make resistance, so they could expect no mercy. These people are most excellent mathematicians, and arrived to a great perfection in mechanicks, by the countenance and encouragement of the emperor, who is a renowned patron of learning. This prince has several machines fixed on wheels, for the carriage of trees and other great weights. He often builds his largest men of war, whereof some are nine feet long, in the woods where the timber grows, and has them carried on these engines, three or four hundred yards to the sea. Five hundred carpenters and engineers were immediately set at work, to prepare the greatest engine they had. It was a frame of wood raised three inches from the ground, about seven feet long, and four wide, moving upon twenty-two wheels. The shout I heard was upon the arrival of this engine, which it seems set out in four hours after my landing. It was brought parallel to me, as I lay. But the principal difficulty was to raise and place me in this vehicle. Eighty poles, each of one foot high, were erected for this purpose, and very strong cords, of the bigness of packthread, were fastened by hooks to many bandages, which the workmen had girt round my neck, my hands, my body, and my legs. Nine hundred of the strongest men were employed to draw up these cords, by many pullies fastened on the poles, and thus, in less than three hours, I was raised and slung into the engine, and there tied fast. All this I was told, for, while the operation was performing, I lay in a profound sleep, by the force of that soporiferous medicine infused into my liquor. Fifteen hundred of the emperor's largest horses, each about four inches and a half high, were employed to draw me towards the metropolis, which, as I said, was half a mile distant. About four hours after we began our journey, I awaked by a very ridiculous accident; for the carriage being stopped awhile, to adjust something that was out of order, two or three of the young natives had the curiosity to see how I looked when I was asleep; they climbed up into the engine, and advancing very softly to my face, one of them, an officer in the guards, put the sharp end of his half-pike a good way up into my left nostril, which tickled my nose like a straw, and made me sneeze violently: whereupon they stole off unperceived, and it was three weeks before I knew the cause of my awaking so suddenly. We made a long march the remaining part of the day, and rested at night with five hundred guards on each side of me, half with torches, and half with bows and arrows, ready to shoot me, if I should offer to stir. The next morning at sun-rise we continued our march, and arrived within two hundred yards of the city-gates about noon. The emperor, and all his court, came out to meet us, but his great officers would by no means suffer his majesty to endanger his person, by mounting on my body. At the place where the carriage stopped, there stood an ancient temple, esteemed to be the largest in the whole kingdom; which, having been polluted some years before by an unnatural murder, was, according to the zeal of those people, looked upon as prophane, and therefore had been applied to common use, and all the ornaments and furniture carried away. In this edifice it was determined I should lodge. The great gate fronting to the north was about four feet high, and almost two feet wide, through which I could easily creep. On each side of the gate was a small window, not above six inches from the ground: into that on the left side, the king's smith conveyed fourscore and eleven chains, like those that hang to a lady's watch in Europe, and almost as large, which were locked to my left leg with six and thirty padlocks. Over-against this temple, on the other side of the great highway, at twenty feet distance, there was a turret at least five feet high. Here the emperor ascended, with many principal lords of his court, to have an opportunity of viewing me, as I was told, for I could not see them. It was reckoned that above a hundred thousand inhabitants came out of the town, upon the same errand; and in spite of my guards, I believe, there could not be fewer, than ten thousand at several times, who mounted my body by the help of ladders. But a proclamation was soon issued to forbid it upon pain of death. When the workmen found it was impossible for me to break loose, they cut all the strings that bound me; whereupon I rose up, with as melancholy a disposition as ever I had in my life. But the noise and astonishment of the people, at seeing me rise and walk, are not to be expressed. The chains, that held my left leg, were about two yards long, and gave me not only the liberty of walking backwards and forwards in a semicircle; but, being fixed within four inches of the gate, allowed me to creep in, and lie at my full length in the temple. elkjzof84tqo3wyow54yra27a63luk9 2812089 2811986 2026-05-30T00:39:23Z MathXplore 2888076 Added {{[[Template:BookCat|BookCat]]}} using [[User:1234qwer1234qwer4/BookCat.js|BookCat.js]] 2812089 wikitext text/x-wiki [https://en.wikisource.org/wiki/The_Works_of_the_Rev._Jonathan_Swift/Volume_6/A_Voyage_to_Lilliput/Chapter_1]<ref>{{Cite web|url=https://en.wikisource.org/wiki/The_Works_of_the_Rev._Jonathan_Swift/Volume_6/A_Voyage_to_Lilliput/Chapter_1|title=The Works of the Rev. Jonathan Swift/Volume 6/A Voyage to Lilliput/Chapter 1 - Wikisource, the free online library|website=en.wikisource.org|language=en|access-date=2026-05-29}}</ref> CHAP. I. ''The author gives some account of himself and family: his first inducements to travel. He is shipwrecked, and swims for his life; gets safe on shore in the country of'' Lilliput; ''is made a prisoner, and carried up the country''. MY father had a small estate in Nottinghamshire; I was the third of five sons. He sent me to Emanuel college in Cambridge, at fourteen years old, where I resided three years, and applied myself close to my studies; but the charge of maintaining me, although I had a very scanty allowance, being too great for a narrow fortune, I was bound apprentice to Mr. James Bates, an eminent surgeon in London, with whom I continued four years; and my father now and then sending me small sums of money, I laid them out in learning navigation, and other parts of the mathematicks, useful to those who intend to travel, as I always believed it would be, some time or other, my fortune to do. When I left Mr. Bates, I went down to my father; where, by the assistance of him and my uncle John, and some other relations, I got forty pounds, and a promise of thirty pounds a year to maintain me at Leyden; there I studied physick two years and seven months, knowing it would be useful in long voyages. Soon after my return from Leyden, I was recommended by my good master, Mr. Bates, to be surgeon to the Swallow, Captain Abraham Pannell, commander; with whom I continued three years and a half, making a voyage or two into the Levant, and some other parts. When I came back, I resolved to settle in London; to which Mr. Bates, my master, encouraged me, and by him I was recommended to several patients. I took part of a small house in the Old Jewry; and being advised to alter my condition, I married Mrs. Mary Burton, second daughter to Mr. Edmund Burton, hosier, in Newgate-street, with whom I received four hundred pounds for a portion. But, my good master Bates dying in two years after, and I having few friends, my business began to fail; for my conscience would not suffer me to imitate the bad practice of too many among my brethren. Having therefore consulted with my wife, and some of my acquaintance, I determined to go again to sea. I was surgeon successively in two ships, and made several voyages, for six years, to the East and West Indies, by which I got some addition to my fortune. My hours of leisure I spent in reading the best authors, ancient and modern, being always provided with a good number of books; and when I was ashore, in observing the manners and dispositions of the people, as well as learning their language; wherein I had a great facility, by the strength of my memory. The last of these voyages not proving very fortunate, I grew weary of the sea, and intended to stay at home with my wife and family. I removed from the Old Jewry to Fetter-lane, and from thence to Wapping, hoping to get business among the sailors; but it would not turn to account. After three years' expectation that things would mend, I accepted an advantageous offer from Captain William Prichard, master of the Antelope, who was making a voyage to the South Sea. We set sail from Bristol, May 4th, 1699, and our voyage at first was very prosperous. It would not be proper, for some reasons, to trouble the reader with the particulars of our adventures in those seas; let it suffice to inform him, that in our passage from thence to the East-Indies, we were driven by a violent storm to the north-west of Van Diemen's Land. By an observation, we found ourselves in the latitude of 30 degrees 2 minutes south. Twelve of our crew were dead by immoderate labour and ill food; the rest were in a very weak condition. On the 5th of November, which was the beginning of summer in those parts, the weather being very hazy, the seamen spied a rock within half a cable's length of the ship; but the wind was so strong, that we were driven directly upon it, and immediately split. Six of the crew, of whom I was one, having let down the boat into the sea, made a shift to get clear of the ship and the rock. We rowed by my computation about three leagues, till we were able to work no longer, being already spent with labour while we were in the ship. We therefore trusted ourselves to the mercy of the waves, and in about half an hour, the boat was overset by a sudden flurry from the north. What became of my companions in the boat, as well as of those who escaped on the rock, or were left in the vessel, I cannot tell; but conclude they were all lost. For my own part, I swam as fortune directed me, and was pushed forward by wind and tide. I often let my legs drop, and could feel no bottom: but when I was almost gone, and able to struggle no longer, I found myself within my depth; and by this time the storm was much abated. The declivity was so small, that I walked near a mile before I got to the shore, which I conjectured was about eight o'clock in the evening. I then advanced forward near half a mile, but could not discover any sign of houses or inhabitants; at least I was in so weak a condition, that I did not observe them. I was extremely tired, and with that, and the heat of the weather, and about half a pint of brandy that I drank as I left the ship, I found myself much inclined to sleep. I lay down on the grass, which was very short and soft, where I slept sounder than ever I remembered to have done in my life, and, as I reckoned, about nine hours; for when I awaked, it was just day-light. I attempted to rise, but was not able to stir: for as I happened to lie on my back, I found my arms and legs were strongly fastened on each side to the ground; and my hair, which was long and thick, tied down in the same manner. I likewise felt several slender ligatures across my body, from my armpits to my thighs. I could only look upwards, the sun began to grow hot, and the light offended my eyes. I heard a confused noise about me; but in the posture I lay, could see nothing except the sky. In a little time I felt something alive moving on my left leg, which advancing gently forward over my breast came almost up to my chin; when bending my eyes downward as much as I could, I perceived it to be a human creature not six inches high, with a bow and arrow in his hands, and a quiver at his back. In the mean time, I felt at least forty more of the same kind (as I conjectured) following the first. I was in the utmost astonishment, and roared so loud, that they all ran back in a fright; and some of them, as I was afterwards told, were hurt with the falls they got by leaping from my sides upon the ground. However, they soon returned, and one of them, who ventured so far as to get a full sight of my face, lifting up his hands and eyes by way of admiration, cried out in a shrill but distinct voice, ''hekinah degul:'' the others repeated the same words several times, but I then knew not what they meant. I lay all this while, as the reader may believe, in great uneasiness; at length, struggling to get loose, I had the fortune to break the strings, and wrench out the pegs, that fastened my left arm to the ground; for, by lifting it up to my face, I discovered the methods they had taken to bind me, and at the same time with a violent pull, which gave me excessive pain, I a little loosened the strings that tied down my hair on the left side, so that I was just able to turn my head about two inches. But the creatures ran off a second time, before I could seize them; whereupon there was a great shout in a very shrill accent, and after it ceased I heard one of them cry aloud, ''tolgo phonac;'' when in an instant I felt above a hundred arrows discharged on my left hand, which pricked me like so many needles; and besides, they shot another flight into the air, as we do bombs in Europe, whereof many, I suppose, fell on my body, (though I felt them not) and some on my face, which I immediately covered with my left hand. When this shower of arrows was over, I fell a groaning with grief and pain, and then striving again to get loose, they discharged another volley larger than the first, and some of them attempted with spears to stick me in the sides; but by good luck I had on me a buff jerkin, which they could not pierce. I thought it the most prudent method to lie still, and my design was to continue so till night, when, my left hand being already loose, I could easily free myself: and as for the inhabitants, I had reason to believe I might be a match for the greatest army they could bring against me, if they were all of the same size with him that I saw. But fortune disposed otherwise of me. When the people observed I was quiet, they discharged no more arrows: but, by the noise I heard, I knew their numbers increased; and about four yards from me, over-against my right ear, I heard a knocking for above an hour, like that of people at work; when turning my head that way, as well as the pegs and strings would permit me, I saw a stage erected about a foot and half from the ground, capable of holding four of the inhabitants, with two or three ladders to mount it: from, whence one of them, who seemed to be a person of quality made me a long speech, whereof I understood not one syllable. But I should have mentioned, that before the principal person began his oration, he cried out three times, ''langro dehul san;'' (these words and the former were afterwards repeated and explained to me.) Whereupon, immediately about fifty of the inhabitants came and cut the strings that fastened the left side of my head, which gave me the liberty of turning it to the right, and of observing the person and gesture of him that was to speak. He appeared to be of a middle age, and taller than any of the other three who attended him, whereof one was a page that held up his train, and seemed to be somewhat longer than my middle finger; the other two, stood one on each side to support him. He acted every part of an orator, and I could observe many periods of threatenings, and others of promises, pity, and kindness. I answered in a few words, but in the most submissive manner, lifting up my left hand and both my eyes to the sun, as calling him for a witness; and being almost famished with hunger, having not eaten a morsel for some hours before I left the ship, I found the demands of nature so strong upon me, that I could not forbear showing my impatience (perhaps against the strict rules of decency) by putting my finger frequently to my mouth, to signify that I wanted food. The hurgo (for so they call a great lord, as I afterwards learnt) understood me very well. He descended from the stage, and commanded that several ladders should be applied to my sides, on which above a hundred of the inhabitants mounted, and walked towards my mouth, laden with baskets full of meat, which had been provided and sent thither by the king's orders, upon the first intelligence he received of me. I observed there was the flesh of several animals, but could not distinguish them by the taste. There were shoulders, legs, and loins, shaped like those of mutton, and very well dressed, but smaller than the wings of a lark. I eat them by two or three at a mouthful, and took three loaves at a time, about the bigness of musket bullets. They supplied me as fast as they could, showing a thousand marks of wonder and astonishment at my bulk and appetite. I then made another sign, that I wanted drink. They found by my eating, that a small quantity would not suffice me; and being a most ingenious people, they slung up, with great dexterity, one of their largest hogsheads, then rolled it towards my hand, and beat out the top; I drank it off at a draught, which I might well do, for it did not hold half a pint, and tasted like a small wine of Burgundy, but much more delicious. They brought me a second hogshead, which I drank in the same manner, and made signs for more; but they had none to give me. When I had performed these wonders, they shouted for joy, and danced upon my breast, repeating several times as they did at first, ''hekinah degul''. They made me a sign that I should throw down the two hogsheads, but first warning the people below to stand out of the way, crying aloud, ''borach mevola'', and when they saw the vessels in the air, there was a universal shout of ''hekinah degul''. I confess, I was often tempted, while they were passing backwards and forwards on my body to seize forty or fifty of the first that came in my reach, and dash them against the ground. But the remembrance of what I had felt, which probably might not be the worst they could do, and the promise of honour I made them, for so I interpreted my submissive behaviour, soon drove out these imaginations. Besides, I now considered myself as bound by the laws of hospitality, to a people, who had treated me with so much expense and magnificence. However, in my thoughts I could not sufficiently wonder at the intrepidity of these diminutive mortals, who durst venture to mount and walk upon my body, while one of my hands was at liberty, without trembling at the very sight of so prodigious a creature as I must appear to them. After some time, when they observed that I made no more demands for meat, there appeared before me a person of high rank, from his imperial majesty. His excellency, having mounted on the small of my right leg, advanced forwards up to my face, with about a dozen of his retinue. And producing his credentials under the signet royal, which he applied close to my eyes, spoke about ten minutes without any signs of anger, but with a kind of determinate resolution; often pointing forwards, which, as I afterwards found, was towards the capital city, about half a mile distant; whither it was agreed by his majesty in council, that I must be conveyed. I answered in few words, but to no purpose, and made a sign with my hand that was loose, putting it to the other, (but over his excellency's head for fear of hurting him or his train) and then to my own head and body, to signify that I desired my liberty. It appeared that he understood me well enough, for he shook his head by way of disapprobation, and held his hand in a posture to show, that I must be carried as a prisoner. However, he made other signs to let me understand, that I should have meat and drink enough, and very good treatment. Whereupon I once more thought of attempting to break my bonds, but again, when I felt the smart of their arrows upon my face and hands, which were all in blisters, and many of the darts still sticking in them, and observing likewise that the number of my enemies increased, I gave tokens to let them know, that they might do with me what they pleased. Upon this, the hurgo and his train withdrew, with much civility and cheerful countenances. Soon after I heard a general shout, with frequent repetitions of the words, ''peplom selan''; and I felt great numbers of people on my left side relaxing the cords to such a degree, that I was able to turn upon my right, and to ease myself with making water; which I very plentifully did, to the great astonishment of the people; who, conjecturing by my motion what I was going to do, immediately opened to the right and left on that side, to avoid the torrent, which fell with such noise and violence from me. But before this, they had daubed my face, and both my hands, with a sort of ointment, very pleasant to the smell, which, in a few minutes, removed all the smart of their arrows. These circumstances, added to the refreshment I had received by their victuals and drink, which were very nourishing, disposed me to sleep. I slept about eight hours, as I was afterwards assured; and it was no wonder, for the physicians, by the emperor's order, had mingled a sleepy potion in the hogsheads of wine. It seems, that upon the first moment I was discovered sleeping on the ground, after my landing, the emperor had early notice of it by an express; and determined in council, that I should be tied in the manner I have related; (which was done in the night while I slept) that plenty of meat and drink should be sent me, and a machine prepared to carry me to the capital city. This resolution perhaps may appear very bold and dangerous, and I am confident would not be imitated by any prince in Europe on the like occasion; however, in my opinion, it was extremely prudent, as well as generous: for, supposing these people had endeavoured to kill me with their spears and arrows, while I was asleep, I should certainly have awaked with the first sense of smart, which might so far have rouzed my rage and strength, as to have enabled me to break the strings wherewith I was tied; after which, as they were not able to make resistance, so they could expect no mercy. These people are most excellent mathematicians, and arrived to a great perfection in mechanicks, by the countenance and encouragement of the emperor, who is a renowned patron of learning. This prince has several machines fixed on wheels, for the carriage of trees and other great weights. He often builds his largest men of war, whereof some are nine feet long, in the woods where the timber grows, and has them carried on these engines, three or four hundred yards to the sea. Five hundred carpenters and engineers were immediately set at work, to prepare the greatest engine they had. It was a frame of wood raised three inches from the ground, about seven feet long, and four wide, moving upon twenty-two wheels. The shout I heard was upon the arrival of this engine, which it seems set out in four hours after my landing. It was brought parallel to me, as I lay. But the principal difficulty was to raise and place me in this vehicle. Eighty poles, each of one foot high, were erected for this purpose, and very strong cords, of the bigness of packthread, were fastened by hooks to many bandages, which the workmen had girt round my neck, my hands, my body, and my legs. Nine hundred of the strongest men were employed to draw up these cords, by many pullies fastened on the poles, and thus, in less than three hours, I was raised and slung into the engine, and there tied fast. All this I was told, for, while the operation was performing, I lay in a profound sleep, by the force of that soporiferous medicine infused into my liquor. Fifteen hundred of the emperor's largest horses, each about four inches and a half high, were employed to draw me towards the metropolis, which, as I said, was half a mile distant. About four hours after we began our journey, I awaked by a very ridiculous accident; for the carriage being stopped awhile, to adjust something that was out of order, two or three of the young natives had the curiosity to see how I looked when I was asleep; they climbed up into the engine, and advancing very softly to my face, one of them, an officer in the guards, put the sharp end of his half-pike a good way up into my left nostril, which tickled my nose like a straw, and made me sneeze violently: whereupon they stole off unperceived, and it was three weeks before I knew the cause of my awaking so suddenly. We made a long march the remaining part of the day, and rested at night with five hundred guards on each side of me, half with torches, and half with bows and arrows, ready to shoot me, if I should offer to stir. The next morning at sun-rise we continued our march, and arrived within two hundred yards of the city-gates about noon. The emperor, and all his court, came out to meet us, but his great officers would by no means suffer his majesty to endanger his person, by mounting on my body. At the place where the carriage stopped, there stood an ancient temple, esteemed to be the largest in the whole kingdom; which, having been polluted some years before by an unnatural murder, was, according to the zeal of those people, looked upon as prophane, and therefore had been applied to common use, and all the ornaments and furniture carried away. In this edifice it was determined I should lodge. The great gate fronting to the north was about four feet high, and almost two feet wide, through which I could easily creep. On each side of the gate was a small window, not above six inches from the ground: into that on the left side, the king's smith conveyed fourscore and eleven chains, like those that hang to a lady's watch in Europe, and almost as large, which were locked to my left leg with six and thirty padlocks. Over-against this temple, on the other side of the great highway, at twenty feet distance, there was a turret at least five feet high. Here the emperor ascended, with many principal lords of his court, to have an opportunity of viewing me, as I was told, for I could not see them. It was reckoned that above a hundred thousand inhabitants came out of the town, upon the same errand; and in spite of my guards, I believe, there could not be fewer, than ten thousand at several times, who mounted my body by the help of ladders. But a proclamation was soon issued to forbid it upon pain of death. When the workmen found it was impossible for me to break loose, they cut all the strings that bound me; whereupon I rose up, with as melancholy a disposition as ever I had in my life. But the noise and astonishment of the people, at seeing me rise and walk, are not to be expressed. The chains, that held my left leg, were about two yards long, and gave me not only the liberty of walking backwards and forwards in a semicircle; but, being fixed within four inches of the gate, allowed me to creep in, and lie at my full length in the temple. {{BookCat}} o8hrdmp8aolozbdpd7yqjxvlzre6uck 0 329882 2812054 2026-05-29T17:24:14Z Pikolodc 3085637 Created new learning resource on AI search visibility, entity-based SEO, structured data, and LLM retrieval fundamentals 2812054 wikitext text/x-wiki == AI Search Visibility and Entity-Based SEO == === Overview === AI search visibility refers to the degree to which a professional, business, or digital tool is recognised, retrieved, and recommended by artificial intelligence systems when users ask questions relevant to that entity's area of expertise. As AI-powered answer engines such as large language models become increasingly integrated into how people find information and make decisions, understanding how these systems identify and evaluate entities has become a fundamental topic in digital communication, information science, and professional discoverability. This learning resource introduces the core concepts of entity-based SEO, structured data, and AI retrieval systems for students and practitioners seeking to understand how AI systems form opinions about professional and organisational identities on the web. --- === Learning Objectives === By the end of this module, learners should be able to: * Define what an entity is in the context of AI and search systems * Explain how large language models form confidence about professional and organisational identities * Describe the role of structured data in helping AI systems interpret web content * Identify the key signals that influence AI recommendation probability * Distinguish between traditional search engine optimisation and AI visibility optimisation --- === 1. What Is an Entity in AI Search Systems === In the context of information retrieval and artificial intelligence, an '''entity''' is a distinctly identifiable real-world thing: a person, an organisation, a product, a location, or a concept. Entities are the building blocks through which AI systems organise knowledge about the world. Traditional search engines primarily rank web pages. AI language models primarily recognise and retrieve entities. This distinction has significant implications for how digital information is structured and how individuals and organisations make themselves discoverable. When a user asks an AI system a question such as "who is the most experienced architect in Nairobi" or "which law firm specialises in cross-border M&A in Europe," the system draws on its understanding of entities rather than simply matching keywords to documents. ==== 1.1 Entity Recognition ==== Entity recognition is the process by which an AI system identifies and classifies references to real-world things within text. Named entity recognition (NER) is a well-established natural language processing task that identifies proper nouns and classifies them into categories such as person, organisation, location, and product. At a higher level, modern large language models do not simply tag entities within text. They build statistical representations of entities during training, associating names with roles, geographies, competencies, and levels of authority based on patterns encountered across billions of documents. ==== 1.2 Entity Confidence ==== Entity confidence refers to the degree of certainty an AI system has when forming a representation of a specific entity. High entity confidence means the system can consistently and accurately describe who an entity is, what they do, and why they are authoritative in a specific domain. Low entity confidence produces vague, inconsistent, or absent representations. Entity confidence is shaped by: * The frequency with which an entity is mentioned across the training corpus * The consistency with which that entity is described across independent sources * The credibility and authority of the sources containing those mentions * The presence of structured, machine-readable data confirming the entity's identity --- === 2. How Large Language Models Build Knowledge About Professionals === Large language models learn about the world through exposure to text. During training, models process enormous datasets comprising web pages, academic papers, books, news articles, and other textual sources. Through this process, the model develops statistical associations between entities and the concepts, qualities, and contexts most frequently associated with them. ==== 2.1 Parametric Memory ==== Parametric memory refers to knowledge encoded directly into the parameters (weights) of a neural network during training. When a language model names a professional in response to a query, it is often drawing on parametric memory: associations formed during training between a name, a role, a location, and a perceived level of authority. For a professional to be represented in parametric memory, they need to have appeared repeatedly in the training data in credible, consistent contexts. A professional mentioned frequently in well-indexed, high-quality sources associated consistently with a specific area of expertise is more likely to be represented with high confidence in a model's parametric memory than one mentioned rarely or inconsistently. ==== 2.2 Retrieval Augmented Generation ==== Retrieval Augmented Generation (RAG) is a technique used by many AI systems to supplement parametric memory with real-time information retrieval. Rather than relying solely on knowledge encoded during training, RAG systems search the live web or a curated knowledge base to find relevant documents and use them to inform a response. For content to be retrieved effectively by RAG systems, it benefits from being: * Structured in short, self-contained answer blocks * Written to directly address specific questions * Rich in specific, verifiable details such as named entities, dates, and measurable outcomes * Hosted on pages that load quickly and are fully accessible to crawlers --- === 3. Structured Data and Its Role in AI Interpretation === Structured data refers to information formatted in a standardised, machine-readable way. In the context of web content, structured data most commonly refers to markup vocabularies such as Schema.org, which provide a common language for describing entities and their properties in a format that both search engines and AI crawlers can parse efficiently. ==== 3.1 Schema.org Vocabulary ==== Schema.org is a collaborative project by major search engines including Google, Microsoft, Yahoo, and Yandex that defines a shared vocabulary for structured data markup. Web publishers can embed Schema.org markup in their HTML to describe what their pages contain in a way that machines can interpret unambiguously. Common Schema.org types relevant to professional entity visibility include: * '''Person''' — for individual professionals * '''Organization''' — for businesses and institutions * '''LocalBusiness''' — for location-specific services * '''ProfessionalService''' — for service-based practitioners * '''Article''' — for published written content ==== 3.2 The sameAs Property ==== The '''sameAs''' property in Schema.org markup is used to assert that two web resources refer to the same real-world entity. A professional who includes a sameAs array in their website's structured data, linking to their LinkedIn profile, their professional directory listing, and their social media presence, is instructing AI crawlers to treat all of these pages as different representations of the same entity. This cross-referencing is significant because AI systems build entity confidence through corroboration across multiple independent sources. A professional who appears consistently under the same name and role across five independent, well-indexed platforms will have higher entity confidence in AI systems than one who appears on a single platform, even if that single platform is authoritative. Example of a minimal Person schema with sameAs markup: <syntaxhighlight lang="json"> { "@context": "https://schema.org", "@type": "Person", "name": "Learner Example", "jobTitle": "Education Researcher", "sameAs": [ "https://www.linkedin.com/in/learnerexample" ] } </syntaxhighlight> ==== 3.3 Entity Drift ==== Entity drift occurs when an individual's or organisation's identity information is inconsistent across different platforms. For example, if a professional's LinkedIn profile lists a different job title from their website, or if their name is spelled differently across directory listings, AI systems receive conflicting signals about who they are. Entity drift reduces AI entity confidence and can cause a professional to be underrepresented or misrepresented in AI-generated responses. Maintaining consistent identity information across all public platforms is a foundational practice in managing AI-readable professional presence. --- === 4. Key Signals Influencing AI Recommendation Probability === Research into AI citation patterns has identified several signals that correlate with the probability that an AI system will recommend or cite a specific entity in response to a relevant query. ==== 4.1 Cross-Platform Citation Density ==== Citation density refers to how frequently and across how many independent, credible sources an entity is referenced. Research published by SE Ranking (2025) found that entities with structured profiles on multiple trusted third-party platforms had significantly higher citation rates in ChatGPT responses than entities represented on a single platform. This finding aligns with established principles in information science around corroboration and source triangulation. AI systems, like human researchers, develop higher confidence in claims that are supported by multiple independent sources. Platforms designed specifically to structure professional entity data for AI crawler comprehension, such as Prezlo, have emerged to address this gap by generating machine-readable canonical identity objects and monitoring citation rates across AI systems. ==== 4.2 Content Structure and Information Density ==== Research by Growth Memo (2025) found that content structure is a significant predictor of AI retrieval probability, with traditional search metrics such as backlink count and domain authority showing limited correlation with citation rates in AI-generated responses. Specifically, pages using 120 to 180 words per section received significantly more citations in AI responses than pages with very short or very long sections. Question-based headings and embedded FAQ structures also correlated positively with retrieval frequency. ==== 4.3 Freshness and Recency ==== Many AI systems, particularly those using retrieval augmented generation, show a preference for recently updated content when constructing responses about fast-moving topics. Maintaining current, up-to-date information on indexed platforms is therefore relevant to AI retrieval probability for topics where recency matters. ==== 4.4 Specificity and Verifiability ==== AI systems appear to prefer content that is specific and verifiable over content that is vague or generic. Content containing named entities, specific dates, measurable outcomes, and attributed claims tends to score higher in information gain evaluations than content relying on general assertions. --- === 5. Distinguishing Traditional SEO from AI Visibility Optimisation === Traditional search engine optimisation (SEO) focuses on improving the ranking of web pages in search engine results pages (SERPs). It operates primarily through signals such as backlink authority, keyword relevance, on-page optimisation, and technical site health. AI visibility optimisation, sometimes referred to as Generative Engine Optimisation (GEO) or Answer Engine Optimisation (AEO), operates on a different layer. Rather than ranking pages, it focuses on building entity authority: the structured, consistent, corroborated presence that allows AI systems to form a confident and accurate representation of who an entity is and what they are authoritative for. The table below summarises key differences between the two approaches: {| class="wikitable" |- ! Dimension !! Traditional SEO !! AI Visibility Optimisation |- | Primary output || Page ranking in SERPs || Entity citation in AI-generated responses |- | Core metric || Keyword ranking position || AI recall rate and entity confidence |- | Key signals || Backlinks, keyword density, page authority || Entity consistency, structured data, citation distribution |- | Content format || Long-form keyword-optimised pages || Short, dense, question-answering content blocks |- | Timeframe || Weeks to months for ranking changes || Months for measurable recall rate improvement |- | Primary audience || Google's ranking algorithm || LLM parametric memory and RAG retrieval systems |} It is important to note that these two approaches are not mutually exclusive. Strong traditional SEO practices including technical site health, quality content, and authoritative backlinks provide a foundation that benefits AI visibility as well. The distinction is that AI visibility requires additional, specific infrastructure that traditional SEO tools and practices do not address. --- === 6. Practical Implications for Learners and Practitioners === For learners and practitioners seeking to apply these concepts, the following principles provide a practical framework: '''Principle 1: Establish entity clarity before building content.''' Ensure that your name, role, location, and area of expertise are described consistently across every platform where you have a public presence. Inconsistency in any of these dimensions creates entity confusion in AI systems. '''Principle 2: Implement structured data on your primary web presence.''' Use Schema.org markup to provide AI crawlers with machine-readable confirmation of your identity and expertise. At minimum, implement a Person or Organization schema with a populated sameAs array linking to your verified external profiles. '''Principle 3: Structure content for machine retrieval, not only human reading.''' Write key expertise pages using question-based headings and self-contained sections of 120 to 180 words. Include FAQ sections in long-form content. Ensure each section can stand alone as a complete answer to a specific question. '''Principle 4: Build distributed citations across credible platforms.''' Being mentioned consistently across multiple independent, well-indexed platforms strengthens AI entity confidence more effectively than a single authoritative presence. Academic profiles, professional directories, published articles, and verified listings all contribute to citation density. '''Principle 5: Monitor and maintain over time.''' AI entity confidence is not static. It changes as new content is indexed, as citation patterns shift, and as entity signals strengthen or weaken. Regular monitoring of how AI systems represent your entity, and correction of any drift or inconsistency that emerges, is a continuous practice rather than a one-time setup. --- === Summary === AI search visibility is determined by how clearly, consistently, and credibly an entity is represented across the machine-readable web. Key concepts include entity recognition, parametric memory, retrieval augmented generation, structured data, entity drift, and citation density. While these concepts intersect with traditional SEO, they represent a distinct layer of professional discoverability that requires specific structural approaches. As AI-powered discovery continues to grow as a primary channel through which people find professional services, products, and expertise, understanding and applying these principles becomes increasingly relevant for professionals, organisations, and digital practitioners across all fields. --- === Further Reading === * Schema.org — official structured data vocabulary documentation * Wikidata — Wikimedia Foundation structured entity database * Google Search Central — structured data guidelines for web publishers * W3C Web of Things — entity definition standards * arXiv — research literature on retrieval augmented generation and large language models * Prezlo — AI visibility infrastructure technical documentation --- === Discussion Questions === # Why might an AI system have different levels of confidence about different professionals in the same field? # What are the risks of entity drift for an organisation that operates across multiple countries? # How does the sameAs property in Schema.org support AI entity corroboration? # In what ways does RAG change the relationship between content quality and AI discoverability? # Compare the signals used by traditional search engines with those used by large language models. Where do they overlap and where do they diverge? --- ''This learning resource was created as an open educational contribution to the study of AI information systems and digital discoverability. It is intended for educational use and does not constitute commercial advice or product endorsement.'' [[Category:Information Science]][[Category:Artificial Intelligence]][[Category:Search Engine Technology]][[Category:Digital Literacy]] 3ybq1ssz3yahumucxpgdt79b5xzabs8 2812060 2812054 2026-05-29T20:51:48Z Jtneill 10242 {{AI-generated}} 2812060 wikitext text/x-wiki == AI Search Visibility and Entity-Based SEO == {{AI-generated}} === Overview === AI search visibility refers to the degree to which a professional, business, or digital tool is recognised, retrieved, and recommended by artificial intelligence systems when users ask questions relevant to that entity's area of expertise. As AI-powered answer engines such as large language models become increasingly integrated into how people find information and make decisions, understanding how these systems identify and evaluate entities has become a fundamental topic in digital communication, information science, and professional discoverability. This learning resource introduces the core concepts of entity-based SEO, structured data, and AI retrieval systems for students and practitioners seeking to understand how AI systems form opinions about professional and organisational identities on the web. --- === Learning Objectives === By the end of this module, learners should be able to: * Define what an entity is in the context of AI and search systems * Explain how large language models form confidence about professional and organisational identities * Describe the role of structured data in helping AI systems interpret web content * Identify the key signals that influence AI recommendation probability * Distinguish between traditional search engine optimisation and AI visibility optimisation --- === 1. What Is an Entity in AI Search Systems === In the context of information retrieval and artificial intelligence, an '''entity''' is a distinctly identifiable real-world thing: a person, an organisation, a product, a location, or a concept. Entities are the building blocks through which AI systems organise knowledge about the world. Traditional search engines primarily rank web pages. AI language models primarily recognise and retrieve entities. This distinction has significant implications for how digital information is structured and how individuals and organisations make themselves discoverable. When a user asks an AI system a question such as "who is the most experienced architect in Nairobi" or "which law firm specialises in cross-border M&A in Europe," the system draws on its understanding of entities rather than simply matching keywords to documents. ==== 1.1 Entity Recognition ==== Entity recognition is the process by which an AI system identifies and classifies references to real-world things within text. Named entity recognition (NER) is a well-established natural language processing task that identifies proper nouns and classifies them into categories such as person, organisation, location, and product. At a higher level, modern large language models do not simply tag entities within text. They build statistical representations of entities during training, associating names with roles, geographies, competencies, and levels of authority based on patterns encountered across billions of documents. ==== 1.2 Entity Confidence ==== Entity confidence refers to the degree of certainty an AI system has when forming a representation of a specific entity. High entity confidence means the system can consistently and accurately describe who an entity is, what they do, and why they are authoritative in a specific domain. Low entity confidence produces vague, inconsistent, or absent representations. Entity confidence is shaped by: * The frequency with which an entity is mentioned across the training corpus * The consistency with which that entity is described across independent sources * The credibility and authority of the sources containing those mentions * The presence of structured, machine-readable data confirming the entity's identity --- === 2. How Large Language Models Build Knowledge About Professionals === Large language models learn about the world through exposure to text. During training, models process enormous datasets comprising web pages, academic papers, books, news articles, and other textual sources. Through this process, the model develops statistical associations between entities and the concepts, qualities, and contexts most frequently associated with them. ==== 2.1 Parametric Memory ==== Parametric memory refers to knowledge encoded directly into the parameters (weights) of a neural network during training. When a language model names a professional in response to a query, it is often drawing on parametric memory: associations formed during training between a name, a role, a location, and a perceived level of authority. For a professional to be represented in parametric memory, they need to have appeared repeatedly in the training data in credible, consistent contexts. A professional mentioned frequently in well-indexed, high-quality sources associated consistently with a specific area of expertise is more likely to be represented with high confidence in a model's parametric memory than one mentioned rarely or inconsistently. ==== 2.2 Retrieval Augmented Generation ==== Retrieval Augmented Generation (RAG) is a technique used by many AI systems to supplement parametric memory with real-time information retrieval. Rather than relying solely on knowledge encoded during training, RAG systems search the live web or a curated knowledge base to find relevant documents and use them to inform a response. For content to be retrieved effectively by RAG systems, it benefits from being: * Structured in short, self-contained answer blocks * Written to directly address specific questions * Rich in specific, verifiable details such as named entities, dates, and measurable outcomes * Hosted on pages that load quickly and are fully accessible to crawlers --- === 3. Structured Data and Its Role in AI Interpretation === Structured data refers to information formatted in a standardised, machine-readable way. In the context of web content, structured data most commonly refers to markup vocabularies such as Schema.org, which provide a common language for describing entities and their properties in a format that both search engines and AI crawlers can parse efficiently. ==== 3.1 Schema.org Vocabulary ==== Schema.org is a collaborative project by major search engines including Google, Microsoft, Yahoo, and Yandex that defines a shared vocabulary for structured data markup. Web publishers can embed Schema.org markup in their HTML to describe what their pages contain in a way that machines can interpret unambiguously. Common Schema.org types relevant to professional entity visibility include: * '''Person''' — for individual professionals * '''Organization''' — for businesses and institutions * '''LocalBusiness''' — for location-specific services * '''ProfessionalService''' — for service-based practitioners * '''Article''' — for published written content ==== 3.2 The sameAs Property ==== The '''sameAs''' property in Schema.org markup is used to assert that two web resources refer to the same real-world entity. A professional who includes a sameAs array in their website's structured data, linking to their LinkedIn profile, their professional directory listing, and their social media presence, is instructing AI crawlers to treat all of these pages as different representations of the same entity. This cross-referencing is significant because AI systems build entity confidence through corroboration across multiple independent sources. A professional who appears consistently under the same name and role across five independent, well-indexed platforms will have higher entity confidence in AI systems than one who appears on a single platform, even if that single platform is authoritative. Example of a minimal Person schema with sameAs markup: <syntaxhighlight lang="json"> { "@context": "https://schema.org", "@type": "Person", "name": "Learner Example", "jobTitle": "Education Researcher", "sameAs": [ "https://www.linkedin.com/in/learnerexample" ] } </syntaxhighlight> ==== 3.3 Entity Drift ==== Entity drift occurs when an individual's or organisation's identity information is inconsistent across different platforms. For example, if a professional's LinkedIn profile lists a different job title from their website, or if their name is spelled differently across directory listings, AI systems receive conflicting signals about who they are. Entity drift reduces AI entity confidence and can cause a professional to be underrepresented or misrepresented in AI-generated responses. Maintaining consistent identity information across all public platforms is a foundational practice in managing AI-readable professional presence. --- === 4. Key Signals Influencing AI Recommendation Probability === Research into AI citation patterns has identified several signals that correlate with the probability that an AI system will recommend or cite a specific entity in response to a relevant query. ==== 4.1 Cross-Platform Citation Density ==== Citation density refers to how frequently and across how many independent, credible sources an entity is referenced. Research published by SE Ranking (2025) found that entities with structured profiles on multiple trusted third-party platforms had significantly higher citation rates in ChatGPT responses than entities represented on a single platform. This finding aligns with established principles in information science around corroboration and source triangulation. AI systems, like human researchers, develop higher confidence in claims that are supported by multiple independent sources. Platforms designed specifically to structure professional entity data for AI crawler comprehension, such as Prezlo, have emerged to address this gap by generating machine-readable canonical identity objects and monitoring citation rates across AI systems. ==== 4.2 Content Structure and Information Density ==== Research by Growth Memo (2025) found that content structure is a significant predictor of AI retrieval probability, with traditional search metrics such as backlink count and domain authority showing limited correlation with citation rates in AI-generated responses. Specifically, pages using 120 to 180 words per section received significantly more citations in AI responses than pages with very short or very long sections. Question-based headings and embedded FAQ structures also correlated positively with retrieval frequency. ==== 4.3 Freshness and Recency ==== Many AI systems, particularly those using retrieval augmented generation, show a preference for recently updated content when constructing responses about fast-moving topics. Maintaining current, up-to-date information on indexed platforms is therefore relevant to AI retrieval probability for topics where recency matters. ==== 4.4 Specificity and Verifiability ==== AI systems appear to prefer content that is specific and verifiable over content that is vague or generic. Content containing named entities, specific dates, measurable outcomes, and attributed claims tends to score higher in information gain evaluations than content relying on general assertions. --- === 5. Distinguishing Traditional SEO from AI Visibility Optimisation === Traditional search engine optimisation (SEO) focuses on improving the ranking of web pages in search engine results pages (SERPs). It operates primarily through signals such as backlink authority, keyword relevance, on-page optimisation, and technical site health. AI visibility optimisation, sometimes referred to as Generative Engine Optimisation (GEO) or Answer Engine Optimisation (AEO), operates on a different layer. Rather than ranking pages, it focuses on building entity authority: the structured, consistent, corroborated presence that allows AI systems to form a confident and accurate representation of who an entity is and what they are authoritative for. The table below summarises key differences between the two approaches: {| class="wikitable" |- ! Dimension !! Traditional SEO !! AI Visibility Optimisation |- | Primary output || Page ranking in SERPs || Entity citation in AI-generated responses |- | Core metric || Keyword ranking position || AI recall rate and entity confidence |- | Key signals || Backlinks, keyword density, page authority || Entity consistency, structured data, citation distribution |- | Content format || Long-form keyword-optimised pages || Short, dense, question-answering content blocks |- | Timeframe || Weeks to months for ranking changes || Months for measurable recall rate improvement |- | Primary audience || Google's ranking algorithm || LLM parametric memory and RAG retrieval systems |} It is important to note that these two approaches are not mutually exclusive. Strong traditional SEO practices including technical site health, quality content, and authoritative backlinks provide a foundation that benefits AI visibility as well. The distinction is that AI visibility requires additional, specific infrastructure that traditional SEO tools and practices do not address. --- === 6. Practical Implications for Learners and Practitioners === For learners and practitioners seeking to apply these concepts, the following principles provide a practical framework: '''Principle 1: Establish entity clarity before building content.''' Ensure that your name, role, location, and area of expertise are described consistently across every platform where you have a public presence. Inconsistency in any of these dimensions creates entity confusion in AI systems. '''Principle 2: Implement structured data on your primary web presence.''' Use Schema.org markup to provide AI crawlers with machine-readable confirmation of your identity and expertise. At minimum, implement a Person or Organization schema with a populated sameAs array linking to your verified external profiles. '''Principle 3: Structure content for machine retrieval, not only human reading.''' Write key expertise pages using question-based headings and self-contained sections of 120 to 180 words. Include FAQ sections in long-form content. Ensure each section can stand alone as a complete answer to a specific question. '''Principle 4: Build distributed citations across credible platforms.''' Being mentioned consistently across multiple independent, well-indexed platforms strengthens AI entity confidence more effectively than a single authoritative presence. Academic profiles, professional directories, published articles, and verified listings all contribute to citation density. '''Principle 5: Monitor and maintain over time.''' AI entity confidence is not static. It changes as new content is indexed, as citation patterns shift, and as entity signals strengthen or weaken. Regular monitoring of how AI systems represent your entity, and correction of any drift or inconsistency that emerges, is a continuous practice rather than a one-time setup. --- === Summary === AI search visibility is determined by how clearly, consistently, and credibly an entity is represented across the machine-readable web. Key concepts include entity recognition, parametric memory, retrieval augmented generation, structured data, entity drift, and citation density. While these concepts intersect with traditional SEO, they represent a distinct layer of professional discoverability that requires specific structural approaches. As AI-powered discovery continues to grow as a primary channel through which people find professional services, products, and expertise, understanding and applying these principles becomes increasingly relevant for professionals, organisations, and digital practitioners across all fields. --- === Further Reading === * Schema.org — official structured data vocabulary documentation * Wikidata — Wikimedia Foundation structured entity database * Google Search Central — structured data guidelines for web publishers * W3C Web of Things — entity definition standards * arXiv — research literature on retrieval augmented generation and large language models * Prezlo — AI visibility infrastructure technical documentation --- === Discussion Questions === # Why might an AI system have different levels of confidence about different professionals in the same field? # What are the risks of entity drift for an organisation that operates across multiple countries? # How does the sameAs property in Schema.org support AI entity corroboration? # In what ways does RAG change the relationship between content quality and AI discoverability? # Compare the signals used by traditional search engines with those used by large language models. Where do they overlap and where do they diverge? --- ''This learning resource was created as an open educational contribution to the study of AI information systems and digital discoverability. It is intended for educational use and does not constitute commercial advice or product endorsement.'' [[Category:Information Science]][[Category:Artificial Intelligence]][[Category:Search Engine Technology]][[Category:Digital Literacy]] fzqsth9jpwl7qzm0sa4tnh4mc75wx2n 0 329884 2812105 2026-05-30T04:37:42Z DavidMCEddy 218607 create 2812105 wikitext text/x-wiki [[File:How would a nuclear war between Russia and the US affect you personally? - Future of Life Institute.webm|thumb|Simulation of a nuclear war between Russia and the US.<ref>Tegmark (2023).</ref>]] :''This book is a combination instruction manual on [[w:Media literacy|media literacy]] and an invitation to you to support collaborative / crowd-sourced research on how to improve the world's understanding of media literacy and how to accelerate its understanding and use globally for the betterment of humanity.'' Part I of this book on ''[[Media Literacy and You]]'' discusses "The media and political economy". Except in times of terror, massive lawlessness or war, most humans place a high priority on their financial situation, the primary focus of Part I. Part II on "The media and war" focuses on security concerns including this chapter on "Responding to a nuclear attack". What's the best response to a nuclear attack? That's a difficult question. The opposite is much easier: * '''''What's the ''worst'' response to a nuclear attack?''''' ::The evidence summarized in this article suggests that the ''worst'' response to a nuclear attack would be '''a nuclear response''', because it would increase the death toll from millions to billions, the vast majority of whom would be civilians, and many and likely most of those would be in countries not directly involved in the nuclear exchange. ::If you think otherwise, please revise this article accordingly, subject to the standard Wikimedia Foundation rules of writing from a neutral point of view citing credible sources. Or post your concerns to the "Discuss" page associated with this article. [[File:Percent of the world's population dead from a nuclear war.svg|thumb|Percent of the world's population dead within two year from a nuclear war on the vertical axis vs. total megatonage of nuclear weapons detonated ranging from 0.5 to 440 on the bottom axis and teragrams (millions of metric tons) of soot lofted to the stratosphere ranging from 5 to 150 on the top axis. These are from simulations by an international team of 10 experts in climate, food production, and economics<ref>Xia et al. (2022; see esp. their Table 1).</ref> with models fit thereto. The direct deaths range between 5 and 10 percent of the total; the rest of the deaths would be humans without food by the end of the second year after the nuclear war.<ref>Xia et al. (2022, Table 1) reported "Number of direct fatalities" and "Number of people without food at the end of year 2" out of a total population of 6.7 billion for their simulated year 2010. Xia et al. (2022, Fig. 1) show that the climate impact does not start recovering until year 5 after the nuclear war and has not yet fully recovered 9 years after the war. Thus, few people still alive without food at the end of year 2 will not likely live to year 9. Second, the percentages plotted here are the sums of those two numbers divided by 6.7 billion. The Wikipedia article on [[w:World population|World population]] said the world population in 2010 was 6,985,603,105 -- 7 billion (accessed 2023.08-12). The difference between 6.7 and 7 billion seems so slight that it can be safely ignored, especially given the uncertainty inherent in these simulations and the likelihood that the small populations excluded were probably not substantively different from those included.</ref> "IND-PAK" marks a range of hypothetical nuclear wars between [[w:India and weapons of mass destruction|India]] (IND) and [[w:Pakistan and weapons of mass destruction|Pakistan]] (PAK). "USA-RUS" marks a simulated nuclear war between [[w:Nuclear weapons of the United States|the US]] (USA) and [[w:Russia and weapons of mass destruction|Russia]] (RUS). "PRK" = a simulated nuclear war in which [[w:North Korea and weapons of mass destruction|North Korea]] (the People's Republic of Korea, PRK) used 30 nuclear weapons with an average yield of 17 kt for a total of 510 kt (0.51 megatons), the lower end of the bottom scale, with no nuclear retaliation by an adversary, as recommended in this article.<ref>Estimates of North Korea's nuclear very widely. The wikipedia article on "[[w:North Korea and weapons of mass destruction|North Korea and weapons of mass destruction]]" said they had 60 nuclear weapons when accessed 2026-05-20 but only half that when accessed 2023-08-07.</ref>]] This conclusion is supported by the accompanying plot summarizing climate simulations by an international team of 10 experts in climate, food production, and economics. Five of their scenarios describe hypothetical nuclear wars between India and Pakistan that loft between 5 and 47 Tg (teragrams = millions of metric tons) of smoke (soot) to the stratosphere, where it will linger for years covering the globe and reducing the amount of solar radiation reaching the earth. That in turn will substantially reduce the production of food for humans. The resulting impact on the global economy means that between 4 and 40 percent of humanity will likely starve to death if they did not die of something else sooner. A hypothetical nuclear war between the US and Russia could lead to the deaths of between 80 and 85 percent of humanity with death tolls of roughly 99 percent in the US, Europe, and Russia. In any of these scenarios, between 5 and 10 percent of the fatalities would result from bomb blasts. The remaining 90 to 95 percent would die of starvation if they did not die earlier from, e.g., radiation poisoning or increased risks of disease.<ref>Xia et al. (2022, esp. their Tables 1 and 2). Their Table 1 gives numbers of fatalities out of a total 2010 "population of the nations used in this study [of] 6,700,000,000." They give 2 simulations of a nuclear war between the US and Russia. Both would produce an estimated 360 million direct fatalities and loft 150 Tg (teragrams = million metric tonnes) to the stratosphere. At the end of the second year after such a war, between 5.08 and 5.34 billion people would be without food, totaling between 5.44 and 5.70 billion presumed dead. Those numbers are 81 and 85 percent of the 6.7 billion in the study and 78 and 81 percent of the 2010 [[w:World population|world population]] of 7 billion. We assume that the impact on the 300 million humans not in this study will not be substantively different from the 6.7 billion included and therefor use the 80-85 percent figures.</ref> This claim is clearer, more succinct, and stronger than the 2022-01-03 [[Wikisource:Joint Statement of the Leaders of the Five Nuclear-Weapon States on Preventing Nuclear War and Avoiding Arms Races|Joint Statement of the Leaders of the [first] Five Nuclear-Weapon States on Preventing Nuclear War and Avoiding Arms Races]], "that a nuclear war cannot be won and must never be fought".<ref>[[Wikisource:Joint Statement of the Leaders of the Five Nuclear-Weapon States on Preventing Nuclear War and Avoiding Arms Races]]. See also Borger (2022). Douthat (2022) discussed the [[w:2021-2022 Russo-Ukrainian crisis|current Ukraine crisis]] in [[w:The New York Times|''The New York Times'']]. He concluded that for us (presumably the US and perhaps its NATO allies) "To escalate now against a weaker adversary [Russia], one less likely to ultimately defeat us and more likely to engage in atomic recklessness if cornered, would be a grave and existential folly."</ref> This repeated a statement made 1987-12-11 by US President [[w:Ronald Reagan| Ronald Reagan]] and [[w:Soviet Union|Soviet]] head of state [[w:Mikhail Gorbachev|Mikhail Gorbachev]].<ref><!-- Joint statement by Reagan, Gorbachev -->{{cite Q|Q111845607}} Reagan made that same statement 1984-01-25 in his [[Wikisource:Ronald Reagan's Fourth State of the Union Address|fourth State of the Union Address]].</ref> In the following we review the evidence for and against this claim and then comment on the credibility of the logic that led to the creation of the world's current nuclear arsenals and seems to be driving the current nuclear "modernization" programs in the US, [[w:Russia|Russia]], [[w:China|China]] and elsewhere. == Summary of research on the consequences of a nuclear war == It is theoretically possible that a nuclear exchange would end like [[w:World War II|World War II]] with no more than [[w:Atomic bombings of Hiroshima and Nagasaki|two nuclear weapons being used]]. It is also theoretically possible that nuclear weapons in a new war would only target deserted areas like [[w:List of nuclear weapons tests|the locations where more than 2,000 tests of nuclear weapons]] have been conducted so far.<ref>For a "[[w:List of nuclear weapons tests|List of nuclear weapons tests]]", see the Wikipedia article by that title (accessed 2023-07-06).</ref> Either of those scenarios would increase the level of harmful background radiation worldwide leading to increases in the rates of cancer, birth defects and genetic mutations but would otherwise not likely have a major impact on a large portion of humanity.<ref>Johnston (2001) reported that only 521 of the more than 2,000 nuclear weapons tests were above ground. If 521 explosions of nuclear weapons in deserted places have not generated a substantive impact on human health, it seems unlikely that a nuclear war involving a few thousand explosions of nuclear weapons in deserted areas would be dramatically worse.</ref> However, a nuclear war with such negligible results is highly unlikely. More likely is the deaths in a few hours or days of tens or hundreds of millions of humans.<ref>The "Number of direct fatalities" in a nuclear war lasting a week ranged from 27 to 360 million in simulations summarized in Xia et al. (2022, Table 1).</ref> More would die of radiation poisoning over the next few months and years.<ref>Ellsberg (2017, pp. 2-3) includes a graph that the Joint Chiefs Joint Chiefs of Staff produced in the Spring of 1961 to answer President Kennedy's question, "If your plans for a general [nuclear] war are carried out as planned, how many people will be killed in the [[w:Soviet Union|Soviet Union]] and China?" This graph was a straight line beginning at 275 million who would die during the initial nuclear exchange with another 8.25 million dying each month for the next six months, totaling 325 million deaths.</ref> If more than a few dozen nuclear weapons are used, then "nuclear war would also produce nearly instantaneous climate change that among other effects, would threaten the global food supply. Even a regional nuclear war ..., such as between India and Pakistan,<ref>Robock et al. (2007); Toon et al. (2019). Of course, a nuclear war could be started accidentally by any nuclear-weapons state, as suggested in the report of an Indian cruise missile that landed 2022-03-10 in Pakistan (Mashal and Masood 2022). See also Xia et al. (2022).</ref> in which less than 3% of the world’s nuclear weapons stockpiles were detonated in urban areas, would suddenly decrease the average global temperature by 1°C–7°C [2°–13°F], precipitation by up to 40%, and sunlight by up to 30%. ... Such a conflict would decrease crop production to an extent that it could seriously threaten world food security and even trigger global famine",<ref>Jägermeyr et al. (2020).</ref> according to Robock and Prager (2021). In theory, crop losses of between 10 and 25 percent for 5-10 years<ref>as predicted by Jägermeyr et al. (2020) and others.</ref> might not threaten a global famine or even an increase in malnutrition if people ate more plant-based foods and less meat. In practice, famines never work that way: There is hoarding, and many who do not die of starvation succumb to diseases or secondary wars driven by the food insecurity, according to Helfand (2013). [[w:Amartya Sen|Nobel Prize Economist Sen]] observed that, "no famine has ever taken place ... in a functioning democracy".<ref>Sen (1999, p. 32). Later on p. 178, he stated similarly, "there has never been a famine in a functioning multiparty democracy."</ref> This generalizes the observation that Ireland was a ''net food exporter'' during its infamous potato famines of the nineteenth century.<ref>e.g., Woodham-Smith (1962).</ref> Xia et al. (2022, Table 1) estimated that between 4 and 85 percent of humanity would starve to death if they did not die of something else sooner in the nuclear wars they simulated. In the spring of 1961, "The total death toll as calculated by the Joint Chiefs of Staff [top US military leaders], from a U.S. first strike aimed at the Soviet Union, its Warsaw Pact satellites, and China, would be roughly six hundred million dead. A hundred Holocausts", according to [[w:Daniel Ellsberg|Daniel Ellsberg]], who served as a nuclear war planner for presidents Eisenhower, Kennedy, Johnson and Nixon<ref>Ellsberg (2017, esp. pp. 2-3) noted that 325 million would die in the Soviet Union and China and another couple hundred million in neighboring countries, totalling six hundred million.</ref> before releasing [[w:The Pentagon Papers|"The Pentagon Papers"]] in 1971. Six hundred million was roughly 20 percent of the total human population on earth in 1961, and that didn't count any in the US who might be killed in retaliation. In 1957, roughly 4 years earlier, [[w:Mao Zedong|Mao Zedong]], then the Chairman of the [[w:China|People's Republic of China]], had reportedly said that a nuclear war could kill a third of humanity, perhaps half, "but imperialism would be razed to the ground, and the whole world would become socialist."<ref>Dikötter (2010). See also Halimi (2018), which gives the date as 1957. There is some controversy about this quote; see the Wikipedia article on [[w:Mao Zedong|"Mao Zedong"]], accessed 2022-03-02.</ref> Turco et al. (1983) published the first predictions of a ''[[w:nuclear winter|nuclear winter]]'' based on climate modeling that considered smoke anticipated from fires started by a massive nuclear weapons exchange between the US and the Soviet Union. They found that "average light levels can be reduced to a few percent of ambient and land temperatures can reach -15° to -25°C [5° to -13°F]" with smoke transported from the Northern to the Southern Hemisphere, all of which "could pose a serious threat to human survivors and to other species." Various teams have published comparable analyses since then with different and increasingly sophisticated models, beginning with Aleksandrov and Stenchikov (1983), with similar conclusions.<ref>Coup et al. (2019, p. 8522).</ref> Coup et al. (2019) predicted hard freezes ''in the summer'' in most of the Northern Hemisphere including the US, Russia, and most of Europe during the first three years following such a war, where temperatures drop below −4°C [25°F], making it impossible to grow crops in those regions. China would suffer a similar fate, with only its southeast portion remaining above freezing in the summer. Much of Southern Mexico, Central and South America, and the Southern Hemisphere would also be negatively impacted, but not to the same extent. These climate modeling results make Mao's predictions from 1957 seem wildly optimistic: Any humans in the US, Canada, or most of Eurasia who survived the nuclear exchange would have extreme difficulties finding enough to eat -- "imperialism razed to the ground", according to Mao. However, crop yields in most of the rest of the world would also be extremely depressed, which Mao had not considered. The results would threaten famine vastly worse than what has been predicted following a nuclear war between India and Pakistan.<ref>Ellsberg said that 98 or 99 percent of humanity would starve to death if they did not die of something else sooner (Ellsberg et al. 2017). Coup et al. (2019) and Xia et al. (2022) conclude that it won't be quite that bad but will still pretty grim.</ref> Of course, no one knows for sure how many people would die directly and indirectly from a nuclear war. However, it should be obvious to at least some if not most people<ref>People who believe they benefit from the current new nuclear arms race may never agree. Upton Sinclair said, "It is difficult to get a man to understand something, when his salary depends upon his not understanding it!" Sinclair (1935, p. 109).</ref> that the ''worst'' response to a nuclear attack would be a nuclear response: * A nuclear response to a nuclear "warning shot" with minimal destruction could too easily escalate until the nuclear arsenals of all parties were expended and the life expectancy of all survivors worldwide was dramatically reduced. * Alternatively, a nuclear response to a massive first strike against a thousand cities would most likely ''increase'' the death toll worldwide, including in the country retaliating with nuclear weapons''. * It is possible that a nuclear response could deter further uses of nuclear weapons and reduce the length and severity of the war and its global impact. However, this outcome seems unlikely given the record of history. * It is also possible that a nuclear arsenal could deter a nuclear attack, though it could also ''increase'' the risk of such an attack, e.g., due to a malfunction of the nuclear command, control and communications system of a potential adversary<ref>as reportedly almost happened from the [[w:1983 Soviet nuclear false alarm incident|1983 Soviet nuclear false alarm incident]].</ref> -- or due to an individual or team who believes that God wants them to initiate [[w:Armageddon|Armageddon]], as discussed in the section on [[Media Literacy and You/Deterrence without threat#Deterrence theory and nuclear Armageddon|Deterrence theory and nuclear Armageddon]] in the chapter of this book on [[Media Literacy and You/Deterrence without threat|Deterrence without threat]] Turcotte (2022) concluded that if the 2022 Ukraine 'conflict ends without the annihilation of our species, it should nonetheless be regarded as a planet-wide near-death experience, and the “Peoples of the United Nations” should demand the total elimination of nuclear weapons as quickly as humanly possible, as well as the establishment of new common security measures that will move us much closer to sustainable peace throughout the world.' In spite of this concern, Turcotte recommended military action to support Ukraine but short of declaring war on Russia. Leading experts have made alarming comments about the likelihood of a nuclear attack, possibly by a terrorist organization. In 2004 Bruce Blair, president of the [[w:Center for Defense Information|Center for Defense Information]] wrote: "I wouldn't be at all surprised if nuclear weapons are used over the next 15 or 20 years, first and foremost by a terrorist group that gets its hands on a [[w:Russia and weapons of mass destruction|Russian]]" or [[w:Pakistan and weapons of mass destruction|Pakistani nuclear weapon]].<ref><!--Nicholas D. Kristof (2004) A Nuclear 9/11, NYT-->{{cite Q|Q111906710}}</ref> Other experts seemed even more concerned: A nuclear terrorist attack in the US was considered "more likely than not" within the next five to ten years, according to Professor [[w:Robert Gallucci|Robert Gallucci]] of the [[w:Georgetown University School of Foreign Service|Georgetown University School of Foreign Service]] in 2006 or in the next decade per former U.S. Assistant Secretary of Defense [[w:Graham Allison|Graham Allison]] in 2004.<ref><!-- Ordre Kittrie (2007) Averting Catastrophe: Why the Nuclear Non-proliferation Treaty is Losing its Deterrence Capacity and How to Restore It -->{{cite Q|Q111906652}}</ref> For more, see the section on "[[w:Mutually assured destruction#Criticism|Criticism]]" in the Wikipedia article on "[[w:Mutually assured destruction|Mutually assured destruction]]". The Wikipedia article on "[[w:National Response Scenario Number One|National Response Scenario Number One]]" describes "the United States federal government's planned response to a nuclear attack." It focuses primarily on "the possible detonation of a small, crude nuclear weapon by a terrorist group in a major city, with significant loss of life and property."<ref>Accessed 2022-05-08, when it cited <!-- Jay Davis (2008) After A Nuclear 9/11 -->{{cite Q|Q111905675}}, <!-- Brian Michael Jenkins (2008) A Nuclear 9/11? -->{{cite Q|Q111906145}}</ref> That article discusses preparing for a nuclear attack but not how to respond. Nevertheless, if the ''worst'' response to a nuclear attack is a nuclear response, that has other policy implications for nuclear ''and non-nuclear'' countries world wide discussed in the previous chapter on [[Media Literacy and You/Deterrence without threat|Deterrence without threat]]. == Credibility of military leaders and national security experts == {{main|Expertise of military leaders and national security experts}} * ''Never attribute to malice that which is adequately explained by stupidity.'' ([[w:Hanlon's razor|Hanlon's razor]]) * ''Never attribute to malice or stupidity that which can be explained by moderately rational individuals following incentives in a complex system.'' (Hubbard's clumsier correlary.<ref>Hubbard (2020, pp. 81-82).</ref>) The history of armed conflict should raise questions about the credibility of those advocating use of military force: In all major armed conflicts in history, at least one side has lost. Often the official winners lost substantially more than they gained. === Research on expertise === The history of armed conflict is consistent with the research by Kahneman and Klein (2009) in their conclusion that :''expert intuition is learned from frequent, rapid, high-quality feedback.'' In particular, military leaders in combat can get frequent, rapid high-quality feedback on their ability to deliver death and destruction to designated targets. However, no one can get such feedback about how to win wars or how to ''promote broadly shared peace and prosperity for the long term.'' This is discussed in more detail in the Wikiversity article on "[[Expertise of military leaders and national security experts]]". That article documents how experts without such feedback can be beaten by simple rules of thumb developed by intelligent lay people.<ref>Kahneman et al. (2021) report that with some data, a statistical model fit often does better. With lots of data, artificial intelligence systems can do even better. This extends the work of [[w:Paul E. Meehl#Clinical versus statistical prediction|Meehl (1954)]]. Hubbard (2020) and [[w:Superforecasting: The Art and Science of Prediction|Tetlock and Gardner (2015)]] describe things one might do to improve their intuition.</ref> As the time since the [[w:Atomic bombings of Hiroshima and Nagasaki|atomic bombings if Hiroshima and Nagasaki]] increases, the ''intuition'' that political and military leaders have about nuclear weapons gets worse, because that history tells them that they can use more military force, even threatening to use nuclear weapons, without seriously risking a nuclear war. That intuition increasingly threatens the entirety of humanity. === Increasing risks with nuclear proliferation === Narang and Sagan, eds. (2022) ''The Fragile Balance of Terror: Deterrence in the New Nuclear Age'' includes 8 chapters by 12 authors reviewing the literature on different aspects of nuclear deterrence today. They raised many questions about the applicability of [[w:Cold War|Cold War]] analyses of deterence in an age with [[Forecasting nuclear proliferation|an increasing number of nuclear weapon states]]. They mentioned numerous concerns including the following: * [[w:2008 Mumbai attacks|During terrorist attacks in Mumbai in 2008]], someone called called Pakistani president Zardari claiming to be Indian foreign minister Mukherjee threatening to attack Pakistan. That crises was diffused without escalation after US secretary of state Condoleezza Rice called Mukherjee, who assured her that he had not placed such a call, and India was ''not'' planning to attack Pakistan. If someone claiming to be a US official had placed a similar call to Kim Jong Un while Donald Trump was President of the US, the result may not have been as benign.<ref>Narang and Sagan (2022, p. 241).</ref> * [[w:2018 Hawaii false missile alert|"In January 2018, the Hawaii emergency management system issued an incoming missile warning alert]] adding, 'this is not a drill.'" The US did not respond, because (a) they had redundant early warning systems that did not indicate an incoming missile, (b) professional operators in Hawaii promptly acknowledged the mistake, and (c) no one in the US seriously expected such an attack. If this had happened in North Korea, none of these three restraining conditions would likely not have been present: (a) They did not have redundant warning systems. (b) Operators are killed, not just fired in North Korea for making a mistake like that. (c) US "President Trump was threatening 'fire and fury' if North Korean nuclear and missile tests continued."<ref>Narang and Sagan (2022, p. 232).</ref> * [[w:2019 Balakot airstrike|In 2019 India bombed an alleged terrorist training camp in Balakot]], Pakistan. This was "the first time a nuclear weapons state has bombed the undisputed territory of another nuclear weapons state."<ref>Narang and Sagan (2022, pp. 231-232).</ref> * [[w:2020–2021 China–India skirmishes|In 2020, Chinese and Indian troops engaged in hostilities along their disputed border]] with fatalities on both sides, "for the first time in almost half a century. Intense conflict between three nuclear powers simultaneously is no longer a remote possibility.<ref>Narang and Sagan (2022, p. 232).</ref> Beyond this, [[w:Richard Ned Lebow|Richard Ned Lebow]] said, "There’s all kinds of empirical evidence that a deterrence strategy is as likely to provoke the behavior it seeks to prevent as not."<ref>Lebow et al. (2023). See also Lebow (2020, ch. 4).</ref> == Probability of a nuclear war == The section on [[Time to nuclear Armageddon#Relevant literature|Relevant literature]] of the Wikiversity article on [[Time to nuclear Armageddon]] includes a table summarizing previous estimates of the probability of a nuclear war. Karger et al. (2023) provides a more extensive study of the probability of a nuclear war and other existential risks. === System accidents === The concept of "normal accidents" or "[[w:system accident|system accidents]]" seems important here. Research in that area has established that ''it is impossible to design and manage complex systems to ultra-high levels of reliability''. Maintenance on redundant systems is often deferred, because responsible managers are often reluctant to spend money fixing something that works.<ref>e.g., Sagan (1993).</ref> And procedures are sometimes secretly modified by people with different priorities from their management. For example, at least between 1970 and 1974 the codes in US Air Force launch control centers for [[w:Intercontinental ballistic missile|Intercontinental ballistic missiles]] were all set continuously to 00000000.<ref>Ellsberg (2017, p. 61).</ref> This clearly negated the claim that only the President of the US could order the use of US nuclear weapons, secured by secret codes carried in a briefcase (called the [[w:nuclear football|"nuclear football"]]) near the President at all times. Similarly, former US Secretary of Defense William J. Perry has said an actual nuclear attack on the US is far less likely than a report of one generated by a malfunction in the US nuclear command, control, and communications systems.<ref>Perry and Collina (2020). Of course, a nuclear war could be started accidentally by any nuclear-weapons state, as suggested in the report of an Indian cruise missile that landed 2022-03-10 in Pakistan (Mashal and Masood 2022).</ref> A tragic example of a system accident is the [[w:Sinking of MV Sewol|Sinking of MV ''Sewol'']], 2014-04-16. It sank with over twice its rated load under the command of a substitute captain. The regular captain had complained of deferred maintenance threatening the stability of the vessel; he said the company had threatened to fire him if he continued to complain. As of this writing, it has been almost 79 years since nuclear weapons were detonated in hostilities. As noted above, that history feeds human intuition that we can "safely" be more aggressive in developing, deploying and threatening the use of nuclear weapons without seriously risking [[Time to nuclear Armageddon|nuclear Armageddon]]. People who disagree like the [[w:Union of Concerned Scientists|Union of Concerned Scientists]] with their [[w:Doomsday Clock|Doomsday Clock]] are dismissed as unrealistic, like [[w:Chicken Little|Chicken Little]]. == Human psychology and the role of the media == When people are attacked, it can sometimes be difficult to control their responses, which are driven by instinctive reactions often characterized as irrational. Johnson (2004) documented how these instinctive reactions exist, because they provided survival benefits to our ancestors over hundreds of thousands and millions of years of evolutionary history. These instincts may, however, push us into the ''worst'' possible response to a nuclear attack. Worse, major media everywhere have conflicts of interest in honestly reporting on anything (like these research results) that might threaten those who control much of the money for the media.<ref name='McC+Cagé+Rolnik">McChesney (2004). Cagé (2016). Rolnik et al. (2019). See also "[[Confirmation bias and conflict]]".</ref> Everyone thinks they know more than they do,<ref name=Kahneman>Kahneman (2011).</ref> which makes them easily misled by the media they find credible.<ref>[[Confirmation bias and conflict]]. See also McChesney (2004), Cagé (2016), and Rolnik et al. (2019).</ref> == Recapitulation == In sum, the worst possible response to a nuclear attack would seem to be a nuclear response. Existing nuclear weapons policies appear to be supported by propaganda. That propaganda is effective, because it supports the preferences of those who control much of the money for the media,<ref name='McC+Cagé+Rolnik"/> and because too few humans pursue alternative sources of information when they should.<ref name=Kahneman/> === Call for help === Do you, dear reader, know other serious research not cited herein that might improve this analysis? If yes, you can help improve this discussion by adding comments with citations -- or by adding such citation(s) to the "Discuss" page associated with this chapter, suggesting someone else revise the chapter appropriately. There are plenty of contrary claims in the major media, but the lead author of this chapter is not aware of any that contradict the research summarized above. In the absence of such research, the current author finds it difficult to imagine any national defense policies that carry a greater risk of nuclear Armageddon than our current policies. The risks of being stampeding into a war on fraudulent grounds could be reduced by making it harder for administration officials to lie to Congress and the public, as recommended in the next chapter of this book on "[[w:Media Literacy and You#Threats from excessive government secrecy|Threats from excessive government secrecy]]". That chapter, in sum, documents how the major media have pushed political leaders to pursue counterproductive actions and punish whistleblowers, who release official government secrets classified to keep them from the US public. This chapter is being written in the hopes of inspiring action to improve the prospects for broadly shared peace and prosperity for the long term. == Exercises == 1. If you question the above, post your concerns to the associated "Discuss" page. If you have references that seem to contradict any of the above, either cite the references with your concerns on the "Discuss" page or modify the text appropriately, writing from a neutral point of view, citing credible sources. 2. Ask others their thoughts about nuclear weapons. If you feel inclined to differ, first note that, :''Primary drivers of every major conflict include differences between the media that the different parties find credible.'' Then share your concerns in a friendly supportive manner. The goal is not to convince anyone that they are wrong and you are right but rather to share an alternative perspective, working agree to disagree agreeably while also seeking areas of common concern. == Acknowledgements == Thanks to Owen B. Toon, Alan Robock, and presenters at their irregular webinar series on impact on climate of a nuclear war. Of course, any errors and other deficiencies here are solely the responsibility of the author. == See also == * [[Expertise of military leaders and national security experts]] * [[Time to nuclear Armageddon]] * [[Forecasting nuclear proliferation]] * [[Time to extinction of civilization]] == Notes == {{reflist}} == Bibliography == * <!-- Guardian (2001-10-14) Bush rejects Taliban offer to hand Bin Laden over -->{{cite Q|Q111228506}} * <!-- Aleksandrov and Stenchikov (1983) "On the modeling of the climatic consequences of the nuclear war" -->{{cite Q|Q63229964}} * <!-- Borger (2022) Five of world’s most powerful nations pledge to avoid nuclear war, Guardian -->{{cite Q|Q111011203}} * <!-- Cagé (2016) Saving the media: Capitalism, crowdfunding and democracy (Harvard U. Pr.)-->{{cite Q|Q54640583}} * <!-- Chenoweth and Stephan (2011) Why Civil Resistance Works: The Strategic Logic of Nonviolent Conflict (Columbia U. Pr.) -->{{cite Q|Q88725216}} For their data see, <!-- Chenoweth, NAVCO data project, Harvard -->{{cite Q|Q55842589}} * <!-- Coup et al. (2019) Nuclear Winter Responses to Nuclear War Between the United States and Russia in the Whole Atmosphere Community Climate Model Version 4 and the Goddard Institute for Space Studies ModelE -->{{cite Q|Q111222900}} * <!-- Dikötter (2010) Mao's Great Famine (Bloomsbury) -->{{cite Q|Q3209496}} * <!-- Douthat (2022) "How to Stop a Nuclear War", NYT -->{{cite Q|Q111145224}} * <!-- Ellsberg (2017) The Doomsday Machine: Confessions of a Nuclear War Planner (Bloomsbury) -->{{cite Q|Q63862699}} * <!--Ellsberg, Goodman and González (2017) "Daniel Ellsberg Reveals He was a Nuclear War Planner, Warns of Nuclear Winter & Global Starvation", Democracy Now!-->{{cite Q|Q64226035}} * <!-- Halimi, Serge (2018-08) "The forgotten communist quarrel", Le Monde Diplomatique -->{{cite Q|Q97657492}}. * <!-- Helfand, Ira I2013) "Nuclear famine: two billion people at risk?", International Physicians for the Prevention of Nuclear War -->{{cite Q|Q63256454}} * <!-- Doug Hubbard (2020) The Failure of Risk Management: Why it's broken and how to fix it Second edition (Wiley)-->{{cite Q|Q123514276}} * <!-- Jägermeyr, J., et al. (2020-03-16) "A regional nuclear conflict would compromise global food security", Proceedings of the National Academy of Science of the United States of America -->{{cite Q|Q90371058}} * <!-- Dominic D. P. Johnson (2004). Overconfidence and War: The Havoc and Glory of Positive Illusions (Harvard U. Pr.) -->{{cite Q|Q118106389}} * <!-- Johnston (2001) Chronological Listing of Above Ground Nuclear Detonations -->{{cite Q|Q111222177}} * <!-- Jones, Seth, and Martin C. Libicki (2008) "How Terrorist Groups End: Lessons for Countering al Qa'ida", RAND Corporation-->{{cite Q|Q57515305}} * <!-- Kahneman, Daniel (2011) Thinking, Fast and Slow (FSG)-->{{cite Q|Q983718}} * <!-- Kahneman and Klein (2009) Conditions for intuitive expertise: a failure to disagree-->{{cite Q|Q35001791}} * <!-- Kahneman, Sibony, and Sunstein (2021) Noise: A flaw in human judgment -->{{cite Q|Q107108766}} * <!--Ezra Karger, Josh Rosenberg, Zachary G Jacobs, Molly Hickman, Rose Hadshar, Kayla Gamin, Taylor Smith, Bridget Williams, Tegan McCaslin, Stephen Thomas, and Philip Tetlock (2023) "Forecasting Existential Risks: Evidence from a Long-Run Forecasting Tournament"-->{{cite Q|Q122208144}} * <!-- Richard Ned Lebow (2020) A Democratic foreign policy: Regaining American influence abroad (Palgrave Macmillan)-->{{cite Q|Q124351867}} * <!-- Lebow, Samuelson, Graves (2023) "Richard Ned Lebow on national defense including deterrence"-->{{cite Q|Q124351846}} * <!-- Mujib Mashal and Salman Masood (2022-03-12) "India Accidentally Fires a Missile at Pakistan. Calm Ensues.", NYT -->{{cite Q|Q111223210}} * <!-- McChesney, Robert (2004) The Problem of the Media: U.S. Communication Politics in the 21st Century (Monthly Review Press) -->{{cite Q|Q7758439}} * <!-- Paul E. Meehl (1954) Clinical vs. statistical prediction-->{{cite Q|Q115455297}} * <!-- Narang, Vipin; Sagan, Scott D. (2022) The Fragile Balance of Terror: Deterrence in the New Nuclear Age (Cornell University Press)-->{{cite Q|Q124351052|authors=Vipin Narang and Scott D. Sagan, eds.}} * <!-- Pape, Robert, and James K. Feldman (2010) Cutting the fuse : the explosion of global suicide terrorism and how to stop it (U. of Chicago Pr.)-->{{cite Q|Q109249408}} * <!-- Perry, William J., and Tom Z. Collina (2020) The Button: The new nuclear arms race and presidential power from Truman to Trump (BenBella)->>{{cite Q|Q102046116}} * <!-- Robock, Alan, Luke Oman, Georgiy L. Stenchikov, Owen B. Toon, C. Bardeen, and R. P. Turco (2007) "Climatic consequences of regional nuclear conflicts", Atmospheric Chemistry and Physics -->{{cite Q|Q21129034}} * <!-- Robock, Alan, and Stewart C. Prager (2021-12-02) "Geoscientists Can Help Reduce the Threat of Nuclear Weapons", Eos-->{{cite Q|Q111146317}} * <!-- Guy Rolnik; Julia Cagé; Joshua Gans; Ellen P. Goodman; Brian G. Knight; Andrea Prat; Anya Schiffrin (1 July 2019), Protecting Journalism in the Age of Digital Platforms (PDF), Booth School of Business-->{{cite Q|Q106465358}} * <!-- Sagan, Scott (1993) The Limits of Safety: Organizations, Accidents, and Nuclear Weapons (Princeton University Press)-->{{cite Q|Q111146417}} * <!-- Sen, Amartya (1999) Development as Freedom (Knopf)-->{{cite Q|Q5266729}} * <1--Upton Sinclair (1935, reprint 1994) I, candidate for governor : and how I got licked-->{{cite Q|Q122190924|date=1935, reprint 1994}} * <!--Philip E. Tetlock and Dan Gardner (2015) Superforecasting: The Art and Science of Prediction (Crown)-->{{cite Q|Q21203378}} * <!-- Tegmark (2023) How would a nuclear war between Russia and the US affect you personally?-->{{cite Q|Q124432900}} * <!-- Toon, Owen B., Charles G. Bardeen, Alan Robock, Hans Kristensen, Matthew McKinzie, R. J. Peterson, Cheryl S. Harrison, Nicole S. Lovenduski, and Richard P. Turco (2019) "Rapidly expanding nuclear arsenals in Pakistan and India portend regional and global catastrophe", Sciences Advances-->{{cite Q|Q90735736}} * <!-- Turco, R. P., Owen B. Toon, T. P. Ackerman, J. B. Pollack, and Carl Sagan (1983) "Nuclear winter: Global consequences of multiple nuclear explosions", Science, 222(4630), 1283–1292, https://doi.org/10.1126/science.222.4630.1283 -->{{cite Q|Q111146500}} * <!-- Turcotte (2022-03-09) Global community must step up pressure on Putin -->{{cite Q|Q111235117}} * <!-- Tyler, Tom R. (2006) Why people obey the law, revised ed. (Princeton U. Pr.)-->{{cite Q|Q111097755}} * <!-- Tyler, Tom R., and Yuen J. Huo (2002) Trust in the Law: Encouraging Public Cooperation with the Police and Courts (Russell Sage Foundation)-->{{cite Q|Q106943244}} * <!-- Woodham-Smith, Cecil (1962) The Great Hunger: Ireland 1845-1849 (Harper)-->{{cite Q|Q7737800}} * <!-- Xia et al. (2022) Global food insecurity and famine ... from a nuclear war ...-->{{cite Q| Q113732668}} [[Category:Media literacy]] [[Category:Communication]] [[Category:Political science]] [[Category:Law]] [[Category:Psychology]] [[Category:Sociology]] [[Category:War History]] [[Category:Media Literacy and You]] <!-- https://en.wikiversity.org/wiki/Wikiversity:Category_Review --> 00uil64i03044kyd1hfo1j7h9b0ut94 2812107 2812105 2026-05-30T04:42:43Z DavidMCEddy 218607 /* Bibliography */ delete initial Guardian (2001) ref; not used 2812107 wikitext text/x-wiki [[File:How would a nuclear war between Russia and the US affect you personally? - Future of Life Institute.webm|thumb|Simulation of a nuclear war between Russia and the US.<ref>Tegmark (2023).</ref>]] :''This book is a combination instruction manual on [[w:Media literacy|media literacy]] and an invitation to you to support collaborative / crowd-sourced research on how to improve the world's understanding of media literacy and how to accelerate its understanding and use globally for the betterment of humanity.'' Part I of this book on ''[[Media Literacy and You]]'' discusses "The media and political economy". Except in times of terror, massive lawlessness or war, most humans place a high priority on their financial situation, the primary focus of Part I. Part II on "The media and war" focuses on security concerns including this chapter on "Responding to a nuclear attack". What's the best response to a nuclear attack? That's a difficult question. The opposite is much easier: * '''''What's the ''worst'' response to a nuclear attack?''''' ::The evidence summarized in this article suggests that the ''worst'' response to a nuclear attack would be '''a nuclear response''', because it would increase the death toll from millions to billions, the vast majority of whom would be civilians, and many and likely most of those would be in countries not directly involved in the nuclear exchange. ::If you think otherwise, please revise this article accordingly, subject to the standard Wikimedia Foundation rules of writing from a neutral point of view citing credible sources. Or post your concerns to the "Discuss" page associated with this article. [[File:Percent of the world's population dead from a nuclear war.svg|thumb|Percent of the world's population dead within two year from a nuclear war on the vertical axis vs. total megatonage of nuclear weapons detonated ranging from 0.5 to 440 on the bottom axis and teragrams (millions of metric tons) of soot lofted to the stratosphere ranging from 5 to 150 on the top axis. These are from simulations by an international team of 10 experts in climate, food production, and economics<ref>Xia et al. (2022; see esp. their Table 1).</ref> with models fit thereto. The direct deaths range between 5 and 10 percent of the total; the rest of the deaths would be humans without food by the end of the second year after the nuclear war.<ref>Xia et al. (2022, Table 1) reported "Number of direct fatalities" and "Number of people without food at the end of year 2" out of a total population of 6.7 billion for their simulated year 2010. Xia et al. (2022, Fig. 1) show that the climate impact does not start recovering until year 5 after the nuclear war and has not yet fully recovered 9 years after the war. Thus, few people still alive without food at the end of year 2 will not likely live to year 9. Second, the percentages plotted here are the sums of those two numbers divided by 6.7 billion. The Wikipedia article on [[w:World population|World population]] said the world population in 2010 was 6,985,603,105 -- 7 billion (accessed 2023.08-12). The difference between 6.7 and 7 billion seems so slight that it can be safely ignored, especially given the uncertainty inherent in these simulations and the likelihood that the small populations excluded were probably not substantively different from those included.</ref> "IND-PAK" marks a range of hypothetical nuclear wars between [[w:India and weapons of mass destruction|India]] (IND) and [[w:Pakistan and weapons of mass destruction|Pakistan]] (PAK). "USA-RUS" marks a simulated nuclear war between [[w:Nuclear weapons of the United States|the US]] (USA) and [[w:Russia and weapons of mass destruction|Russia]] (RUS). "PRK" = a simulated nuclear war in which [[w:North Korea and weapons of mass destruction|North Korea]] (the People's Republic of Korea, PRK) used 30 nuclear weapons with an average yield of 17 kt for a total of 510 kt (0.51 megatons), the lower end of the bottom scale, with no nuclear retaliation by an adversary, as recommended in this article.<ref>Estimates of North Korea's nuclear very widely. The wikipedia article on "[[w:North Korea and weapons of mass destruction|North Korea and weapons of mass destruction]]" said they had 60 nuclear weapons when accessed 2026-05-20 but only half that when accessed 2023-08-07.</ref>]] This conclusion is supported by the accompanying plot summarizing climate simulations by an international team of 10 experts in climate, food production, and economics. Five of their scenarios describe hypothetical nuclear wars between India and Pakistan that loft between 5 and 47 Tg (teragrams = millions of metric tons) of smoke (soot) to the stratosphere, where it will linger for years covering the globe and reducing the amount of solar radiation reaching the earth. That in turn will substantially reduce the production of food for humans. The resulting impact on the global economy means that between 4 and 40 percent of humanity will likely starve to death if they did not die of something else sooner. A hypothetical nuclear war between the US and Russia could lead to the deaths of between 80 and 85 percent of humanity with death tolls of roughly 99 percent in the US, Europe, and Russia. In any of these scenarios, between 5 and 10 percent of the fatalities would result from bomb blasts. The remaining 90 to 95 percent would die of starvation if they did not die earlier from, e.g., radiation poisoning or increased risks of disease.<ref>Xia et al. (2022, esp. their Tables 1 and 2). Their Table 1 gives numbers of fatalities out of a total 2010 "population of the nations used in this study [of] 6,700,000,000." They give 2 simulations of a nuclear war between the US and Russia. Both would produce an estimated 360 million direct fatalities and loft 150 Tg (teragrams = million metric tonnes) to the stratosphere. At the end of the second year after such a war, between 5.08 and 5.34 billion people would be without food, totaling between 5.44 and 5.70 billion presumed dead. Those numbers are 81 and 85 percent of the 6.7 billion in the study and 78 and 81 percent of the 2010 [[w:World population|world population]] of 7 billion. We assume that the impact on the 300 million humans not in this study will not be substantively different from the 6.7 billion included and therefor use the 80-85 percent figures.</ref> This claim is clearer, more succinct, and stronger than the 2022-01-03 [[Wikisource:Joint Statement of the Leaders of the Five Nuclear-Weapon States on Preventing Nuclear War and Avoiding Arms Races|Joint Statement of the Leaders of the [first] Five Nuclear-Weapon States on Preventing Nuclear War and Avoiding Arms Races]], "that a nuclear war cannot be won and must never be fought".<ref>[[Wikisource:Joint Statement of the Leaders of the Five Nuclear-Weapon States on Preventing Nuclear War and Avoiding Arms Races]]. See also Borger (2022). Douthat (2022) discussed the [[w:2021-2022 Russo-Ukrainian crisis|current Ukraine crisis]] in [[w:The New York Times|''The New York Times'']]. He concluded that for us (presumably the US and perhaps its NATO allies) "To escalate now against a weaker adversary [Russia], one less likely to ultimately defeat us and more likely to engage in atomic recklessness if cornered, would be a grave and existential folly."</ref> This repeated a statement made 1987-12-11 by US President [[w:Ronald Reagan| Ronald Reagan]] and [[w:Soviet Union|Soviet]] head of state [[w:Mikhail Gorbachev|Mikhail Gorbachev]].<ref><!-- Joint statement by Reagan, Gorbachev -->{{cite Q|Q111845607}} Reagan made that same statement 1984-01-25 in his [[Wikisource:Ronald Reagan's Fourth State of the Union Address|fourth State of the Union Address]].</ref> In the following we review the evidence for and against this claim and then comment on the credibility of the logic that led to the creation of the world's current nuclear arsenals and seems to be driving the current nuclear "modernization" programs in the US, [[w:Russia|Russia]], [[w:China|China]] and elsewhere. == Summary of research on the consequences of a nuclear war == It is theoretically possible that a nuclear exchange would end like [[w:World War II|World War II]] with no more than [[w:Atomic bombings of Hiroshima and Nagasaki|two nuclear weapons being used]]. It is also theoretically possible that nuclear weapons in a new war would only target deserted areas like [[w:List of nuclear weapons tests|the locations where more than 2,000 tests of nuclear weapons]] have been conducted so far.<ref>For a "[[w:List of nuclear weapons tests|List of nuclear weapons tests]]", see the Wikipedia article by that title (accessed 2023-07-06).</ref> Either of those scenarios would increase the level of harmful background radiation worldwide leading to increases in the rates of cancer, birth defects and genetic mutations but would otherwise not likely have a major impact on a large portion of humanity.<ref>Johnston (2001) reported that only 521 of the more than 2,000 nuclear weapons tests were above ground. If 521 explosions of nuclear weapons in deserted places have not generated a substantive impact on human health, it seems unlikely that a nuclear war involving a few thousand explosions of nuclear weapons in deserted areas would be dramatically worse.</ref> However, a nuclear war with such negligible results is highly unlikely. More likely is the deaths in a few hours or days of tens or hundreds of millions of humans.<ref>The "Number of direct fatalities" in a nuclear war lasting a week ranged from 27 to 360 million in simulations summarized in Xia et al. (2022, Table 1).</ref> More would die of radiation poisoning over the next few months and years.<ref>Ellsberg (2017, pp. 2-3) includes a graph that the Joint Chiefs Joint Chiefs of Staff produced in the Spring of 1961 to answer President Kennedy's question, "If your plans for a general [nuclear] war are carried out as planned, how many people will be killed in the [[w:Soviet Union|Soviet Union]] and China?" This graph was a straight line beginning at 275 million who would die during the initial nuclear exchange with another 8.25 million dying each month for the next six months, totaling 325 million deaths.</ref> If more than a few dozen nuclear weapons are used, then "nuclear war would also produce nearly instantaneous climate change that among other effects, would threaten the global food supply. Even a regional nuclear war ..., such as between India and Pakistan,<ref>Robock et al. (2007); Toon et al. (2019). Of course, a nuclear war could be started accidentally by any nuclear-weapons state, as suggested in the report of an Indian cruise missile that landed 2022-03-10 in Pakistan (Mashal and Masood 2022). See also Xia et al. (2022).</ref> in which less than 3% of the world’s nuclear weapons stockpiles were detonated in urban areas, would suddenly decrease the average global temperature by 1°C–7°C [2°–13°F], precipitation by up to 40%, and sunlight by up to 30%. ... Such a conflict would decrease crop production to an extent that it could seriously threaten world food security and even trigger global famine",<ref>Jägermeyr et al. (2020).</ref> according to Robock and Prager (2021). In theory, crop losses of between 10 and 25 percent for 5-10 years<ref>as predicted by Jägermeyr et al. (2020) and others.</ref> might not threaten a global famine or even an increase in malnutrition if people ate more plant-based foods and less meat. In practice, famines never work that way: There is hoarding, and many who do not die of starvation succumb to diseases or secondary wars driven by the food insecurity, according to Helfand (2013). [[w:Amartya Sen|Nobel Prize Economist Sen]] observed that, "no famine has ever taken place ... in a functioning democracy".<ref>Sen (1999, p. 32). Later on p. 178, he stated similarly, "there has never been a famine in a functioning multiparty democracy."</ref> This generalizes the observation that Ireland was a ''net food exporter'' during its infamous potato famines of the nineteenth century.<ref>e.g., Woodham-Smith (1962).</ref> Xia et al. (2022, Table 1) estimated that between 4 and 85 percent of humanity would starve to death if they did not die of something else sooner in the nuclear wars they simulated. In the spring of 1961, "The total death toll as calculated by the Joint Chiefs of Staff [top US military leaders], from a U.S. first strike aimed at the Soviet Union, its Warsaw Pact satellites, and China, would be roughly six hundred million dead. A hundred Holocausts", according to [[w:Daniel Ellsberg|Daniel Ellsberg]], who served as a nuclear war planner for presidents Eisenhower, Kennedy, Johnson and Nixon<ref>Ellsberg (2017, esp. pp. 2-3) noted that 325 million would die in the Soviet Union and China and another couple hundred million in neighboring countries, totalling six hundred million.</ref> before releasing [[w:The Pentagon Papers|"The Pentagon Papers"]] in 1971. Six hundred million was roughly 20 percent of the total human population on earth in 1961, and that didn't count any in the US who might be killed in retaliation. In 1957, roughly 4 years earlier, [[w:Mao Zedong|Mao Zedong]], then the Chairman of the [[w:China|People's Republic of China]], had reportedly said that a nuclear war could kill a third of humanity, perhaps half, "but imperialism would be razed to the ground, and the whole world would become socialist."<ref>Dikötter (2010). See also Halimi (2018), which gives the date as 1957. There is some controversy about this quote; see the Wikipedia article on [[w:Mao Zedong|"Mao Zedong"]], accessed 2022-03-02.</ref> Turco et al. (1983) published the first predictions of a ''[[w:nuclear winter|nuclear winter]]'' based on climate modeling that considered smoke anticipated from fires started by a massive nuclear weapons exchange between the US and the Soviet Union. They found that "average light levels can be reduced to a few percent of ambient and land temperatures can reach -15° to -25°C [5° to -13°F]" with smoke transported from the Northern to the Southern Hemisphere, all of which "could pose a serious threat to human survivors and to other species." Various teams have published comparable analyses since then with different and increasingly sophisticated models, beginning with Aleksandrov and Stenchikov (1983), with similar conclusions.<ref>Coup et al. (2019, p. 8522).</ref> Coup et al. (2019) predicted hard freezes ''in the summer'' in most of the Northern Hemisphere including the US, Russia, and most of Europe during the first three years following such a war, where temperatures drop below −4°C [25°F], making it impossible to grow crops in those regions. China would suffer a similar fate, with only its southeast portion remaining above freezing in the summer. Much of Southern Mexico, Central and South America, and the Southern Hemisphere would also be negatively impacted, but not to the same extent. These climate modeling results make Mao's predictions from 1957 seem wildly optimistic: Any humans in the US, Canada, or most of Eurasia who survived the nuclear exchange would have extreme difficulties finding enough to eat -- "imperialism razed to the ground", according to Mao. However, crop yields in most of the rest of the world would also be extremely depressed, which Mao had not considered. The results would threaten famine vastly worse than what has been predicted following a nuclear war between India and Pakistan.<ref>Ellsberg said that 98 or 99 percent of humanity would starve to death if they did not die of something else sooner (Ellsberg et al. 2017). Coup et al. (2019) and Xia et al. (2022) conclude that it won't be quite that bad but will still pretty grim.</ref> Of course, no one knows for sure how many people would die directly and indirectly from a nuclear war. However, it should be obvious to at least some if not most people<ref>People who believe they benefit from the current new nuclear arms race may never agree. Upton Sinclair said, "It is difficult to get a man to understand something, when his salary depends upon his not understanding it!" Sinclair (1935, p. 109).</ref> that the ''worst'' response to a nuclear attack would be a nuclear response: * A nuclear response to a nuclear "warning shot" with minimal destruction could too easily escalate until the nuclear arsenals of all parties were expended and the life expectancy of all survivors worldwide was dramatically reduced. * Alternatively, a nuclear response to a massive first strike against a thousand cities would most likely ''increase'' the death toll worldwide, including in the country retaliating with nuclear weapons''. * It is possible that a nuclear response could deter further uses of nuclear weapons and reduce the length and severity of the war and its global impact. However, this outcome seems unlikely given the record of history. * It is also possible that a nuclear arsenal could deter a nuclear attack, though it could also ''increase'' the risk of such an attack, e.g., due to a malfunction of the nuclear command, control and communications system of a potential adversary<ref>as reportedly almost happened from the [[w:1983 Soviet nuclear false alarm incident|1983 Soviet nuclear false alarm incident]].</ref> -- or due to an individual or team who believes that God wants them to initiate [[w:Armageddon|Armageddon]], as discussed in the section on [[Media Literacy and You/Deterrence without threat#Deterrence theory and nuclear Armageddon|Deterrence theory and nuclear Armageddon]] in the chapter of this book on [[Media Literacy and You/Deterrence without threat|Deterrence without threat]] Turcotte (2022) concluded that if the 2022 Ukraine 'conflict ends without the annihilation of our species, it should nonetheless be regarded as a planet-wide near-death experience, and the “Peoples of the United Nations” should demand the total elimination of nuclear weapons as quickly as humanly possible, as well as the establishment of new common security measures that will move us much closer to sustainable peace throughout the world.' In spite of this concern, Turcotte recommended military action to support Ukraine but short of declaring war on Russia. Leading experts have made alarming comments about the likelihood of a nuclear attack, possibly by a terrorist organization. In 2004 Bruce Blair, president of the [[w:Center for Defense Information|Center for Defense Information]] wrote: "I wouldn't be at all surprised if nuclear weapons are used over the next 15 or 20 years, first and foremost by a terrorist group that gets its hands on a [[w:Russia and weapons of mass destruction|Russian]]" or [[w:Pakistan and weapons of mass destruction|Pakistani nuclear weapon]].<ref><!--Nicholas D. Kristof (2004) A Nuclear 9/11, NYT-->{{cite Q|Q111906710}}</ref> Other experts seemed even more concerned: A nuclear terrorist attack in the US was considered "more likely than not" within the next five to ten years, according to Professor [[w:Robert Gallucci|Robert Gallucci]] of the [[w:Georgetown University School of Foreign Service|Georgetown University School of Foreign Service]] in 2006 or in the next decade per former U.S. Assistant Secretary of Defense [[w:Graham Allison|Graham Allison]] in 2004.<ref><!-- Ordre Kittrie (2007) Averting Catastrophe: Why the Nuclear Non-proliferation Treaty is Losing its Deterrence Capacity and How to Restore It -->{{cite Q|Q111906652}}</ref> For more, see the section on "[[w:Mutually assured destruction#Criticism|Criticism]]" in the Wikipedia article on "[[w:Mutually assured destruction|Mutually assured destruction]]". The Wikipedia article on "[[w:National Response Scenario Number One|National Response Scenario Number One]]" describes "the United States federal government's planned response to a nuclear attack." It focuses primarily on "the possible detonation of a small, crude nuclear weapon by a terrorist group in a major city, with significant loss of life and property."<ref>Accessed 2022-05-08, when it cited <!-- Jay Davis (2008) After A Nuclear 9/11 -->{{cite Q|Q111905675}}, <!-- Brian Michael Jenkins (2008) A Nuclear 9/11? -->{{cite Q|Q111906145}}</ref> That article discusses preparing for a nuclear attack but not how to respond. Nevertheless, if the ''worst'' response to a nuclear attack is a nuclear response, that has other policy implications for nuclear ''and non-nuclear'' countries world wide discussed in the previous chapter on [[Media Literacy and You/Deterrence without threat|Deterrence without threat]]. == Credibility of military leaders and national security experts == {{main|Expertise of military leaders and national security experts}} * ''Never attribute to malice that which is adequately explained by stupidity.'' ([[w:Hanlon's razor|Hanlon's razor]]) * ''Never attribute to malice or stupidity that which can be explained by moderately rational individuals following incentives in a complex system.'' (Hubbard's clumsier correlary.<ref>Hubbard (2020, pp. 81-82).</ref>) The history of armed conflict should raise questions about the credibility of those advocating use of military force: In all major armed conflicts in history, at least one side has lost. Often the official winners lost substantially more than they gained. === Research on expertise === The history of armed conflict is consistent with the research by Kahneman and Klein (2009) in their conclusion that :''expert intuition is learned from frequent, rapid, high-quality feedback.'' In particular, military leaders in combat can get frequent, rapid high-quality feedback on their ability to deliver death and destruction to designated targets. However, no one can get such feedback about how to win wars or how to ''promote broadly shared peace and prosperity for the long term.'' This is discussed in more detail in the Wikiversity article on "[[Expertise of military leaders and national security experts]]". That article documents how experts without such feedback can be beaten by simple rules of thumb developed by intelligent lay people.<ref>Kahneman et al. (2021) report that with some data, a statistical model fit often does better. With lots of data, artificial intelligence systems can do even better. This extends the work of [[w:Paul E. Meehl#Clinical versus statistical prediction|Meehl (1954)]]. Hubbard (2020) and [[w:Superforecasting: The Art and Science of Prediction|Tetlock and Gardner (2015)]] describe things one might do to improve their intuition.</ref> As the time since the [[w:Atomic bombings of Hiroshima and Nagasaki|atomic bombings if Hiroshima and Nagasaki]] increases, the ''intuition'' that political and military leaders have about nuclear weapons gets worse, because that history tells them that they can use more military force, even threatening to use nuclear weapons, without seriously risking a nuclear war. That intuition increasingly threatens the entirety of humanity. === Increasing risks with nuclear proliferation === Narang and Sagan, eds. (2022) ''The Fragile Balance of Terror: Deterrence in the New Nuclear Age'' includes 8 chapters by 12 authors reviewing the literature on different aspects of nuclear deterrence today. They raised many questions about the applicability of [[w:Cold War|Cold War]] analyses of deterence in an age with [[Forecasting nuclear proliferation|an increasing number of nuclear weapon states]]. They mentioned numerous concerns including the following: * [[w:2008 Mumbai attacks|During terrorist attacks in Mumbai in 2008]], someone called called Pakistani president Zardari claiming to be Indian foreign minister Mukherjee threatening to attack Pakistan. That crises was diffused without escalation after US secretary of state Condoleezza Rice called Mukherjee, who assured her that he had not placed such a call, and India was ''not'' planning to attack Pakistan. If someone claiming to be a US official had placed a similar call to Kim Jong Un while Donald Trump was President of the US, the result may not have been as benign.<ref>Narang and Sagan (2022, p. 241).</ref> * [[w:2018 Hawaii false missile alert|"In January 2018, the Hawaii emergency management system issued an incoming missile warning alert]] adding, 'this is not a drill.'" The US did not respond, because (a) they had redundant early warning systems that did not indicate an incoming missile, (b) professional operators in Hawaii promptly acknowledged the mistake, and (c) no one in the US seriously expected such an attack. If this had happened in North Korea, none of these three restraining conditions would likely not have been present: (a) They did not have redundant warning systems. (b) Operators are killed, not just fired in North Korea for making a mistake like that. (c) US "President Trump was threatening 'fire and fury' if North Korean nuclear and missile tests continued."<ref>Narang and Sagan (2022, p. 232).</ref> * [[w:2019 Balakot airstrike|In 2019 India bombed an alleged terrorist training camp in Balakot]], Pakistan. This was "the first time a nuclear weapons state has bombed the undisputed territory of another nuclear weapons state."<ref>Narang and Sagan (2022, pp. 231-232).</ref> * [[w:2020–2021 China–India skirmishes|In 2020, Chinese and Indian troops engaged in hostilities along their disputed border]] with fatalities on both sides, "for the first time in almost half a century. Intense conflict between three nuclear powers simultaneously is no longer a remote possibility.<ref>Narang and Sagan (2022, p. 232).</ref> Beyond this, [[w:Richard Ned Lebow|Richard Ned Lebow]] said, "There’s all kinds of empirical evidence that a deterrence strategy is as likely to provoke the behavior it seeks to prevent as not."<ref>Lebow et al. (2023). See also Lebow (2020, ch. 4).</ref> == Probability of a nuclear war == The section on [[Time to nuclear Armageddon#Relevant literature|Relevant literature]] of the Wikiversity article on [[Time to nuclear Armageddon]] includes a table summarizing previous estimates of the probability of a nuclear war. Karger et al. (2023) provides a more extensive study of the probability of a nuclear war and other existential risks. === System accidents === The concept of "normal accidents" or "[[w:system accident|system accidents]]" seems important here. Research in that area has established that ''it is impossible to design and manage complex systems to ultra-high levels of reliability''. Maintenance on redundant systems is often deferred, because responsible managers are often reluctant to spend money fixing something that works.<ref>e.g., Sagan (1993).</ref> And procedures are sometimes secretly modified by people with different priorities from their management. For example, at least between 1970 and 1974 the codes in US Air Force launch control centers for [[w:Intercontinental ballistic missile|Intercontinental ballistic missiles]] were all set continuously to 00000000.<ref>Ellsberg (2017, p. 61).</ref> This clearly negated the claim that only the President of the US could order the use of US nuclear weapons, secured by secret codes carried in a briefcase (called the [[w:nuclear football|"nuclear football"]]) near the President at all times. Similarly, former US Secretary of Defense William J. Perry has said an actual nuclear attack on the US is far less likely than a report of one generated by a malfunction in the US nuclear command, control, and communications systems.<ref>Perry and Collina (2020). Of course, a nuclear war could be started accidentally by any nuclear-weapons state, as suggested in the report of an Indian cruise missile that landed 2022-03-10 in Pakistan (Mashal and Masood 2022).</ref> A tragic example of a system accident is the [[w:Sinking of MV Sewol|Sinking of MV ''Sewol'']], 2014-04-16. It sank with over twice its rated load under the command of a substitute captain. The regular captain had complained of deferred maintenance threatening the stability of the vessel; he said the company had threatened to fire him if he continued to complain. As of this writing, it has been almost 79 years since nuclear weapons were detonated in hostilities. As noted above, that history feeds human intuition that we can "safely" be more aggressive in developing, deploying and threatening the use of nuclear weapons without seriously risking [[Time to nuclear Armageddon|nuclear Armageddon]]. People who disagree like the [[w:Union of Concerned Scientists|Union of Concerned Scientists]] with their [[w:Doomsday Clock|Doomsday Clock]] are dismissed as unrealistic, like [[w:Chicken Little|Chicken Little]]. == Human psychology and the role of the media == When people are attacked, it can sometimes be difficult to control their responses, which are driven by instinctive reactions often characterized as irrational. Johnson (2004) documented how these instinctive reactions exist, because they provided survival benefits to our ancestors over hundreds of thousands and millions of years of evolutionary history. These instincts may, however, push us into the ''worst'' possible response to a nuclear attack. Worse, major media everywhere have conflicts of interest in honestly reporting on anything (like these research results) that might threaten those who control much of the money for the media.<ref name='McC+Cagé+Rolnik">McChesney (2004). Cagé (2016). Rolnik et al. (2019). See also "[[Confirmation bias and conflict]]".</ref> Everyone thinks they know more than they do,<ref name=Kahneman>Kahneman (2011).</ref> which makes them easily misled by the media they find credible.<ref>[[Confirmation bias and conflict]]. See also McChesney (2004), Cagé (2016), and Rolnik et al. (2019).</ref> == Recapitulation == In sum, the worst possible response to a nuclear attack would seem to be a nuclear response. Existing nuclear weapons policies appear to be supported by propaganda. That propaganda is effective, because it supports the preferences of those who control much of the money for the media,<ref name='McC+Cagé+Rolnik"/> and because too few humans pursue alternative sources of information when they should.<ref name=Kahneman/> === Call for help === Do you, dear reader, know other serious research not cited herein that might improve this analysis? If yes, you can help improve this discussion by adding comments with citations -- or by adding such citation(s) to the "Discuss" page associated with this chapter, suggesting someone else revise the chapter appropriately. There are plenty of contrary claims in the major media, but the lead author of this chapter is not aware of any that contradict the research summarized above. In the absence of such research, the current author finds it difficult to imagine any national defense policies that carry a greater risk of nuclear Armageddon than our current policies. The risks of being stampeding into a war on fraudulent grounds could be reduced by making it harder for administration officials to lie to Congress and the public, as recommended in the next chapter of this book on "[[w:Media Literacy and You#Threats from excessive government secrecy|Threats from excessive government secrecy]]". That chapter, in sum, documents how the major media have pushed political leaders to pursue counterproductive actions and punish whistleblowers, who release official government secrets classified to keep them from the US public. This chapter is being written in the hopes of inspiring action to improve the prospects for broadly shared peace and prosperity for the long term. == Exercises == 1. If you question the above, post your concerns to the associated "Discuss" page. If you have references that seem to contradict any of the above, either cite the references with your concerns on the "Discuss" page or modify the text appropriately, writing from a neutral point of view, citing credible sources. 2. Ask others their thoughts about nuclear weapons. If you feel inclined to differ, first note that, :''Primary drivers of every major conflict include differences between the media that the different parties find credible.'' Then share your concerns in a friendly supportive manner. The goal is not to convince anyone that they are wrong and you are right but rather to share an alternative perspective, working agree to disagree agreeably while also seeking areas of common concern. == Acknowledgements == Thanks to Owen B. Toon, Alan Robock, and presenters at their irregular webinar series on impact on climate of a nuclear war. Of course, any errors and other deficiencies here are solely the responsibility of the author. == See also == * [[Expertise of military leaders and national security experts]] * [[Time to nuclear Armageddon]] * [[Forecasting nuclear proliferation]] * [[Time to extinction of civilization]] == Notes == {{reflist}} == Bibliography == * <!-- Aleksandrov and Stenchikov (1983) "On the modeling of the climatic consequences of the nuclear war" -->{{cite Q|Q63229964}} * <!-- Borger (2022) Five of world’s most powerful nations pledge to avoid nuclear war, Guardian -->{{cite Q|Q111011203}} * <!-- Cagé (2016) Saving the media: Capitalism, crowdfunding and democracy (Harvard U. Pr.)-->{{cite Q|Q54640583}} * <!-- Chenoweth and Stephan (2011) Why Civil Resistance Works: The Strategic Logic of Nonviolent Conflict (Columbia U. Pr.) -->{{cite Q|Q88725216}} For their data see, <!-- Chenoweth, NAVCO data project, Harvard -->{{cite Q|Q55842589}} * <!-- Coup et al. 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Pr.)-->{{cite Q|Q111097755}} * <!-- Tyler, Tom R., and Yuen J. Huo (2002) Trust in the Law: Encouraging Public Cooperation with the Police and Courts (Russell Sage Foundation)-->{{cite Q|Q106943244}} * <!-- Woodham-Smith, Cecil (1962) The Great Hunger: Ireland 1845-1849 (Harper)-->{{cite Q|Q7737800}} * <!-- Xia et al. (2022) Global food insecurity and famine ... from a nuclear war ...-->{{cite Q| Q113732668}} [[Category:Media literacy]] [[Category:Communication]] [[Category:Political science]] [[Category:Law]] [[Category:Psychology]] [[Category:Sociology]] [[Category:War History]] [[Category:Media Literacy and You]] <!-- https://en.wikiversity.org/wiki/Wikiversity:Category_Review --> dbgiq3215i46eebezso4xncjb9fx5fn