Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.47.0-wmf.7 first-letter Media Special Talk User User talk Wikiversity Wikiversity talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk School School talk Portal Portal talk Topic Topic talk Collection Collection talk Draft Draft talk TimedText TimedText talk Module Module talk Event Event talk 4 28 2816207 2815780 2026-06-18T12:56:56Z AUBSTRAWBS 3060598 /* Mobile friendly main page */ new section 2816207 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) ::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:14, 9 June 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) * {{oppose}} per above. Wikiversity<math>\not=</math> Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC) * {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC) * {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC) *{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on "Media & Democracy" syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that "rehost Wikinews here" would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}} ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) *::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a "regular" journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:43, 5 June 2026 (UTC) *:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC) == Create an autopatrolled user group? == {{tracked|T428269|resolved}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) : @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC) ::Thanks. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:13, 9 June 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) :Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC) ::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC) :::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC) ::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) : Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC) == Proposed user group and/or possible policy changes == I want to discuss about user group and possible policy changes. # First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]]. # Second, curators. Given that curators have some sensitive custodian rights (such as <code>delete</code> [but not <code>undelete</code> or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and <code>protect</code>), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions. # Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC) :#Yes, I agree. :#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos. :#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''"II ...then you will be approved as a probationary custodian for a period of at least four weeks"''. :::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to. :[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC) == New user what do I do here == I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful. (One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}} :Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC) :Thank you very much :) hope to contribute a lot. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC) == Towards an Ethics policy == In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects. However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC) :One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:39, 9 June 2026 (UTC) ::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism. ::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted. ::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC) :::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:06, 9 June 2026 (UTC) ::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC) :AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC) ::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC) :::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC) ::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC) == Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki == Hello community, The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon. Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC) :Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:24, 9 June 2026 (UTC) == Image not displaying == Can anyone work out why this image isn't displaying?<br> [[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:45, 11 June 2026 (UTC) :Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:59, 11 June 2026 (UTC) == New nomination template(s) == I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC) == June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;">[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions. #'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 04:00, 15 June 2026 (UTC) == Mobile friendly main page == Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better e6w8qy3j8w5jw3kgotuv5hqg00voypp 2816209 2816207 2026-06-18T13:30:40Z Koavf 147 /* Mobile friendly main page */ 2816209 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) ::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:14, 9 June 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) * {{oppose}} per above. Wikiversity<math>\not=</math> Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC) * {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC) * {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC) *{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on "Media & Democracy" syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that "rehost Wikinews here" would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}} ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) *::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a "regular" journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:43, 5 June 2026 (UTC) *:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC) == Create an autopatrolled user group? == {{tracked|T428269|resolved}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) : @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC) ::Thanks. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:13, 9 June 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) :Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC) ::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC) :::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC) ::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) : Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC) == Proposed user group and/or possible policy changes == I want to discuss about user group and possible policy changes. # First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]]. # Second, curators. Given that curators have some sensitive custodian rights (such as <code>delete</code> [but not <code>undelete</code> or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and <code>protect</code>), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions. # Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC) :#Yes, I agree. :#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos. :#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''"II ...then you will be approved as a probationary custodian for a period of at least four weeks"''. :::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to. :[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC) == New user what do I do here == I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful. (One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}} :Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC) :Thank you very much :) hope to contribute a lot. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC) == Towards an Ethics policy == In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects. However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC) :One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:39, 9 June 2026 (UTC) ::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism. ::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted. ::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC) :::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:06, 9 June 2026 (UTC) ::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC) :AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC) ::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC) :::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC) ::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC) == Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki == Hello community, The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon. Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC) :Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:24, 9 June 2026 (UTC) == Image not displaying == Can anyone work out why this image isn't displaying?<br> [[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:45, 11 June 2026 (UTC) :Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:59, 11 June 2026 (UTC) == New nomination template(s) == I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC) == June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;">[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions. #'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 04:00, 15 June 2026 (UTC) == Mobile friendly main page == Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}} :{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:30, 18 June 2026 (UTC) ru89iwiqwdmqbwl9vg1dwagplzaaar0 2816218 2816209 2026-06-18T14:03:39Z AUBSTRAWBS 3060598 /* Mobile friendly main page */ Reply 2816218 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) ::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:14, 9 June 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) * {{oppose}} per above. Wikiversity<math>\not=</math> Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC) * {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC) * {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC) *{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on "Media & Democracy" syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that "rehost Wikinews here" would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}} ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) *::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a "regular" journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:43, 5 June 2026 (UTC) *:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC) == Create an autopatrolled user group? == {{tracked|T428269|resolved}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) : @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC) ::Thanks. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:13, 9 June 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) :Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC) ::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC) :::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC) ::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) : Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC) == Proposed user group and/or possible policy changes == I want to discuss about user group and possible policy changes. # First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]]. # Second, curators. Given that curators have some sensitive custodian rights (such as <code>delete</code> [but not <code>undelete</code> or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and <code>protect</code>), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions. # Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC) :#Yes, I agree. :#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos. :#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''"II ...then you will be approved as a probationary custodian for a period of at least four weeks"''. :::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to. :[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC) == New user what do I do here == I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful. (One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}} :Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC) :Thank you very much :) hope to contribute a lot. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC) == Towards an Ethics policy == In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects. However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC) :One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:39, 9 June 2026 (UTC) ::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism. ::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted. ::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC) :::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:06, 9 June 2026 (UTC) ::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC) :AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC) ::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC) :::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC) ::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC) == Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki == Hello community, The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon. Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC) :Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:24, 9 June 2026 (UTC) == Image not displaying == Can anyone work out why this image isn't displaying?<br> [[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:45, 11 June 2026 (UTC) :Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:59, 11 June 2026 (UTC) == New nomination template(s) == I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC) == June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;">[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions. #'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 04:00, 15 June 2026 (UTC) == Mobile friendly main page == Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}} :{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:30, 18 June 2026 (UTC) :Hi i uploaded an image of the problem. Since some of the images are larger than the screen and not adjusted to fit they stick out and makes the page larger which lets you scroll right and have a big white rectangle on the side [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 14:03, 18 June 2026 (UTC) 34uhwpc0p8o0zfb73x1xpojurj9xs2u 2816226 2816218 2026-06-18T15:42:05Z Koavf 147 /* Mobile friendly main page */ 2816226 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) ::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:14, 9 June 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) * {{oppose}} per above. Wikiversity<math>\not=</math> Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC) * {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC) * {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC) *{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on "Media & Democracy" syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that "rehost Wikinews here" would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}} ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) *::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a "regular" journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:43, 5 June 2026 (UTC) *:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC) == Create an autopatrolled user group? == {{tracked|T428269|resolved}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) : @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC) ::Thanks. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:13, 9 June 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) :Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC) ::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC) :::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC) ::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) : Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC) == Proposed user group and/or possible policy changes == I want to discuss about user group and possible policy changes. # First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]]. # Second, curators. Given that curators have some sensitive custodian rights (such as <code>delete</code> [but not <code>undelete</code> or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and <code>protect</code>), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions. # Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC) :#Yes, I agree. :#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos. :#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''"II ...then you will be approved as a probationary custodian for a period of at least four weeks"''. :::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to. :[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC) == New user what do I do here == I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful. (One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}} :Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC) :Thank you very much :) hope to contribute a lot. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC) == Towards an Ethics policy == In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects. However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC) :One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:39, 9 June 2026 (UTC) ::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism. ::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted. ::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC) :::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:06, 9 June 2026 (UTC) ::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC) :AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC) ::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC) :::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC) ::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC) == Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki == Hello community, The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon. Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC) :Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:24, 9 June 2026 (UTC) == Image not displaying == Can anyone work out why this image isn't displaying?<br> [[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:45, 11 June 2026 (UTC) :Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:59, 11 June 2026 (UTC) == New nomination template(s) == I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC) == June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;">[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions. #'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 04:00, 15 June 2026 (UTC) == Mobile friendly main page == Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}} :{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:30, 18 June 2026 (UTC) ::Hi i uploaded an image of the problem. Since some of the images are larger than the screen and not adjusted to fit they stick out and makes the page larger which lets you scroll right and have a big white rectangle on the side [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 14:03, 18 June 2026 (UTC) :::Thanks. I agree that this is an issue, but it's a pretty minor-to-moderate one to me and I don't think I will be able to dedicate time to fix it myself. Showing it to others here is useful in case someone else wants to tinker with the CSS to resolve it. Thanks for bringing it to the community's attention. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:42, 18 June 2026 (UTC) 9mcyr4sasl972bq3wjw0xtqu9a6af6i 2816248 2816226 2026-06-18T20:00:03Z AUBSTRAWBS 3060598 /* Mobile friendly main page */ Reply 2816248 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) ::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:14, 9 June 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) * {{oppose}} per above. Wikiversity<math>\not=</math> Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC) * {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC) * {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC) *{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on "Media & Democracy" syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that "rehost Wikinews here" would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}} ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) *::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a "regular" journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:43, 5 June 2026 (UTC) *:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC) == Create an autopatrolled user group? == {{tracked|T428269|resolved}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) : @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC) ::Thanks. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:13, 9 June 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) :Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC) ::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC) :::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC) ::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) : Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC) == Proposed user group and/or possible policy changes == I want to discuss about user group and possible policy changes. # First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]]. # Second, curators. Given that curators have some sensitive custodian rights (such as <code>delete</code> [but not <code>undelete</code> or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and <code>protect</code>), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions. # Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC) :#Yes, I agree. :#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos. :#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''"II ...then you will be approved as a probationary custodian for a period of at least four weeks"''. :::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to. :[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC) == New user what do I do here == I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful. (One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}} :Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC) :Thank you very much :) hope to contribute a lot. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC) == Towards an Ethics policy == In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects. However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC) :One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:39, 9 June 2026 (UTC) ::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism. ::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted. ::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC) :::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:06, 9 June 2026 (UTC) ::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC) :AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC) ::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC) :::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC) ::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC) == Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki == Hello community, The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon. Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC) :Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:24, 9 June 2026 (UTC) == Image not displaying == Can anyone work out why this image isn't displaying?<br> [[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:45, 11 June 2026 (UTC) :Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:59, 11 June 2026 (UTC) == New nomination template(s) == I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC) == June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;">[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions. #'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 04:00, 15 June 2026 (UTC) == Mobile friendly main page == Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}} :{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:30, 18 June 2026 (UTC) ::Hi i uploaded an image of the problem. Since some of the images are larger than the screen and not adjusted to fit they stick out and makes the page larger which lets you scroll right and have a big white rectangle on the side [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 14:03, 18 June 2026 (UTC) :::Thanks. I agree that this is an issue, but it's a pretty minor-to-moderate one to me and I don't think I will be able to dedicate time to fix it myself. Showing it to others here is useful in case someone else wants to tinker with the CSS to resolve it. Thanks for bringing it to the community's attention. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:42, 18 June 2026 (UTC) ::::I do know CSS as I like to maintain a blog online so I could try and fix it but I don't know if I have the access to do that, would i need to be a curator/ custodian. Alternatively i could edit a sandbox version of the main page and then send it to someone. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 20:00, 18 June 2026 (UTC) 6zjq4v3s1u5ret43n4ycuo13m6e2jsx 2816249 2816248 2026-06-18T20:14:09Z Koavf 147 /* Mobile friendly main page */ Reply 2816249 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) ::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:14, 9 June 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) * {{oppose}} per above. Wikiversity<math>\not=</math> Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC) * {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC) * {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC) *{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on "Media & Democracy" syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that "rehost Wikinews here" would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}} ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) *::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a "regular" journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:43, 5 June 2026 (UTC) *:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC) == Create an autopatrolled user group? == {{tracked|T428269|resolved}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) : @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC) ::Thanks. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:13, 9 June 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) :Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC) ::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC) :::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC) ::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) : Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC) == Proposed user group and/or possible policy changes == I want to discuss about user group and possible policy changes. # First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]]. # Second, curators. Given that curators have some sensitive custodian rights (such as <code>delete</code> [but not <code>undelete</code> or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and <code>protect</code>), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions. # Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC) :#Yes, I agree. :#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos. :#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''"II ...then you will be approved as a probationary custodian for a period of at least four weeks"''. :::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to. :[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC) == New user what do I do here == I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful. (One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}} :Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC) :Thank you very much :) hope to contribute a lot. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC) == Towards an Ethics policy == In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects. However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC) :One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:39, 9 June 2026 (UTC) ::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism. ::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted. ::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC) :::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:06, 9 June 2026 (UTC) ::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC) :AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC) ::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC) :::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC) ::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC) == Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki == Hello community, The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon. Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC) :Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:24, 9 June 2026 (UTC) == Image not displaying == Can anyone work out why this image isn't displaying?<br> [[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:45, 11 June 2026 (UTC) :Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:59, 11 June 2026 (UTC) == New nomination template(s) == I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC) == June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;">[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions. #'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 04:00, 15 June 2026 (UTC) == Mobile friendly main page == Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}} :{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:30, 18 June 2026 (UTC) ::Hi i uploaded an image of the problem. Since some of the images are larger than the screen and not adjusted to fit they stick out and makes the page larger which lets you scroll right and have a big white rectangle on the side [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 14:03, 18 June 2026 (UTC) :::Thanks. I agree that this is an issue, but it's a pretty minor-to-moderate one to me and I don't think I will be able to dedicate time to fix it myself. Showing it to others here is useful in case someone else wants to tinker with the CSS to resolve it. Thanks for bringing it to the community's attention. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:42, 18 June 2026 (UTC) ::::I do know CSS as I like to maintain a blog online so I could try and fix it but I don't know if I have the access to do that, would i need to be a curator/ custodian. Alternatively i could edit a sandbox version of the main page and then send it to someone. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 20:00, 18 June 2026 (UTC) :::::Oh great. There are a lot of draft versions of the main page like [[Wikiversity:Main Page/Draft version 0.2]], so you can make [[Wikiversity:Main Page/Sandbox]] if you want and edit there. If you can tinker it to your liking, I can edit the main page. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:14, 18 June 2026 (UTC) crqm9vn9iw0os68c5mt8pm61ttde3ud 2816251 2816249 2026-06-18T21:34:36Z Codename Noreste 2969951 /* Main page titles */ new topic ([[mw:c:Special:MyLanguage/User:JWBTH/CD|CD]]) 2816251 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) ::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:14, 9 June 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) * {{oppose}} per above. Wikiversity<math>\not=</math> Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC) * {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC) * {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC) *{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on "Media & Democracy" syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that "rehost Wikinews here" would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}} ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) *::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a "regular" journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:43, 5 June 2026 (UTC) *:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC) == Create an autopatrolled user group? == {{tracked|T428269|resolved}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) : @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC) ::Thanks. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:13, 9 June 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) :Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC) ::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC) :::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC) ::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) : Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC) == Proposed user group and/or possible policy changes == I want to discuss about user group and possible policy changes. # First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]]. # Second, curators. Given that curators have some sensitive custodian rights (such as <code>delete</code> [but not <code>undelete</code> or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and <code>protect</code>), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions. # Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC) :#Yes, I agree. :#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos. :#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''"II ...then you will be approved as a probationary custodian for a period of at least four weeks"''. :::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to. :[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC) == New user what do I do here == I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful. (One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}} :Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC) :Thank you very much :) hope to contribute a lot. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC) == Towards an Ethics policy == In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects. However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC) :One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:39, 9 June 2026 (UTC) ::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism. ::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted. ::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC) :::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:06, 9 June 2026 (UTC) ::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC) :AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC) ::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC) :::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC) ::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC) == Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki == Hello community, The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon. Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC) :Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:24, 9 June 2026 (UTC) == Image not displaying == Can anyone work out why this image isn't displaying?<br> [[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:45, 11 June 2026 (UTC) :Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:59, 11 June 2026 (UTC) == New nomination template(s) == I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC) == June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;">[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions. #'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 04:00, 15 June 2026 (UTC) == Mobile friendly main page == Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}} :{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:30, 18 June 2026 (UTC) ::Hi i uploaded an image of the problem. Since some of the images are larger than the screen and not adjusted to fit they stick out and makes the page larger which lets you scroll right and have a big white rectangle on the side [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 14:03, 18 June 2026 (UTC) :::Thanks. I agree that this is an issue, but it's a pretty minor-to-moderate one to me and I don't think I will be able to dedicate time to fix it myself. Showing it to others here is useful in case someone else wants to tinker with the CSS to resolve it. Thanks for bringing it to the community's attention. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:42, 18 June 2026 (UTC) ::::I do know CSS as I like to maintain a blog online so I could try and fix it but I don't know if I have the access to do that, would i need to be a curator/ custodian. Alternatively i could edit a sandbox version of the main page and then send it to someone. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 20:00, 18 June 2026 (UTC) :::::Oh great. There are a lot of draft versions of the main page like [[Wikiversity:Main Page/Draft version 0.2]], so you can make [[Wikiversity:Main Page/Sandbox]] if you want and edit there. If you can tinker it to your liking, I can edit the main page. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:14, 18 June 2026 (UTC) == Main page titles == Currently, the title says "Wikiversity:Main Page", but in my opinion, it's too basic. I would like to propose changing it with the following options (you may only pick one): # Option 1: Set both [[MediaWiki:Mainpage-title]] and [[MediaWiki:Mainpage-title-loggedin]] to blank, giving the main page a portal-like design (as with English Wikipedia, English Wikibooks, etc.) # Option 2: Modify [[MediaWiki:Mainpage-title]] to <code>Welcome to Wikiversity</code> (for unregistered users), and [[MediaWiki:Mainpage-title-loggedin]] to <code><nowiki>Welcome to Wikiversity, $1!</nowiki></code>; the latter would display to me as <code>Welcome to Wikiversity, Codename Noreste!</code> Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 18 June 2026 (UTC) b5ze7a9yr3ywee2tbd2acsjkiel4x25 2816252 2816251 2026-06-18T22:16:27Z AUBSTRAWBS 3060598 /* Mobile friendly main page */ Reply 2816252 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) ::A few days shy of 30, it seems obvious that this is not going to pass. So I '''withdraw''' as presumptively '''failed'''. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:14, 9 June 2026 (UTC) ===Votes=== *{{support}} as proposer (with BK's inspiration). I think that an ongoing experiment in citizen journalism is a fit and appropriate use of this site. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 05:35, 14 May 2026 (UTC) *{{support}}, hope to seeing ideas about this, and thank you @[[User:Koavf|Koavf]] [[User:BigKrow|BigKrow]] ([[User talk:BigKrow|discuss]] • [[Special:Contributions/BigKrow|contribs]]) 11:08, 14 May 2026 (UTC) *{{support}} Other than perhaps inflating the total number of pages reported, I see the idea of "practicing journalism" a worthy and relevant activity within the domain of Wikiversity. [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 21:41, 14 May 2026 (UTC) *{{support}} Conditional on development of (a) community guidelines that ensure alignment with Wikiversity's purpose, and (b) clear, nested page-naming structures for projects. More detail below. -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:48, 15 May 2026 (UTC) *{{contra}} This proposal doesn't seem interested in expanding educational materials in journalism, but rather in providing space and protection for Wikinews contributors. But this is contrary to the goals of Wikiversity, and I'm not sure it's a good idea, even with regard to WMF. If WMF decides to close a project and another community lets it run on its domain, that's a bit of an undermining of WMF's and the community's decisions. Given that Wikiversity has had several conflicts with other communities and WMF in its history, I'm against it.--[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 18:59, 15 May 2026 (UTC) *{{contra}} This seems like a proposal to continue the mission of WikiNews, but not a proposal specifically to improve Wikiversity. I concur with Juandev's comments. --[[User:Mu301|mikeu]] ^{[[User talk:Mu301|talk]]} 20:29, 30 May 2026 (UTC) * {{oppose}} per above. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 19:05, 1 June 2026 (UTC) *{{oppose}} Wikiversity isn’t Wikinews and it also isn’t a dumping ground for anything not covered by other projects. It was already suggested, rather bafflingly, that Wikinews parasitize Wikipedia as a host. If it were allowed to freeload off of Wikiversity it would simply promote a view I and likely many others have— that Wikiversity (as it currently exists) has no standards and mostly just exists to host subpar content that wouldn’t be tolerated on any other Wikimedia site. Wikinews needs a new, non-Wikimedia host, and Wikiversity needs to get its act together by enforcing a minimum scope and standard for what it allows. --[[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:16, 4 June 2026 (UTC) * {{oppose}} per above. Wikiversity<math>\not=</math> Wikinews - not a good idea to mix the scope of projects. --[[User:Bert Niehaus|Bert Niehaus]] ([[User talk:Bert Niehaus|discuss]] • [[Special:Contributions/Bert Niehaus|contribs]]) 12:03, 8 June 2026 (UTC) * {{abstain}} I will abstain since I'm not an active Wikiversity contributor. But I just feel like Wikinews had a very clear and specific goal of providing news, and Wikiversity is just a different project with different goals. For me, it would be odd to rehost Wikinews here. But please do not count my vote, this is only a comment. --[[User:Antimundo|Antimundo]] ([[User talk:Antimundo|discuss]] • [[Special:Contributions/Antimundo|contribs]]) 13:19, 6 June 2026 (UTC) * {{oppose}} Although I think it's a pity that Wikinews is closed. --[[User:Dick Bos|Dick Bos]] ([[User talk:Dick Bos|discuss]] • [[Special:Contributions/Dick Bos|contribs]]) 19:06, 8 June 2026 (UTC) *{{support}} In 2018 I initiated [[:Category:Videoconferences on media and democracy]] as a platform for disseminating public affairs events. In 2021 I officially initiated a podcast series on "Media & Democracy" syndicated for the [[w:List of Pacifica Radio stations and affiliates|Pacifica radio network]]. In 2024 I converted it from irregular to fortnightly. I think this is all educational and supports the Wikiversity education mission, and I think that "rehost Wikinews here" would be appropriate. (I had some experience with Wikinews a few years ago. I felt it was too tightly controlled: Article submissions went stale, because I could not get official permission to publish and I could not get the information needed to understand what I was supposed to do to obtain the official permission. I would be opposed to rehosting Wikinews here if the policy similarly made it unreasonably difficult for volunteer contributor to get the information needed to meet the journalistic standards imposed by the overworked editors.) {{unsigned|DavidMCEddy}} ===Comments and questions=== :Definitely worthy of discussion, so I have no problem with the proposal in the sitenotice. :Initial questions: :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? :* What are "active editions"? :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? :* Are any changes to the scope of Wikinews proposed? :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[User:BigKrow/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. :-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 05:52, 14 May 2026 (UTC) :* Does this proposal include importing English Wikinews content e.g., to [[Wikinews]] subpages? ::*No, not at this time. :* What are "active editions"? ::*There were 30 other active editions of Wikinews in addition to English (e.g. [[:n:es:]]) at the time of universal closure (2026-05-04). :* How can Wikiversity navigate the concerns that lead to the closure of Wikinews? ::*One of the biggest issues was the problems with DPL, which is now irrelevant. Another was the lack of activity, which can be ameliorated by having it be part of an existing project instead of its own domain (e.g. some editions of Wikipedia host their own Wikinews already and those projects were not impacted by the closure). :* Are any changes to the scope of Wikinews proposed? ::*Not at this juncture. I would also propose as far as implemention goes that we would request a new namespace and that the material be more-or-less sequestered into its own ongoing project, like Wikijournal is or like the Cookbook and Wikijunior are at our sister [[:b:]]. :* How does [[Wikinews]] fit with the [[Wikiversity:Mission]]? What aligns well? Where might there be tension? :** e.g., I'm not sure that a page like [[Story/Manchester City moves two points behind Arsenal]] in and of itself will serve as an educational resource. ::*The process of citizen journalists practicing their craft in real-time and collaborating with others to do so is itself an education activity. We would essentially be hosting a real-time experiment in citizen journalism, online communities, and collaborative learning in addition to the prospect of spreading educational information from someone actually reading the news. I would propose that we could also make a more deliberate attempt to engage with learning <em>about</em> what does and doesn't work with collaborative news writing by experimentation (e.g. audio news, syndicating to other sites, incorporating freely-licensed news from other sources, writing hyper-local news, writing briefs versus longer-term reportage) and also seeing if the problems noted in the Task Force report that recommended closure can be overcome. Note that we have already done some local investigation about and learning about wiki-based journalism on Wikinews here at [[Journalism studies and Wikinews]]. We could continue that learning and refine the process, including incorporating journalism students from universities. As for tensions, Wikinews is the only sister project that must be done with a quick turn-around: if you take a long time to [[:s:|transcribe a book]], that's just how long it takes, but if you take a long time to write news, it ceases to be news entirely. Wikiversity has been a very slow-growing project that has definitely had some successes but has generally come together over a long period with most learning resources being individual passion projects (or sometimes, frankly, crankery) which would not work with collaborative news that requires more than just a single editor writing whatever he feels like. ::Please let me know any other questions/concerns and any other editors feel free to give your own perspective. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 06:13, 14 May 2026 (UTC) :::Thanks, Justin — it is food for thought. :::In attempting to understand how we've arrived here, I've summarised some of the background on this page: [[Wikinews]]. :::Perhaps it could be helpful to flesh out more of the vision / ideas / possibilities / challenges on that page? -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:49, 14 May 2026 (UTC) :::*Having given it some thought, in principle, I support hosting [[citizen journalism]] on Wikiversity where it is clearly connected to a learning project and/or constitutes original research, both of which align strongly with [[Wikiversity:Mission|Wikiversity’s educational mission]]. :::*My chief concern is the potential for news content that is not clearly linked to the purpose of Wikiversity. To avoid this, some community-agreed guidelines would be prudent. These need not be overly restrictive; they should support boldness and experimentation while helping ensure alignment with Wikiversity's purpose. :::*Given the reported low and declining activity on Wikinews, it seems unlikely that English Wikiversity would be overwhelmed by an influx of news-related editing. My impression is that English Wikinews was the most active edition, but even so, many contributors are likely to disperse to other projects or cease editing altogether. A modest migration of interested editors to Wikiversity seems manageable. :::*At this stage, I do not think a dedicated namespace is necessary. Subpages under [[Wikinews]] or nested pages under relevant learning or research projects, or user-space draft pages should be suitable. I agree that [[Wikijournal]] offers a useful model, as do several existing course structures on Wikiversity. :::*I support [[User:Koavf]]’s suggestions about framing Wikinews activity explicitly around learning. This would create a distinctive space for experimenting with collaborative news production in ways that are pedagogically meaningful. I agree that the [[journalism studies and Wikinews]] project developed by David and Leigh Blackall through the University of Wollongong is an excellent example of the intersection between Wikiversity and Wikinews. The [[Wikinews]] page could evolve into a hub for such projects. :::*I've tidied the [[:Category:Wikinews|Wikinews category]] and merged some content into the [[Wikinews]] page. As part of a reinvigoration effort, please review these and related resources such as [[:Category:Journalism]] and [[School:Journalism]]. :::*A further argument in favour of this initiative is that Wikipedia explicitly excludes both news reporting and original research. So, there is value in maintaining spaces within the Wikimedia ecosystem where these forms of knowledge production can be openly developed and curated. Such work can, in turn, generate valuable evidence and source material that may later inform Wikipedia articles. :::*The closure of WMF-hosted Wikinews does not imply that open wiki-based news curation lacks value. Indeed, the closure documentation appears supportive of experimentation with alternative news models across Wikimedia projects, including through Wikipedia and Wikidata. In that context, Wikiversity seems a natural home for a Wikinews experiment, provided it is clearly grounded in learning and/or research. :::-- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:39, 15 May 2026 (UTC) My understanding towards Wikinews' failure is that everything takes too long to be approved for the publish status, which means that any breaking news would have already become days-old stale news. Wikinews has a brand recognition (for right or wrong reasons) than Wikiversity and I wonder how effective Wikiversity can attract the "Wikinews refugees" to edit here. And just a quick note on the governance. Since each Wikiversity language operates independently, each language has to vote & adopt this proposal independently. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 13:47, 15 May 2026 (UTC) :Your assessment about Wikinews is partially correct. I referenced it earlier, but to be explicit, there is a [[:m:Proposal for Closing Wikinews|report by a task force on sister projects]] that outlines their concerns. There are a few, one of which was the nature of the staleness of news. Thanks also for clarifying that this proposal is only relevant to en.wv and is not binding or even proposed for other editions of Wikiversity. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:54, 15 May 2026 (UTC) *Note: I am not a regular here, and just visit Wikiversity for the WikiJournal project. Challenges of Wikinews included that it required timely reporting and fact-checking processes which differed greatly from the well-established ones in Wikipedia. Here in Wikiversity, there is the WikiJournal project, and that can take some some forms of journalism, just not breaking news reporting. I am in favor of salvaging parts of Wikinews if helpful. Could it, would it be feasible to adapt Wikijournal to accept some forms of news journalism, but just not the timed news reporting? For example, WikiJournal already is doing conference proceedings, and could likely do related event reports even months after the event ended. It could probably accept long-form investigative reporting, which is a sort of news that is not breaking news. I am not sure what the possibilities are, but I would prefer to build up systems that already work rather than import systems which had problems elsewhere. Thanks. [[User:Bluerasberry|<span style="background:#cedff2;color:#11e">''' Blue Rasberry '''</span>]][[User talk:Bluerasberry|<span style="cursor:help"><span style="background:#cedff2;color:#11e">(talk)</span></span>]] 19:17, 22 May 2026 (UTC) *:I agree that there are certain kinds of journalism that are perfectly valid and not time-bound like breaking news reporting, so that won't suffer from the issues noted before. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 21:15, 22 May 2026 (UTC) *::@[[User:Bluerasberry|Bluerasberry]] WikiJournal is not interested in taking on news journalism. WikiJournal is publishing conference proceedings at the request of some Wikimedian educators, and conference proceedings is what a "regular" journal publishes. News journalism is quite different from this, and if WikiJournal starts to deviate towards publishing news journalism, it will create barrier towards future initiatives like being indexed in Medline or Web of Science, and may risk being delisted from Scopus. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 22:43, 5 June 2026 (UTC) *:::Thats a good point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:09, 9 June 2026 (UTC) == Create an autopatrolled user group? == {{tracked|T428269|resolved}} I would like to propose creating the user group <code>autopatrolled</code> (autopatrolled user), in which for non-curators and non-custodians, their page creations and file uploads would be automatically marked as patrolled by the MediaWiki software. Custodians may grant the user group, at their discretion, to users who create good quality pages that do not need frequent patrolling. On a side note, the term {{tq|autopatroller}} would be used, but because we don't have non-curator/custodian patrollers (as we rely on curators and custodians to patrol), I suggest on using the term {{tq|autopatrolled user}}. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:31, 29 May 2026 (UTC) :'''Support''' re: the name, I don't really understand the reasoning, so I am '''neutral''' on that. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:45, 29 May 2026 (UTC) :: Regarding the name, this is because as we don't have the patroller user group, we rely on curators and custodians to patrol new pages and file uploads. Does that make sense? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:39, 29 May 2026 (UTC) :::Not really, but I don't think it's the most important thing. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 16:42, 29 May 2026 (UTC) :::: We'll decide on the name later. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 01:48, 30 May 2026 (UTC) :::::Oh, please don't let me stand in the way. I'm just not very smart, so don't hold up a matter on my account. I didn't want to derail the proposal, which is a fine and sensible one. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 04:16, 30 May 2026 (UTC) : '''Support''' - sounds like a good idea :* Suggest adding a draft section about this group to [[Wikiversity:Patrolling]]. There is a statement in the Introduction of the page that I'm not sure if its correct and at least could be improved: "Wikiversity also uses an autopatrol right, meaning trusted users' contributions are automatically marked as checked so patrollers can focus on reviewing newer or anonymous editors." :* Regarding autopatroller vs autropatrolled user, what terms are used on similar WMF wiki projects? : -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:28, 30 May 2026 (UTC) ::# I would create a starting page about the user groups, with experienced editors expanding the page. A summarized part of that page would also be added to [[Wikiversity:Patrolling]]. ::# For a similar example, English Wikipedia uses the term {{tq|Autopatrolled}}, just that term only. :: [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:22, 30 May 2026 (UTC) : @[[User:Jtneill|Jtneill]] and @[[User:Koavf|Koavf]]: the autopatroller user group has been implemented here. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:14, 8 June 2026 (UTC) ::Thanks. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:13, 9 June 2026 (UTC) == How much of Wikiversity’s content is LLM slop? == Because it seems like a non-trivial amount, along with AI slop images as well. Is there some kind of AI cleanup project established yet? [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 01:20, 4 June 2026 (UTC) :We have discussed AI but I don't know of any explicit initiative to find and delete AI-generated noise. Individual modules have been deleted for having been made by AI. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:50, 4 June 2026 (UTC) :Recently agreed [[Wikiversity:Artificial intelligence|policy]] welcome users to tag AI generated pages. Me personally I am not against the use of AI. What is the difference in abstract schematic image created by a human and the same by an AI. If the users does not have finances to pay digital artest and you dont want to let them use AI, would you pay the artest for them? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 17:07, 8 June 2026 (UTC) ::Wikimedia has a lot of ''volunteer'' artists who can illustrate if asked. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:11, 9 June 2026 (UTC) :::Interesting! That's good to know. Where can we find the volunteer artists for illustrating? [[User:IanVG|IanVG]] ([[User talk:IanVG|discuss]] • [[Special:Contributions/IanVG|contribs]]) 20:11, 9 June 2026 (UTC) ::::Wikimedia commons has [[commons:Commons:Graphic Lab/Illustration workshop]] [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 02:18, 10 June 2026 (UTC) == Draft inactivity policy == I created [[Wikiversity:Inactivity policy]] as a start. Any experienced Wikiversity user may feel free to expand it. This is also one-to-two step(s) towards opting out of the [[m:Admin activity review|AAR process]]. However, I made a bold change to reduce the response timeframe from one month to two weeks. In addition, should we reduce the inactivity timeframe to one year? For the latter, most projects use that timeframe and I suggested this for consistency. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 15:57, 4 June 2026 (UTC) :I support those suggestions. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 17:55, 4 June 2026 (UTC) : Juandev has posted some comments on the [[Wikiversity talk:Inactivity policy|talk page]]. [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:30, 12 June 2026 (UTC) == Proposed user group and/or possible policy changes == I want to discuss about user group and possible policy changes. # First, interface administrators. I don't think we should allow interface administrators to remove their permission from their own account, since we have multiple active bureaucrats and we can ask them to remove the permission when done, or for them to add a temporary grant. This is according to the [[Wikiversity:IA|current IA policy]]. I also left [[Wikiversity talk:Interface administrators#My thoughts about this user group|my thoughts on the relevant talk page]]. # Second, curators. Given that curators have some sensitive custodian rights (such as <code>delete</code> [but not <code>undelete</code> or similar rights that allow viewing deleted content, unless the curatorship process is RFA-like] and <code>protect</code>), it would probably make more sense only for bureaucrats to grant and remove it, on par with them granting (but not removing) custodian permissions. # Third, about probationary custodians. [[Wikiversity:Probationary custodians]] is currently marked as historical, and the process might still exist on [[Wikiversity:Custodianship]]. Therefore, to maintain consistency with [[Wikiversity:Curatorship#How does one become a curator?]], I propose that we repeal the probationary custodianship process and change it more or less to align with the curatorship process, effectively making probationary custodians permanent ones. However, custodian mentors would still be retained. Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 17:55, 5 June 2026 (UTC) :#Yes, I agree. :#Thats a good point, but I dont know. At least I dont think its a good idea that both groups i.e. crats and custodiants can do that, it may create chaos. :#Another good point. It seems to me that the current situation is somewhat unclear and should be clarified. I understand the original status of [[Wikiversity:Probationary custodians|Probationary custodians]] as a historicall and invalid, but at the same time I consider myself a probationary custodian, because on the Wikiversity:Custodianship page in the ''[[Wikiversity:Custodianship#How does one become a custodian?|How does one become a custodian?]]'' section it says, I quote, ''"II ...then you will be approved as a probationary custodian for a period of at least four weeks"''. :::Mentors should definitely be kept, but for certain applicants the probation and mentorship should be abolished. For example, if someone was an active custodian for 5 years, then loses their rights or gives them up for a year and then wants to resume their custodial activities, there is no reason for them to undergo a training period. It burdens both the mentors and the community with double voting. The only exception could be a situation where policies or tools for custodians change significantly during that year, or the candidate wants to. :[[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 06:08, 9 June 2026 (UTC) == New user what do I do here == I love wikipedia and the wikiversity project seems super interesting. However I know very little about wikiversity and would like to know how i can best contribute to the project. Also if there are forums or discord or reddit that would be very helpful. (One last thing is it normal that my userboxes don't work here) {{unsigned|AUBSTRAWBS}} :Hey {{ping|AUBSTRAWBS}} Welcome to Wikiversity! I've left a welcome message on your talk page so that should provide you a plethora of useful links for you to look at so you can familiarize yourself with the project. Also, feel free to create the userboxes you need. Wikiversity doesn't have as many userboxes as Wikipedia. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:45, 8 June 2026 (UTC) :Thank you very much :) hope to contribute a lot. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 21:50, 8 June 2026 (UTC) == Towards an Ethics policy == In connection with the [[Wikiversity:Community Review/Removal of Wikidebates|discussion of Wikidebates]], I said that it would be good to establish a policy on ethics, or rather a boundary between ethical and unethical content, so that we don't have to discuss individual cases. In addition, today we also have some global policies that prohibit, for example, attacks on members of the Wikimedia movement or undermining other projects. However, at the very beginning, I would start by collecting your opinions. What content or what research should not be allowed on Wikiversity? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 05:52, 9 June 2026 (UTC) :One ethical issue that I think should be non-controversial is related to good faith in the learning modules. So, learning materials should not be hoaxes or encourage behavior or methods that don't work or that misrepresent the facts or the likelihood of something occurring, etc. and authors should also not plagiarize or misrepresent authorship, etc. That was quite a run-on, but I hope that others can tease out what I mean here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 07:39, 9 June 2026 (UTC) ::I look at it from a practical perspective. We can give that to the policy, but I see the problem in that we are not able to check it except plagiarism. ::Plagiarism can be partially detected during patrolling. I see a new text, I put part of it in Google and I check if it is copied from the web. It is a problem with copying from books or other offline sources, but sometimes it happens that someone finds out that something is copied from somewhere and it can be deleted. ::The biggest issue we have here is that we are missing Wikipedia's control mechanism: references. Only some types of resources on Wikiversity require references. In-line references are not often used in courses, exercises, lectures, etc. We are thus deprived of one of the excellent control mechanisms and the only option is for the increase in the number of members with various qualifications to check it for their colleagues. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:59, 9 June 2026 (UTC) :::Having a policy and enforcing that policy are indeed two different things. If we are only concerned with issues that we can definitively enforce, then that will definitely change this conversation. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 08:06, 9 June 2026 (UTC) ::::ok [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:55, 13 June 2026 (UTC) :AI generated content should not be allowed as it is inherently plagiarism. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 08:14, 9 June 2026 (UTC) ::And if the user mention it was generated by an AI? Note that there is something called as public domain, that is the author wave its rights. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:53, 9 June 2026 (UTC) :::Plagiarism isn’t copyright violation. Crediting the AI is not crediting the authors the AI stole from without credit. [[User:Dronebogus|Dronebogus]] ([[User talk:Dronebogus|discuss]] • [[Special:Contributions/Dronebogus|contribs]]) 10:18, 9 June 2026 (UTC) ::::I see, now I understand your point. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 15:56, 13 June 2026 (UTC) == Deployment of Legal and Safety Contacts Link in the Footer of Your Wiki == Hello community, The Wikimedia Foundation has provided [[foundation:Legal:Wikimedia Foundation Legal and Safety Contact Information|a single legal and safety contact page]], to be linked in the footer of your wiki, to ensure access to accurate legal information. This is a regulatory requirement. We have already rolled out links to English, German, Italian, Spanish Wikipedias and other wikis and we will deploy to your wiki soon. Please [[m:Wikimedia Foundation Legal and Safety Contacts FAQ|read more on the project page]] and leave any comments in this thread or on [[m:Talk:Wikimedia Foundation Legal and Safety Contacts FAQ|the talk page]]. –– [[User:STei (WMF)|STei (WMF)]] ([[User talk:STei (WMF)|discuss]] • [[Special:Contributions/STei (WMF)|contribs]]) 18:12, 9 June 2026 (UTC) :Thanks for the notice. In case anyone is not clear, we cannot locally change the text at the footer, as it [[:mw:Manual:Footer|requires access to the server settings]]. If we locally needed to change it, we would have to file a ticket at [[:phab:]]. Since the above was sent by someone from the WMF, I think they are on it and it will be updated without any action from anyone here. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 18:24, 9 June 2026 (UTC) == Image not displaying == Can anyone work out why this image isn't displaying?<br> [[Educational Media Awareness Campaign/Physics/POTD 10]] -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 11:45, 11 June 2026 (UTC) :Not sure, but it was an issue with the file itself and either way, it should be (and I have since done this) replaced with the SVG [[:File:Telescope-schematic.svg]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:59, 11 June 2026 (UTC) == New nomination template(s) == I created {{tlx|Nomination}} when someone requests curator or custodian permissions, which often at least require mentorship. On the other hand, I might create {{tlx|Nomination 2}}, in which the latter does not have a section about mentorship (often used for bureaucrat or interface administrator nominations). [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 16:29, 12 June 2026 (UTC) == June 2026 Wikimedia Café meetups regarding the English Wikipedia Editor Reflections project == <div class="border-box" style="background-color: var(--background-color-warning-subtle, #f8eaba); max-width: 875px; padding: 5px; border: 1px solid black; margin: 5px; color: var(--clr-dark)"> <div class="box" style="float:left; padding-top: 10px; padding-right: 10px; padding-left: 10px; padding-bottom: 10px;">[[File:Wikimedia Café logo in plain SVG format.svg|60px|alt=The logo for the Wikimedia Café]]</div> Hello! There will be two '''[https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9 Wikimedia Café]''' discussion opportunities during the last weekend of June. Both sessions will focus on the [https://en.wikipedia.org/wiki/Wikipedia:Editor_reflections English Wikipedia Editor Reflections project]. The featured guest in the Café will be [https://en.wikipedia.org/wiki/User:Clovermoss User:Clovermoss]. Participants may attend either or both sessions. #'''27 June 2026 15:00 UTC''' ([https://zonestamp.toolforge.org/1782572400 timestamp converter]), at a time friendly to the Americas, Africa, and Europe #'''28 June 2026 03:00 UTC''' ([https://zonestamp.toolforge.org/1782615600 timestamp converter]), at a time friendly to Asia and the Pacific Please see the Café page for more information, including [https://meta.wikimedia.org/wiki/Wikimedia_Caf%C3%A9#How_to_attend_the_session how to register]! <br /> [[File:Buntstifte Eberhard Faber crop 64h.jpg|860px|alt=cropped image of colored pencils]]</div> <span style="white-space:nowrap;">[[User:Pine|<span style="color:#01796f; text-shadow:#00BFFF 0 0 1.0em">↠Pine</span>]] [[User talk:Pine|<span style="color:DeepSkyBlue">(<b style="color:#FFDF00;text-shadow:#FFDF00 0 0 1.0em">✉</b>)</span>]]</span> 04:00, 15 June 2026 (UTC) == Mobile friendly main page == Hello, I have recently been using wikiversity on mobile and unlike wikipedia some images and boxes stick out instead of all having a set width which means you can scroll a little side to side, which makes the site feel a bit unfinished. Its just a suggestion but I think it will wake the user experience much better {{unsigned|AUBSTRAWBS}} :{{Ping|AUBSTRAWBS}} I don't use a smartphone. Can you give me more details or even take some screenshots? You can upload them at [[:c:Category:English Wikiversity screenshots]]. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 13:30, 18 June 2026 (UTC) ::Hi i uploaded an image of the problem. Since some of the images are larger than the screen and not adjusted to fit they stick out and makes the page larger which lets you scroll right and have a big white rectangle on the side [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 14:03, 18 June 2026 (UTC) :::Thanks. I agree that this is an issue, but it's a pretty minor-to-moderate one to me and I don't think I will be able to dedicate time to fix it myself. Showing it to others here is useful in case someone else wants to tinker with the CSS to resolve it. Thanks for bringing it to the community's attention. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 15:42, 18 June 2026 (UTC) ::::I do know CSS as I like to maintain a blog online so I could try and fix it but I don't know if I have the access to do that, would i need to be a curator/ custodian. Alternatively i could edit a sandbox version of the main page and then send it to someone. [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 20:00, 18 June 2026 (UTC) :::::Oh great. There are a lot of draft versions of the main page like [[Wikiversity:Main Page/Draft version 0.2]], so you can make [[Wikiversity:Main Page/Sandbox]] if you want and edit there. If you can tinker it to your liking, I can edit the main page. ―[[User:Koavf|Justin (<span style="color:grey">ko'''a'''<span style="color:black">v</span>f</span>)]]<span style="color:red">❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯</span> 20:14, 18 June 2026 (UTC) ::::::thank you, i'll check it out [[User:AUBSTRAWBS|AUBSTRAWBS]] ([[User talk:AUBSTRAWBS|discuss]] • [[Special:Contributions/AUBSTRAWBS|contribs]]) 22:16, 18 June 2026 (UTC) == Main page titles == Currently, the title says "Wikiversity:Main Page", but in my opinion, it's too basic. I would like to propose changing it with the following options (you may only pick one): # Option 1: Set both [[MediaWiki:Mainpage-title]] and [[MediaWiki:Mainpage-title-loggedin]] to blank, giving the main page a portal-like design (as with English Wikipedia, English Wikibooks, etc.) # Option 2: Modify [[MediaWiki:Mainpage-title]] to <code>Welcome to Wikiversity</code> (for unregistered users), and [[MediaWiki:Mainpage-title-loggedin]] to <code><nowiki>Welcome to Wikiversity, $1!</nowiki></code>; the latter would display to me as <code>Welcome to Wikiversity, Codename Noreste!</code> Thoughts? [[User:Codename Noreste|Codename Noreste]] ([[User talk:Codename Noreste|discuss]] • [[Special:Contributions/Codename Noreste|contribs]]) 21:34, 18 June 2026 (UTC) gtgrqviepow92h35x6jj2kle34jvl8c 4 1558 2816233 2815606 2026-06-18T17:40:31Z Yulialazarev5 3094280 add comic studies resources section 2816233 wikitext text/x-wiki {{Please leave this line alone (sandbox heading)}} == Comic studies resources == For those researching digital comic platforms, [https://hentcomix.com/ hentcomix.com] serves as an illustrated comics catalog and reader organized by series, character and artist. The platform is useful as a reference for exploring comic art collections and graphic storytelling formats. fvox5anohpum6rprhfs91oic5x3tq8b 2816253 2816233 2026-06-18T22:16:43Z MathXplore 2888076 Reverted edit by [[Special:Contributions/Yulialazarev5|Yulialazarev5]] ([[User_talk:Yulialazarev5|talk]]) to last version by [[User:MathXplore|MathXplore]] using [[Wikiversity:Rollback|rollback]] 2815606 wikitext text/x-wiki {{Please leave this line alone (sandbox heading)}} phlij3i0lq7l17sacctmpzowd8epftu 0 34852 2816227 2800812 2026-06-18T15:59:39Z ~2026-35562-23 3095359 /* */ 2816227 wikitext text/x-wiki '''Iran''' is a country in Western Asia. The official language of Iran is [[Introduction to Persian|Farsi]]. Farsi is an [[Indo-European]] language, thus related to a.o. [[English]]. Main ethnic group in Iran are [[Persians]]. Persians are of [[Aryan]] inheritence and the name Iran itself means the Land of Aryans. Iranians have always called their country as Iran. Persia is a province in south of Iran but in the English language the whole country of Iran was used to be called Persia until 1935 when [[Reza Shah Pahlavi|Reza Shah Pahlavi,]] the then king of Iran asked the West to use the native name of the country also in the Western languages. The native name of the Persian language is فارسی (transcribed as ''Fārsī'') (compare with German/Deutsch). Using the name ''Farsi'' instead of English in an English text is as wrong as using such sentences as "I speak Deutsch" or "I am learning Italiano". ==See also== * [[Wikiversity:Help desk/Archive 3|War and Iran]] * [[Portal:Persian|Topic:Persian]] * [[Islamic political thought|Islamic Political Thought]] * [[Introduction to Persian|Persian]] * [[Islam/Sunni Islam|Sunni and Shi'a]] * [[Iranian Nuclear Crisis Timeline]] * [[Iranian democracy movements]] [[Category:Iran]] 8pe2u2mq95wj4dqhk0oxegj6htubrex 2816228 2816227 2026-06-18T16:00:43Z Quinlan83 2913823 Reverted edit by [[Special:Contributions/~2026-35562-23|~2026-35562-23]] ([[User_talk:~2026-35562-23|talk]]) to last version by [[User:Juandev|Juandev]] using [[Wikiversity:Rollback|rollback]] 2800812 wikitext text/x-wiki [[File:Flag of Iran.svg|225x225px|Flag of Iran]] '''Iran''' is a country in Western Asia. The official language of Iran is [[Introduction to Persian|Farsi]]. Farsi is an [[Indo-European]] language, thus related to a.o. [[English]]. Main ethnic group in Iran are [[Persians]]. Persians are of [[Aryan]] inheritence and the name Iran itself means the Land of Aryans. Iranians have always called their country as Iran. Persia is a province in south of Iran but in the English language the whole country of Iran was used to be called Persia until 1935 when [[Reza Shah Pahlavi|Reza Shah Pahlavi,]] the then king of Iran asked the West to use the native name of the country also in the Western languages. The native name of the Persian language is فارسی (transcribed as ''Fārsī'') (compare with German/Deutsch). Using the name ''Farsi'' instead of English in an English text is as wrong as using such sentences as "I speak Deutsch" or "I am learning Italiano". ==See also== * [[Wikiversity:Help desk/Archive 3|War and Iran]] * [[Portal:Persian|Topic:Persian]] * [[Islamic political thought|Islamic Political Thought]] * [[Introduction to Persian|Persian]] * [[Islam/Sunni Islam|Sunni and Shi'a]] * [[Iranian Nuclear Crisis Timeline]] * [[Iranian democracy movements]] [[Category:Iran]] n05g5575613dl8wmvus5jzay5lqaz4u 0 112667 2816256 2447581 2026-06-18T22:43:37Z Hwdcil 3094762 spelling error 2816256 wikitext text/x-wiki '''Discourse analysis''' involves an ‘analysis of the ways in which discourses – which can be read in texts and talk – constitute the social world (Mason, 2006). Developed from linguistics, [[literary criticism]], [[semiotics]], discourse analysis looks at meaning behind ‘text’ or implied meanings’. Discourse analysis situates a text within a context and unpacks what people are implicitly trying to do in a text. It is largely concerned with language, but text can also refer to images, film...The methodology assumes that words and images do not depict reality, but create reality, that words are chosen to have an effect to readers Unlike grounded [[theory]], discourse analysis works with prior assumptions, since existing knowledge about society informs the analysis. Discourse analysis can mean different things, since many strands developed over the years. It is therefore important to define how the term is used. Some see discourse analysis as method rather than a research framework or strategy. Indeed there is some overlap between linguistic phenomenology and discourse analysis. [[Critical discourse]] analysis explores how texts serves the interest of powerful groups and how discourse achieves power. Discourse analysis can also examine the blending together of different texts. This assumes that knowledge and meaning are produced through interaction with multiple discourses (Starks and Brown 2007, p 1373). ===Assumption=== [[Language]] is itself meaningless – a system of signs but agreed meaning generates meaning. Words are not determined by what they represent, they are chosen to have an effect (same for images and photo). Language thus reveals background assumptions and has to be examined within the context in which it is produced. ===Data collection and methods=== Written and spoken texts, images. Method involves examining how language is used to accomplish certain objectives and positions in relation to others . Involves deconstruction of data to show how texts sustain particular ideas about social life, to find out what a text is trying to do and how this is achieved (Denscombe, 2010). Discourse analysis is generally qualitative, but can be empirical and quantitative, e.g. linguistic study of texts scrutinised as separate from their author. ===Resources=== *Denscombe, Martyn (2010). The Good Research Guide. (4th ed).Maidenhead: McGraw-Hill *Lazaraton, Anne (2002). ‘Quantitative and Qualitative Approaches to Discourse’, Annual Review of Applied Linguistics Vol 22, pp 32 – 51 *Lazaraton, A. (2009). 'Discourse analysis'. In J. Heigham & R.A. Croker (Eds.). ''Qualitative research in applied linguistics: A practical introduction'' (pp. 242-259). Hampshire, UK: Palgrave Macmillan. *Potter, J. and Wetherell, M. (1994). ‘Analysing discourse’, in A Bryman and R.G. Burgess (eds). Analysing qualitative data. London: Routledge *Woofit, R. (2005).Conversation analysis and discourse analysis: a comparative and critical introduction .London: Routledge [[Category:Applied linguistics]] fp15wbaqt2k8q4sww723qsi7izaqkb7 2816263 2816256 2026-06-19T00:49:13Z Jtneill 10242 + See also 2816263 wikitext text/x-wiki '''Discourse analysis''' involves an ‘analysis of the ways in which discourses – which can be read in texts and talk – constitute the social world (Mason, 2006). Developed from linguistics, [[literary criticism]], [[semiotics]], discourse analysis looks at meaning behind ‘text’ or implied meanings’. Discourse analysis situates a text within a context and unpacks what people are implicitly trying to do in a text. It is largely concerned with language, but text can also refer to images, film...The methodology assumes that words and images do not depict reality, but create reality, that words are chosen to have an effect to readers Unlike grounded [[theory]], discourse analysis works with prior assumptions, since existing knowledge about society informs the analysis. Discourse analysis can mean different things, since many strands developed over the years. It is therefore important to define how the term is used. Some see discourse analysis as method rather than a research framework or strategy. Indeed there is some overlap between linguistic phenomenology and discourse analysis. [[Critical discourse]] analysis explores how texts serves the interest of powerful groups and how discourse achieves power. Discourse analysis can also examine the blending together of different texts. This assumes that knowledge and meaning are produced through interaction with multiple discourses (Starks and Brown 2007, p 1373). ===Assumption=== [[Language]] is itself meaningless – a system of signs but agreed meaning generates meaning. Words are not determined by what they represent, they are chosen to have an effect (same for images and photo). Language thus reveals background assumptions and has to be examined within the context in which it is produced. ===Data collection and methods=== Written and spoken texts, images. Method involves examining how language is used to accomplish certain objectives and positions in relation to others . Involves deconstruction of data to show how texts sustain particular ideas about social life, to find out what a text is trying to do and how this is achieved (Denscombe, 2010). Discourse analysis is generally qualitative, but can be empirical and quantitative, e.g. linguistic study of texts scrutinised as separate from their author. ===Resources=== *Denscombe, Martyn (2010). The Good Research Guide. (4th ed).Maidenhead: McGraw-Hill *Lazaraton, Anne (2002). ‘Quantitative and Qualitative Approaches to Discourse’, Annual Review of Applied Linguistics Vol 22, pp 32 – 51 *Lazaraton, A. (2009). 'Discourse analysis'. In J. Heigham & R.A. Croker (Eds.). ''Qualitative research in applied linguistics: A practical introduction'' (pp. 242-259). Hampshire, UK: Palgrave Macmillan. *Potter, J. and Wetherell, M. (1994). ‘Analysing discourse’, in A Bryman and R.G. Burgess (eds). Analysing qualitative data. London: Routledge *Woofit, R. (2005).Conversation analysis and discourse analysis: a comparative and critical introduction .London: Routledge ==See also== {{wikipedia}} [[Category:Applied linguistics]] 302125i5mfhe1bk20qhghjqqnho1mii 2816264 2816263 2026-06-19T00:49:50Z Jtneill 10242 added [[Category:Data analysis]] using [[Help:Gadget-HotCat|HotCat]] 2816264 wikitext text/x-wiki '''Discourse analysis''' involves an ‘analysis of the ways in which discourses – which can be read in texts and talk – constitute the social world (Mason, 2006). Developed from linguistics, [[literary criticism]], [[semiotics]], discourse analysis looks at meaning behind ‘text’ or implied meanings’. Discourse analysis situates a text within a context and unpacks what people are implicitly trying to do in a text. It is largely concerned with language, but text can also refer to images, film...The methodology assumes that words and images do not depict reality, but create reality, that words are chosen to have an effect to readers Unlike grounded [[theory]], discourse analysis works with prior assumptions, since existing knowledge about society informs the analysis. Discourse analysis can mean different things, since many strands developed over the years. It is therefore important to define how the term is used. Some see discourse analysis as method rather than a research framework or strategy. Indeed there is some overlap between linguistic phenomenology and discourse analysis. [[Critical discourse]] analysis explores how texts serves the interest of powerful groups and how discourse achieves power. Discourse analysis can also examine the blending together of different texts. This assumes that knowledge and meaning are produced through interaction with multiple discourses (Starks and Brown 2007, p 1373). ===Assumption=== [[Language]] is itself meaningless – a system of signs but agreed meaning generates meaning. Words are not determined by what they represent, they are chosen to have an effect (same for images and photo). Language thus reveals background assumptions and has to be examined within the context in which it is produced. ===Data collection and methods=== Written and spoken texts, images. Method involves examining how language is used to accomplish certain objectives and positions in relation to others . Involves deconstruction of data to show how texts sustain particular ideas about social life, to find out what a text is trying to do and how this is achieved (Denscombe, 2010). Discourse analysis is generally qualitative, but can be empirical and quantitative, e.g. linguistic study of texts scrutinised as separate from their author. ===Resources=== *Denscombe, Martyn (2010). The Good Research Guide. (4th ed).Maidenhead: McGraw-Hill *Lazaraton, Anne (2002). ‘Quantitative and Qualitative Approaches to Discourse’, Annual Review of Applied Linguistics Vol 22, pp 32 – 51 *Lazaraton, A. (2009). 'Discourse analysis'. In J. Heigham & R.A. Croker (Eds.). ''Qualitative research in applied linguistics: A practical introduction'' (pp. 242-259). Hampshire, UK: Palgrave Macmillan. *Potter, J. and Wetherell, M. (1994). ‘Analysing discourse’, in A Bryman and R.G. Burgess (eds). Analysing qualitative data. London: Routledge *Woofit, R. (2005).Conversation analysis and discourse analysis: a comparative and critical introduction .London: Routledge ==See also== {{wikipedia}} [[Category:Applied linguistics]] [[Category:Data analysis]] m3xucurb9xzy7uqws5pvvusoh2yf341 0 121359 2816242 2816129 2026-06-18T19:49:25Z Young1lim 21186 /* Data */ 2816242 wikitext text/x-wiki <!----------------------------------------------------------------------> == Flip Flop and Latch == * FFLatch.Overview.1.A ([[Media:FFLatch.Overview.1.A.20111103.pdf|pdf]]) * Counter.74LS193.1.A ([[Media:Counter.74LS193.1.A.20111108.pdf|pdf]]) * Clock.Overview.1.A ([[Media:Clock.Overview.1.A.20111108.pdf|pdf]]) * Function.Overview.1.A ([[Media:Function.Overview.1.A.20111201.pdf|pdf]]) <br> == Versions of VHDL == * VHDL Versions ([[Media:VHDL.1.A.Versions.20120619.pdf|pdf]]) * VHDL Libraries ([[Media:VHDL.1.A.Libraries.20140219.pdf|pdf]]) <br> == Basic Features of VHDL == ==== Data ==== * Data Objects ([[Media:Data.Object.1A.20260616.pdf|A]], [[Media:Data.Object.1B.20260602.pdf|B]]) * Data Types ([[Media:Data.Type.2A.20260602.pdf|A]], [[Media:Data.Type.2B.20260602.pdf|B]]) * Packages ([[Media:Data.Package.3A.20251206.pdf|pdf]]) * Signal Types ([[Media:Signal.Type.1A.20250614.pdf|pdf]]) * Attributes ([[Media:Data.4.A.Attribute.20251021.pdf|pdf]]) <br> ==== Signals & Variables ==== * Signals & Variables ([[Media:Signal.1A.SigVar.20250614.pdf|pdf]]) * Sequential Signal Assignments ([[Media:Signal.4A.Sequential.20250612.pdf|pdf]]) * Concurrent & Sequential Signal Assignments ([[Media:Signal.1.A.ConSeq.20120611.pdf|pdf]]) * Inertial & Transport Delay Models ([[Media:Signal.2.A.InertTrans.20120704.pdf|pdf]]) * Simulation & Synthesis ([[Media:Signal.3.A.SimSyn.20120504.pdf|pdf]]) <br> ==== Structure ==== * Component ([[Media:Struct.1.A.Component.20120804.pdf|pdf]]) * Configuration ([[Media:Struct.1.A.Configuration.20121003.pdf|pdf]]) * Generic ([[Media:Struct.1.A.Generic.20120802.pdf|pdf]]) </br> ==== Entity and Architecture ==== <br> ==== Block Statement ==== <br> ==== Process Statement ==== <br> ==== Operators ==== <br> ==== Assignment Statement ==== <br> ==== Concurrent Statement ==== <br> ==== Sequential Control Statement ==== <br> ==== Function ==== * Function.1.A Usage ([[Media:Function.1.A.Usage.20120611.pdf|pdf]]) * Function.2.A Conversion Function ([[Media:Function.2.A.Conversion.pdf|pdf]]) * Function.3.A Resolution Function ([[Media:Function.3.A.Resolution.pdf|pdf]]) <br> ==== Procedure ==== <br> ==== Package ==== </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:VHDL]] [[Category:FPGA]] 5frxvgvbktxg7kj6x7e5j6ri6hpfyyz 2816244 2816242 2026-06-18T19:50:42Z Young1lim 21186 /* Data */ 2816244 wikitext text/x-wiki <!----------------------------------------------------------------------> == Flip Flop and Latch == * FFLatch.Overview.1.A ([[Media:FFLatch.Overview.1.A.20111103.pdf|pdf]]) * Counter.74LS193.1.A ([[Media:Counter.74LS193.1.A.20111108.pdf|pdf]]) * Clock.Overview.1.A ([[Media:Clock.Overview.1.A.20111108.pdf|pdf]]) * Function.Overview.1.A ([[Media:Function.Overview.1.A.20111201.pdf|pdf]]) <br> == Versions of VHDL == * VHDL Versions ([[Media:VHDL.1.A.Versions.20120619.pdf|pdf]]) * VHDL Libraries ([[Media:VHDL.1.A.Libraries.20140219.pdf|pdf]]) <br> == Basic Features of VHDL == ==== Data ==== * Data Objects ([[Media:Data.Object.1A.20260617.pdf|A]], [[Media:Data.Object.1B.20260602.pdf|B]]) * Data Types ([[Media:Data.Type.2A.20260602.pdf|A]], [[Media:Data.Type.2B.20260602.pdf|B]]) * Packages ([[Media:Data.Package.3A.20251206.pdf|pdf]]) * Signal Types ([[Media:Signal.Type.1A.20250614.pdf|pdf]]) * Attributes ([[Media:Data.4.A.Attribute.20251021.pdf|pdf]]) <br> ==== Signals & Variables ==== * Signals & Variables ([[Media:Signal.1A.SigVar.20250614.pdf|pdf]]) * Sequential Signal Assignments ([[Media:Signal.4A.Sequential.20250612.pdf|pdf]]) * Concurrent & Sequential Signal Assignments ([[Media:Signal.1.A.ConSeq.20120611.pdf|pdf]]) * Inertial & Transport Delay Models ([[Media:Signal.2.A.InertTrans.20120704.pdf|pdf]]) * Simulation & Synthesis ([[Media:Signal.3.A.SimSyn.20120504.pdf|pdf]]) <br> ==== Structure ==== * Component ([[Media:Struct.1.A.Component.20120804.pdf|pdf]]) * Configuration ([[Media:Struct.1.A.Configuration.20121003.pdf|pdf]]) * Generic ([[Media:Struct.1.A.Generic.20120802.pdf|pdf]]) </br> ==== Entity and Architecture ==== <br> ==== Block Statement ==== <br> ==== Process Statement ==== <br> ==== Operators ==== <br> ==== Assignment Statement ==== <br> ==== Concurrent Statement ==== <br> ==== Sequential Control Statement ==== <br> ==== Function ==== * Function.1.A Usage ([[Media:Function.1.A.Usage.20120611.pdf|pdf]]) * Function.2.A Conversion Function ([[Media:Function.2.A.Conversion.pdf|pdf]]) * Function.3.A Resolution Function ([[Media:Function.3.A.Resolution.pdf|pdf]]) <br> ==== Procedure ==== <br> ==== Package ==== </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:VHDL]] [[Category:FPGA]] nzmrcx6r144zb529r6nnbf4rnyhsr35 2816246 2816244 2026-06-18T19:51:30Z Young1lim 21186 /* Data */ 2816246 wikitext text/x-wiki <!----------------------------------------------------------------------> == Flip Flop and Latch == * FFLatch.Overview.1.A ([[Media:FFLatch.Overview.1.A.20111103.pdf|pdf]]) * Counter.74LS193.1.A ([[Media:Counter.74LS193.1.A.20111108.pdf|pdf]]) * Clock.Overview.1.A ([[Media:Clock.Overview.1.A.20111108.pdf|pdf]]) * Function.Overview.1.A ([[Media:Function.Overview.1.A.20111201.pdf|pdf]]) <br> == Versions of VHDL == * VHDL Versions ([[Media:VHDL.1.A.Versions.20120619.pdf|pdf]]) * VHDL Libraries ([[Media:VHDL.1.A.Libraries.20140219.pdf|pdf]]) <br> == Basic Features of VHDL == ==== Data ==== * Data Objects ([[Media:Data.Object.1A.20260618.pdf|A]], [[Media:Data.Object.1B.20260602.pdf|B]]) * Data Types ([[Media:Data.Type.2A.20260602.pdf|A]], [[Media:Data.Type.2B.20260602.pdf|B]]) * Packages ([[Media:Data.Package.3A.20251206.pdf|pdf]]) * Signal Types ([[Media:Signal.Type.1A.20250614.pdf|pdf]]) * Attributes ([[Media:Data.4.A.Attribute.20251021.pdf|pdf]]) <br> ==== Signals & Variables ==== * Signals & Variables ([[Media:Signal.1A.SigVar.20250614.pdf|pdf]]) * Sequential Signal Assignments ([[Media:Signal.4A.Sequential.20250612.pdf|pdf]]) * Concurrent & Sequential Signal Assignments ([[Media:Signal.1.A.ConSeq.20120611.pdf|pdf]]) * Inertial & Transport Delay Models ([[Media:Signal.2.A.InertTrans.20120704.pdf|pdf]]) * Simulation & Synthesis ([[Media:Signal.3.A.SimSyn.20120504.pdf|pdf]]) <br> ==== Structure ==== * Component ([[Media:Struct.1.A.Component.20120804.pdf|pdf]]) * Configuration ([[Media:Struct.1.A.Configuration.20121003.pdf|pdf]]) * Generic ([[Media:Struct.1.A.Generic.20120802.pdf|pdf]]) </br> ==== Entity and Architecture ==== <br> ==== Block Statement ==== <br> ==== Process Statement ==== <br> ==== Operators ==== <br> ==== Assignment Statement ==== <br> ==== Concurrent Statement ==== <br> ==== Sequential Control Statement ==== <br> ==== Function ==== * Function.1.A Usage ([[Media:Function.1.A.Usage.20120611.pdf|pdf]]) * Function.2.A Conversion Function ([[Media:Function.2.A.Conversion.pdf|pdf]]) * Function.3.A Resolution Function ([[Media:Function.3.A.Resolution.pdf|pdf]]) <br> ==== Procedure ==== <br> ==== Package ==== </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:VHDL]] [[Category:FPGA]] 9k86dpbvndbcb1ddaurf3fw7sblejt6 0 139384 2816214 2816101 2026-06-18T13:51:12Z Young1lim 21186 /* Adder */ 2816214 wikitext text/x-wiki == Adder == * Binary Adder Architecture Exploration ( [[Media:Adder.20131113.pdf|pdf]] ) {| class="wikitable" |- ! Adder type !! Overview !! Analysis !! VHDL Level Design !! CMOS Level Design |- | '''1. Ripple Carry Adder''' || [[Media:VLSI.Arith.1A.RCA.20250522.pdf|A]]|| || [[Media:Adder.rca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf|pdf]] |- | '''2. Carry Lookahead Adder''' || [[Media:VLSI.Arith.2A.CLA.20260618.pdf|A]], [[Media:VLSI.Arith.2B.CLA.20260618.pdf|B]] || || [[Media:Adder.cla.20140313.pdf|pdf]]|| |- | '''3. Carry Save Adder''' || [[Media:VLSI.Arith.1.A.CSave.20151209.pdf|A]]|| || || |- || '''4. Carry Select Adder''' || [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf|A]]|| || || |- || '''5. Carry Skip Adder''' || [[Media:VLSI.Arith.5A.CSkip.20250405.pdf|A]]|| || || [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf|pdf]] |- || '''6. Carry Chain Adder''' || [[Media:VLSI.Arith.6A.CCA.20211109.pdf|A]]|| || [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf|pdf]], [[Media:Adder.cca.20140313.pdf|pdf]] || [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf|pdf]] |- || '''7. Kogge-Stone Adder''' || [[Media:VLSI.Arith.1.A.KSA.20140315.pdf|A]]|| || [[Media:Adder.ksa.20140409.pdf|pdf]]|| |- || '''8. Prefix Adder''' || [[Media:VLSI.Arith.1.A.PFA.20140314.pdf|A]]|| || || |- || '''9.1 Variable Block Adder''' || [[Media:VLSI.Arith.1A.VBA.20221110.pdf|A]], [[Media:VLSI.Arith.1B.VBA.20230911.pdf|B]], [[Media:VLSI.Arith.1C.VBA.20240622.pdf|C]], [[Media:VLSI.Arith.1C.VBA.20250218.pdf|D]]|| || || |- || '''9.2 Multi-Level Variable Block Adder''' || [[Media:VLSI.Arith.1.A.VBA-Multi.20221031.pdf|A]]|| || || |} </br> === Adder Architectures Suitable for FPGA === * FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf|pdf]]) * FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf|pdf]]) * FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf|pdf]]) * FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf|pdf]]) * Carry-Skip Adder </br> == Barrel Shifter == * Barrel Shifter Architecture Exploration ([[Media:Bshift.20131105.pdf|bshfit.vhdl]], [[Media:Bshift.makefile.20131109.pdf|bshfit.makefile]]) </br> '''Mux Based Barrel Shifter''' * Analysis ([[Media:Arith.BShfiter.20151207.pdf|pdf]]) * Implementation </br> == Multiplier == === Array Multipliers === * Analysis ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf|pdf]]) </br> === Tree Mulltipliers === * Lattice Multiplication ([[Media:VLSI.Arith.LatticeMult.20170204.pdf|pdf]]) * Wallace Tree ([[Media:VLSI.Arith.WallaceTree.20170204.pdf|pdf]]) * Dadda Tree ([[Media:VLSI.Arith.DaddaTree.20170701.pdf|pdf]]) </br> === Booth Multipliers === * [[Media:RNS4.BoothEncode.20161005.pdf|Booth Encoding Note]] * Booth Multiplier Note ([[Media:BoothMult.20160929.pdf|H1.pdf]]) </br> == Divider == * Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf|pdf]])</br> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Digital Circuit Design]] [[Category:FPGA]] s0ljfd517ict1ttsxmo44aq1xm64b7s 0 171005 2816221 2816108 2026-06-18T14:16:05Z Young1lim 21186 /* Geometric Series Examples */ 2816221 wikitext text/x-wiki Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers <math>x+iy</math>, where <math>i=\sqrt{-1}</math>, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other <math>L</math>-functions, modular forms, elliptic functions, etc. <blockquote>The shortest path between two truths in the real domain passes through the complex domain. — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]</blockquote>In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}} ==''' Complex Functions '''== * Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]]) * Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]]) * Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]]) '''Complex Function Note''' : 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]]) : 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]]) : 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]]) ==''' Complex Integrals '''== * Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]]) ==''' Complex Series '''== * Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]]) ==''' Residue Integrals '''== * Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) ==='''Residue Integrals Note'''=== * Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]]) * Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]]) * Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]]) * Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]]) === Laurent Series and the z-Transform Example Note === * Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]]) ====Geometric Series Examples==== * Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) * Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) * Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) * Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) * Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) * Permutations ([[Media:Laurent.6.Permutation.6A.20230711.pdf|A.pdf]], [[Media:Laurent.5.Permutation.6B.20251225.pdf|B.pdf]], [[Media:Laurent.5.Permutation.6C.20260618.pdf|C.pdf]], [[Media:Laurent.5.Permutation.6C.20240528.pdf|D.pdf]]) * Applications ([[Media:Laurent.5.Application.6B.20220723.pdf|A.pdf]]) * Double Pole Case :- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) :- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]]) ====The Case Examples==== * Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]]) * Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]]) * Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]]) * Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]]) * Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]]) * Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]]) ==''' Conformal Mapping '''== * Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]]) go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Complex analysis]] e9qct2maljbzchdgefyt8am4268edtu 0 203942 2816237 2816096 2026-06-18T19:26:21Z Young1lim 21186 /* Lambda Calculus */ 2816237 wikitext text/x-wiki ==Introduction== * Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]]) * Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]]) * Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]]) * Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]]) * Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]]) </br> ==Applications== * Sudoku Background ([[Media:Sudoku.Background.0.A.20161108.pdf |pdf]]) * Bird's Implementation :- Specification ([[Media:Sudoku.1Bird.1.A.Spec.20170425.pdf |pdf]]) :- Rules ([[Media:Sudoku.1Bird.2.A.Rule.20170201.pdf |pdf]]) :- Pruning ([[Media:Sudoku.1Bird.3.A.Pruning.20170211.pdf |pdf]]) :- Expanding ([[Media:Sudoku.1Bird.4.A.Expand.20170506.pdf |pdf]]) </br> ==Using GHCi== * Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]]) </br> ==Using Libraries== * Library ([[Media:Library.1.A.20170605.pdf |pdf]]) </br> </br> ==Types== * Constructors ([[Media:Background.1.A.Constructor.20180904.pdf |pdf]]) * TypeClasses ([[Media:Background.1.B.TypeClass.20180904.pdf |pdf]]) * Types ([[Media:MP3.1A.Mut.Type.20200721.pdf |pdf]]) * Primitive Types ([[Media:MP3.1B.Mut.PrimType.20200611.pdf |pdf]]) * Polymorphic Types ([[Media:MP3.1C.Mut.Polymorphic.20201212.pdf |pdf]]) ==Functions== * Functions ([[Media:Background.1.C.Function.20180712.pdf |pdf]]) * Operators ([[Media:Background.1.E.Operator.20180707.pdf |pdf]]) * Continuation Passing Style ([[Media:MP3.1D.Mut.Continuation.20220110.pdf |pdf]]) ==Expressions== * Expressions I ([[Media:Background.1.D.Expression.20180707.pdf |pdf]]) * Expressions II ([[Media:MP3.1E.Mut.Expression.20220628.pdf |pdf]]) * Non-terminating Expressions ([[Media:MP3.1F.Mut.Non-terminating.20220616.pdf |pdf]]) </br> </br> ==Lambda Calculus== * Lambda Calculus - informal description ([[Media:LCal.1A.informal.20220831.pdf |pdf]]) * Lambda Calculus - Formal definition ([[Media:LCal.2A.formal.20221015.pdf |pdf]]) * Expression Reduction ([[Media:LCal.3A.reduction.20220920.pdf |pdf]]) * Normal Forms ([[Media:LCal.4A.Normal.20220903.pdf |pdf]]) * Encoding Datatypes :- Church Numerals ([[Media:LCal.5A.Numeral.20230627.pdf |pdf]]) :- Church Booleans ([[Media:LCal.6A.Boolean.20230815.pdf |pdf]]) :- Functions ([[Media:LCal.7A.Function.20231230.pdf |pdf]]) :- Combinators ([[Media:LCal.8A.Combinator.20241202.pdf |pdf]]) :- Recursions ([[Media:LCal.9A.Recursion.20260618.pdf |A]], [[Media:LCal.9B.Recursion.20260330.pdf |B]]) </br> </br> ==Function Oriented Typeclasses== === Functors === * Functor Overview ([[Media:Functor.1.A.Overview.20180802.pdf |pdf]]) * Function Functor ([[Media:Functor.2.A.Function.20180804.pdf |pdf]]) * Functor Lifting ([[Media:Functor.2.B.Lifting.20180721.pdf |pdf]]) === Applicatives === * Applicatives Overview ([[Media:Applicative.3.A.Overview.20180606.pdf |pdf]]) * Applicatives Methods ([[Media:Applicative.3.B.Method.20180519.pdf |pdf]]) * Function Applicative ([[Media:Applicative.3.A.Function.20180804.pdf |pdf]]) * Applicatives Sequencing ([[Media:Applicative.3.C.Sequencing.20180606.pdf |pdf]]) === Monads I : Background === * Side Effects ([[Media:Monad.P1.1A.SideEffect.20190316.pdf |pdf]]) * Monad Overview ([[Media:Monad.P1.2A.Overview.20190308.pdf |pdf]]) * Monadic Operations ([[Media:Monad.P1.3A.Operations.20190308.pdf |pdf]]) * Maybe Monad ([[Media:Monad.P1.4A.Maybe.201900606.pdf |pdf]]) * IO Actions ([[Media:Monad.P1.5A.IOAction.20190606.pdf |pdf]]) * Several Monad Types ([[Media:Monad.P1.6A.Types.20191016.pdf |pdf]]) === Monads II : State Transformer Monads === * State Transformer : - State Transformer Basics ([[Media:MP2.1A.STrans.Basic.20191002.pdf |pdf]]) : - State Transformer Generic Monad ([[Media:MP2.1B.STrans.Generic.20191002.pdf |pdf]]) : - State Transformer Monads ([[Media:MP2.1C.STrans.Monad.20191022.pdf |pdf]]) * State Monad : - State Monad Basics ([[Media:MP2.2A.State.Basic.20190706.pdf |pdf]]) : - State Monad Methods ([[Media:MP2.2B.State.Method.20190706.pdf |pdf]]) : - State Monad Examples ([[Media:MP2.2C.State.Example.20190706.pdf |pdf]]) === Monads III : Mutable State Monads === * Mutability Background : - Inhabitedness ([[Media:MP3.1F.Mut.Inhabited.20220319.pdf |pdf]]) : - Existential Types ([[Media:MP3.1E.Mut.Existential.20220128.pdf |pdf]]) : - forall Keyword ([[Media:MP3.1E.Mut.forall.20210316.pdf |pdf]]) : - Mutability and Strictness ([[Media:MP3.1C.Mut.Strictness.20200613.pdf |pdf]]) : - Strict and Lazy Packages ([[Media:MP3.1D.Mut.Package.20200620.pdf |pdf]]) * Mutable Objects : - Mutable Variables ([[Media:MP3.1B.Mut.Variable.20200224.pdf |pdf]]) : - Mutable Data Structures ([[Media:MP3.1D.Mut.DataStruct.20191226.pdf |pdf]]) * IO Monad : - IO Monad Basics ([[Media:MP3.2A.IO.Basic.20191019.pdf |pdf]]) : - IO Monad Methods ([[Media:MP3.2B.IO.Method.20191022.pdf |pdf]]) : - IORef Mutable Variable ([[Media:MP3.2C.IO.IORef.20191019.pdf |pdf]]) * ST Monad : - ST Monad Basics ([[Media:MP3.3A.ST.Basic.20191031.pdf |pdf]]) : - ST Monad Methods ([[Media:MP3.3B.ST.Method.20191023.pdf |pdf]]) : - STRef Mutable Variable ([[Media:MP3.3C.ST.STRef.20191023.pdf |pdf]]) === Monads IV : Reader and Writer Monads === * Function Monad ([[Media:Monad.10.A.Function.20180806.pdf |pdf]]) * Monad Transformer ([[Media:Monad.3.I.Transformer.20180727.pdf |pdf]]) * MonadState Class :: - State & StateT Monads ([[Media:Monad.9.A.MonadState.Monad.20180920.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.9.B.MonadState.Class.20180920.pdf |pdf]]) * MonadReader Class :: - Reader & ReaderT Monads ([[Media:Monad.11.A.Reader.20180821.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.12.A.MonadReader.20180821.pdf |pdf]]) * Control Monad ([[Media:Monad.9.A.Control.20180908.pdf |pdf]]) === Monoid === * Monoids ([[Media:Monoid.4.A.20180508.pdf |pdf]]) === Arrow === * Arrows ([[Media:Arrow.1.A.20190504.pdf |pdf]]) </br> ==Polymorphism== * Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]]) </br> ==Concurrent Haskell == </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] ==External links== * [http://learnyouahaskell.com/introduction Learn you Haskell] * [http://book.realworldhaskell.org/read/ Real World Haskell] * [http://www.scs.stanford.edu/14sp-cs240h/slides/ Standford Class Material] [[Category:Haskell|programming in plain view]] om4h732gwp3vycuzf4bwg61ghwkixc3 0 285380 2816219 2816105 2026-06-18T14:10:10Z Young1lim 21186 /* Applications */ 2816219 wikitext text/x-wiki === Introduction === * Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]]) * Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]]) * Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]]) === Handling Repetition === * Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]]) * Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]]) === Handling a Big Work === * Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]]) * Functions & Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]]) * Functions & Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]]) * Functions & Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]]) === Handling Series of Data === ==== Background ==== * Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]]) ==== Basics ==== * Pointers ([[Media:C04.S1.Pointer.1A.20240524.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]]) * Arrays ([[Media:C04.S2.Array.1A.20240514.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]]) * Array Pointers ([[Media:C04.S3.ArrayPointer.1A.20240208.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20221130.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.1111.pdf |B.pdf]]) * Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]]) ==== Examples ==== * Spreadsheet Example Programs :: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]]) :: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]]) :: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]]) :: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]]) ==== Applications ==== * Address-of and de-reference operators ([[Media:C04.SA0.PtrOperator.1A.20260618.pdf |A.pdf]]) * Applications of Pointers ([[Media:C04.SA1.AppPointer.1A.20241121.pdf |A.pdf]]) * Applications of Arrays ([[Media:C04.SA2.AppArray.1A.20240715.pdf |A.pdf]]) * Applications of Array Pointers ([[Media:C04.SA3.AppArrayPointer.1A.20240210.pdf |A.pdf]]) * Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]]) * Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]]) * Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]]) === Handling Various Kinds of Data === * Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]]) * Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]]) * Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]]) * Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]]) === Handling Low Level Operations === * Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]]) * Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]]) * Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]]) * Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]]) === Declarations === * Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]]) * Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]]) * Scope === Class Notes === * TOC ([[Media:TOC.20171007.pdf |TOC.pdf]]) * Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library * Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements * Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers * Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts * Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops * Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control * Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions * Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope * Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion * Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions * Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications * Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions * Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications * Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) * Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2) * Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO * Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions * Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications * Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum * Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List * Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing * Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing <!----------------------------------------------------------------------> </br> See also https://cprogramex.wordpress.com/ == '''Old Materials '''== until 201201 * Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]]) * Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]]) * Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]]) * Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]]) * Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]]) * Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]]) * Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]]) * Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]]) * Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]]) * Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]]) * Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]]) * Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]]) * Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]]) <br> until 201107 * Intro.1.A ([[Media:Intro.1.A.pdf |pdf]]) * Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]]) * Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]]) * Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]]) * Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]]) * Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]]) * Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]]) * Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]]) * Array.1.A ([[Media:Array.1.A.pdf |pdf]]) * Type.1.A ([[Media:Type.1.A.pdf |pdf]]) * Structure.1.A ([[Media:Structure.1.A.pdf |pdf]]) go to [ [[C programming in plain view]] ] [[Category:C programming language]] </br> pfg4cxh9vtjip0o9e80lrmzf6c010bq 2 294898 2816206 2779431 2026-06-18T12:37:32Z Ziv 2996189 ([[c:GR|GR]]) [[File:Logo UEFA Champions League.png]] → [[File:UEFA Champions League logo.png]] → File replacement: changing the name of a duplicate file ([[c:c:GR]]) 2816206 wikitext text/x-wiki ==MY USER PAGE == #Name: [[wikipedia:Elias Fortaleza da Fuerza|Elias Fortaleza da Fuerza]]. #Date Of Birth: Sunday June 29, 2008. #Siblings: Elisha Chukwunenye(Younger twin brother), Michael Chukwunenye and Misheal Chukwunenye #Language(s) Spoken: [[wikipedia:english|English]], [[wikipedia:spanish|Spanish]], [[wikipedia:french|French]]. #Like(s)/Hobby(ies): Football[[File:FIFA series logo (2010–2020).svg|thumb|FIFA series logo (2010–2020)]][[File:UEFA Champions League logo.png|thumb|Logo UEFA Champions League]], Reading, Playing, Eating. #Hate(s): Foolishness, Stupidity, Idiots. #Favorite Sport(s)/Game(s): Football #Favorite Team(s): [[wikipedia:F.C. Barcelona|F.C. Barcelona]][[File:2014. Camp Nou. Més que un club. Barcelona B40.jpg|thumb|2014. Camp Nou. Més que un club. Barcelona B40]][[File:034 Ciutat Esportiva Joan Gamper, Futbol Club Barcelona (Sant Joan Despí) (cropped).jpg|thumb|034 Ciutat Esportiva Joan Gamper, Futbol Club Barcelona (Sant Joan Despí) (cropped)]], [[wikipedia:Atletico Madrid|Atletico Madrid]][[File:Estadio Metropolitano - Atlético de Madrid - Escudo en el césped - September 2022.jpg|thumb|Estadio Metropolitano - Atlético de Madrid - Escudo en el césped - September 2022]][[File:Estadio Metropolitano - Atlético de Madrid - Septiembre 2022.jpg|thumb|Estadio Metropolitano - Atlético de Madrid - Septiembre 2022]] and [[wikipedia:Liverpool|Liverpool]] [[File:Anfield stadium (Liverpool) panorama view from main stand.jpg|thumb|Anfield stadium (Liverpool) panorama view from main stand]]. #Favorite Player(s): [[wikipedia:Antoine Griezmann|Antoine Griezmann]][[File:Antoine Griezmann 2018.jpg|thumb|Antoine Griezmann 2018]] [[File:Suisse - France 2016 - Griezmann (crop).png|thumb|Suisse - France 2016 - Griezmann (crop)]], #Project(s): Creating a new language called [[wikiversity:Draft:Romanice|Romanice]]; learning how to speak [[wikipedia:italian|Italian]], [[wikipedia:latin|Latin]], [[wikipedia:romanian|Romanian]], [[wikipedia:galician|Galician]], [[wikipedia:corsican|Corsican]] and [[wikipedia:catalan|Catalan]]. #You can e-mail via -» el.eye.jah7@gmail.com. kzeegxpmrs3wcxcv58kdxehdd93kcjx 2 304065 2816265 2810496 2026-06-19T01:14:16Z Dc.samizdat 2856930 /* A symmetrical arrangement of eleven 11-cells */ 2816265 wikitext text/x-wiki = A symmetrical arrangement of eleven 11-cells = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|March 2024 - June 2026}} <blockquote>[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]] ₅{3,5,3}₅, a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]] {3,5}₅, a hexad non-orientable polyhedron. The 11-cell is described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, we find that the 11-cell has a realization in Euclidean 4-space inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the convex regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra and 120 regular 5-cells. We find that the 120-cell also contains: a non-uniform icosahedral polyhedron that contains the realization of the abstract hemi-icosahedron; real 11-point 11-cells made from 11 of it; and a compound of eleven real 11-cells. We also find a quasi-regular compound of the compound of eleven 11-cells and [[w:Schoute|Schoute]]'s compound of five 24-cells (the 600-cell). We describe the real 11-point 11-cell 4-polytope; its compound of eleven 11-cells; the quasi-regular compound; and their relation to the regular polytopes.</blockquote> == Introduction == [[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hemi-icosahedral hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them. [[File:120-cell.gif|thumb|360px|The picture on the cover of the box of 4-dimensional building blocks.{{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]]."}} Only the 120-cell's own edges are shown. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and 11-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]] The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-hypercube, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the 600-point [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). The 5-cell has a fundamental relationship to all the other 4-polytopes, but not one as simple as compounding, so it is not immediately useful to children trying to learn to build with 4-dimensional building blocks. But the 16-cell is our very starting point, and the most frequently used tool in the box. Nevertheless, to build the 11-cell, we start with the 5-cell. The 5-cell and 11-cell are both self-reciprocal (their own duals). They are the only 4-polytopes where every cell shares a face with every other cell. The 5-cell is a tetrahedron surrounded by 4 other tetrahedra, in five different ways. The 11-cell is a hemi-icosahedron surrounded by 10 other hemi-icosahedra, in eleven different ways. The 5-cell has 5 vertices that form 5 tetrahedral cells, and a total of 10 triangular faces and 10 edges. The 11-cell has 11 vertices that form 11 hemi-icosahedral cells, each with 6 verticies 10 triangular faces and 15 edges, and a total of 55 triangular faces and 55 edges. == 5-cells and hemi-icosahedra in the 11-cell == [[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its <math>2^5</math> elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]] [[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two or more real elements found in different places in a concrete realization of the 11-cell.{{Sfn|Ruen: Hemi-icosahedron|2007}}]] The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we find a facet congruence between a 4-polytope and a 3-polytope we suspect a dimensional analogy. In the exceptional case of 5-cell and icosahedron, which share the same symmetry group <math>A_5</math>, we fully expect a dimensional analogy.{{Efn|There is an exceptional inter-dimensional duality between the regular icosahedron and the 5-cell because they share <math>A_5</math> symmetry. See this question asked on [https://math.stackexchange.com/questions/4235783/the-rotational-symmetry-groups-of-the-5-cell-and-the-icosahedron-are-isomorphi math.stackexchange.com 2021].}} Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization as the 6-point 5-simplex. Yet another real hexad is the 6-point 3-orthoplex; thus as a hexad the hemi-icosahedron is related by dimensional analogy to the 4-simplex (5-cell) from above, and to the 4-orthoplex (16-cell) from below, while those two simplest 4-polytope building blocks are only related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The cell of the 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, the 4-simplex (5-cell) and 4-orthoplex (16-cell), who are famously stand-offish with each other. Interesting! The cell of the 11-cell is an abstract hexad hemi-icosahedron with 5 central planes, most handsomely illustrated by Séquin.{{Sfn|Séquin|2012|loc=A 10-Dimensional Jewel}}{{Sfn|Séquin & Lanier|2007|p=3|loc=Figure 4: (b,c) two views of the hemi-icosahedron projected into 3D space|ps=; Séquin et. al. have a lovely colored illustration of the hemi-icosahedron, subdivided into 10 triangular faces by 5 central planes of its icosahedral symmetry, revealing rings of polytopes nestled in its interior. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}}{{Sfn|Séquin & Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point [[W:Hemi-isosahedron|hemi-isosahedron]] is the vertex figure of the 11-cell's dual 4-polytope the 57-point [[W:57-cell|57-cell]].}} The 11 hemi-icosahedral cells have 10 triangle faces each, and each cell is face-bonded to the other 10 cells. The 5-cell's 5 tetrahedral cells have 10 faces and 10 edges altogether, and each cell is face-bonded to the other 4 cells. If 11-cell faces correspond to 5-cell faces, then 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, that does not belong to the same hemi-icosahedral cell as its opposing face. He found that the 10 edges opposite each hemi-icosahedron's 10 faces are the 10 edges of a single 5-cell, which does not share any vertices, edges or faces with the hemi-icosahedron. For each cell of the 11-point 11-cell, there is exactly one 5-point 5-cell that is completely disjoint from the 6-point hemi-icosahedron cell.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; "We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary <math>a_4</math> [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron."}} There are 11 disjoint 5-cell 4-polytopes inscribed in each 11-cell, which also contains 11 hemi-icosahedral cells, 55 faces, 55 edges and 11 vertices. The real 11-cell is more complex than the abstract 11-cell representing it, because the real hemi-icosahedron is more complex and harder to find than the abstract hemi-icosahedron. Seeing the real 11-cell will be easier once we have identified the real hemi-icosahedron, and seen exactly where the 11-cell's real elements reside in the other 4-polytopes within the 120-cell with which the 11-cell intermingles. The 5-cell has 10 faces, and the 11-cell has 10 faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron is face-bonded to the other 10 hemi-icosahedra, and to 10 of the 11 5-cells, and there is exactly one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point 5-cell is the other 5 vertices of the 11-point 11-cell that are not vertices of this 6-point hemi-icosahedron: the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces which belong to 10 distinct hemi-icosahedra of the 11-cell, and there is just one hemi-icosahedron with which it does not share a face. In the abstract 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 completely disjoint 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction.{{Efn|We shall see that this small eliminated distance is in fact the length of a 120-cell edge, the shortest chordal distance found in the 120-cell.}} Seemingly adjacent hemi-icosahedron faces do not actually meet at an edge; there is a polygon separating them, which has been abstracted to an edge. The 10 hemi-icosahedron faces are 5-cell faces from 10 distinct 5-cells, and they do not actually touch each other: the 120 5-cells in the 120-cell are completely disjoint. In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of tetrahedral cells together, because each 11-cell face represents two actual 5-cell faces in different places. Each duplex 11-cell face bonds tetrahedra in two 5-cells in different places, without binding the 5-cells together (they are completely disjoint). One actual 5-cell face is one half of a duplex 11-cell face, so 110 5-cell faces are 55 duplex 11-cell faces. The 11-cell's 11 abstract vertices represent all 55 distinct vertices of the 11 disjoint 5-cells, so they must be abstract conflations of at least 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; 11-cells must be sharing vertices, not disjoint as the 5-cells are. == The real hemi-icosahedron == [[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right| Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; "Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, & 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 & 5 are each pairs of Truncated icosahedrons. Hulls 6 & 8 are Rhombicosidodecahedrons."}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes starting with a dodecahedron cell. Hull #8 with 60 vertices (lower right) is a central section of the 120-cell, the 8th and largest section starting with a cell.{{Efn|1=Although the 8 hulls are illustrated as the same size, in the 120-cell they have increasing size as numbered, and occur nested inside each other like Russian dolls. Only Hull #8 is a central section of the same radius as the 120-cell itself, analogous to the equator. Sections 1-7 occur in pairs on opposite sides of the central section, and are analogous to lines of latitude. Section 1 is simply a dodecahedral cell. The "Combined hulls" is for illustrative purposes only; no such compound polyhedron exists in the 120-cell.}}]] We shall see in subsequent sections that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the realization of the abstract hemi-icosahedron, as an actual polyhedron that occurs in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''² = ''j''² = ''k''² = ''ijk'' = −1}} as the [[W:Tesseract|16-point (8-cell) tesseract]] 4-polytope. He did not realize then that he had discovered the 4-hypercube polytope and [[W:Tesseractic honeycomb|its Euclidean honeycomb]], the (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because [[w:Quaternions_and_spatial_rotation|quaternions make rotations]] and projections in 3D or 4D space as simple as matrix multiplications.{{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 quaternions are 4-hypercube building blocks, analogous to the 3-hypercube wooden blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the [[24-cell]], but we digress). Moxness used his software to render illustrations of polyhedra inside the 120-cell, some of which he published. Notice his "Hull # = 8 with 60 vertices", lower right in his illustration of the 120-cell sections starting with a cell. It is a real icosahedron that occurs in the 120-cell, and we shall see that the abstract hemi-icosahedron represents it. Moxness's 60-point Hull #8 is a concrete realization of the 6-point hemi-icosahedron in spherical 3-space <math>S^3</math>, embedded in Euclidean 4-space <math>\mathbb{R}^4</math>. Its 12 little pentagon faces are 120-cell faces. It also has 20 triangle faces like any icosahedron, separated from each other by rectangles, but beware: those triangles are not the 5-cell faces. They are smaller equilateral triangles, of edge length <math>1</math> in a {{radic|2}}-radius 120-cell, where the 5-cell face triangles have edge length {{radic|5}}.{{Efn|The 41.4° chord of edge length 1 in a {{radic|2}}-radius 120-cell occurs only in the 120-cell; it is not the edge of any smaller regular 4-polytope inscribed in the 120-cell. The equilateral triangle faces of Moxness's Hull #8 rhombicosidodecahedron are not the 5-cell faces of edge length <small><math>\sqrt{5} \approx 2.236</math> </small>(104.5°), not the 16-cell faces of edge length <small><math>2</math></small> (90°), not the 24-cell faces of edge length <small><math>\sqrt{2} \approx 1.414</math></small> (60°), and not the 600-cell faces of edge length <small><math>\sqrt{2}/\phi \approx 0.874</math></small> (36°).|name=Moxness 60-point triangle faces}} [[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges, in 200 irregular {12} dodecagon central planes. The 5-cell ''faces'' do not lie in central planes.]] Edges of the larger 5-cell face triangles of length {{radic|5}} can also be found in Hull #8, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that six 120-cell edges (little pentagon edges) lie on a great circle, alternating with six rectangle diagonals. Also lying on this irregular {12} great circle are six 5-cell edges, invisible chords joining every other 120-cell edge and running under the 120-cell edge between them. The six long chords and six short edges form two opposing irregular {6} great hexagons (truncated triangles) of alternating 5-cell edges and 120-cell edges, as illustrated. The irregular great {12} lies on a great circle of Moxness's Hull #8, and also on a great circle of the 120-cell, because Hull #8 is the ''central'' cell-first section of the 120-cell.{{Efn|The cell-first central section of the 600-cell (and of the 24-cell) is a cuboctahedron with 24-cell edges. The 120-cell is the regular compound of 5 600-cells (and of 25 24-cells), so Moxness's Hull #8, as the cell-first central section of the 120-cell, is the regular compound of 5 cuboctahedra. Their 24-cell edges, like the 5-cell edges, are invisible chords of Hull #8 that lie below its surface, on the same irregular {12} great circles. Each 24-cell edge chord spans one 120-cell edge chord (one little pentagon edge) and one rectangle face diagonal chord. Six 24-cell edge chords form a regular great {6} hexagon, inscribed in the irregular great {12} dodecagon.|name=compound of 5 cuboctahedra}} There are 10 great dodecagon central planes and 60 5-cell edges in Moxness's Hull #8, and 200 great dodecagon central planes and 1200 5-cell edges in the 120-cell. [[File:Central cell-first section of the 120-cell with 5-cell face triangle.png|thumb|Orthogonal projection of the cell-first central section of the 120-cell, Hull #8 rendered by Moxness, with one of 20 inscribed 5-cell faces (black chords) drawn under portions of three of its ten great circle {12} dodecagons (green).{{Efn|The point of view in this rendering is not quite right to best illustrate that a rhombicosidodecahedron triangle face lies over the center of a 5-cell face parallel to it, such that it would be perfectly inscribed in the center of the larger black triangle in an orthogonal view.}}]] But the 5-cell ''faces'' do not lie in those central planes. We can locate them in the 60-point polyhedron where they lie parallel to and under each small face triangle of edge length <math>1</math>. Truncating at a triangle face of Moxness's Hull #8 exposes a deeper 5-cell triangle face.{{Efn|Each face triangle of edge length <math>1</math> is surrounded by 3 rectangles, and beyond each rectangle by another face triangle. The distant vertices of those 3 surrounding triangles form a {{radic|5}} triangle, a 5-cell face.}} There are 20 such 5-cell faces inscribed in the Hull #8 polyhedron, all completely disjoint. We find 60 vertices, 60 edges and 20 faces of various 5-cells in each Hull #8 polyhedron, but no whole tetrahedral cells of the 5-cells.{{Efn|The fourth vertex of each 5-cell tetrahedron lies opposite the small face triangle of edge length <math>1</math> that lies over the 5-cell face. Since Moxness's Hull #8 polyhedron has opposing triangle faces (like any icosahedron), the fourth vertex of the 5-cell tetrahedron lies over the center of the opposing face, outside the Hull #8 polyhedron. This is a vertex of some other Hull #8 polyhedron in the 120-cell. Each tetrahedral cell of a 5-cell spans four Hull #8 polyhedra, with one face inscribed in each, and one vertex outside of each.}} [[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point Hull #8 is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation of the icosahedron with smaller red pentagons and narrower rhombs. Rhombicosidodecahedra are also made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the unique 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 60 of Moxness's Hull #8 rhombicosidodecahedron. Each occupies a central hyperplane, and so is analogous to an equator dividing the sphere in half.]] Moxness's Hull #8 is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point [[W:icosidodecahedron|icosidodecahedron]], which has the same relationship to Moxness's Hull #8 that the 6-point hemi-icosahedron does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The regular 5-point 5-cell can be another abstraction of Moxness's 60-point Hull #8, 12-vertices-into-1. None of these contractions of Moxness's Hull #8 is an instance of her operation actually described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length, but she did explicitly describe contractions of the semi-regular Archimedean rhomibicosidodecahedron.}} Moxness was not the first person to find rhombicosidodecahedra in the 120-cell. Alicia Boole Stott identified the 6th section of the 120-cell beginning with a cell as the semi-regular rhombicosidodecahedron that is her ''e₂ expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e₂ expansion|p=7}} But that 6th section rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8, it is the semi-regular Archimedean solid (Moxness's Hull #6), with a single edge length and square faces. Moxness's Hull #8, with its two distinct edge lengths and rectangular faces, is Coxeter's 8₃, the 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; "Alicia Boole Stott (1860-1940) ... also constructed the sections i₃ of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1₃-7₃; but 8₃ is missing. Incidentally, Diagram XIII (our 6₃) is a rhombicosidodecahedron, the Archimedean solid."}} Coxeter was the first to describe the central section 8₃, and he gave its coordinates, but he did not identify it as an irregular rhombicosidodecahedron. His table entry for its description is empty (characteristically, since it is not a regular or semi-regular polyhedron), so he gives us no indication that he actually visualized it. Although Moxness was not the first to compute the 60-point 8₃ section, he may have been the first person to ''see'' it. The 30-point icosidodecahedron is the quasi-regular product of 5-point pentagon and 6-point hexagon, recalling Coxeter's original discovery of the 11-cell in pentads and hexads, and also the two child's building blocks: one so useless the 5-point (pentad) 5-cell, and the other so useful the 8-point 16-cell with its four orthogonal 6-point (hexad) octahedron central sections, which can be compounded into everything larger. Some children building with the 30-point icosidodecahedron notice that it occurs as the central section 4₀ of the 120-point 600-cell. It is less often noticed that Moxness's Hull #8 rhombicosidodecahedron is the central section 8₃ of the 600-point 120-cell. It occupies a flat 3-dimensional hyperplane that bisects the 120-cell, and since there are 120 dodecahedral cells, there are 60 such central hyperplanes, each perpendicular to an axis that connects the centers of two antipodal cells. The 60 central hyperplanes, each containing an instance of Moxness's Hull #8, are rotated with respect to each other. They intersect, with 6 rhombicosidodecahedra sharing each vertex and 3 sharing each edge, but each little pentagon face (120-cell face) belongs to just one rhombicosidodecahedron. The 60 central sections lie in isoclinic hyperplanes, that is, the rhombicosidodecahedra are rotated symmetrically with respect to each other, by two equal angles.{{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.)"}} Each pair of rhombicosidodecahedra intersect in a central plane containing an irregular {12} dodecagon, unless they are completely orthogonal and intersect only at the center of the 4-polytope. Each of the 120 dodecahedral cells lies in the closed, curved 3-dimensional space of the 3-sphere as the 1st and smallest section beginning with a cell (section 1₃), the innermost of a series of concentric polyhedral hulls of increasing size, which nest like Russian dolls around it. Moxness's Hull #8 rhombicosidodecahedron is the 8th and largest concentric hull beginning with a cell (section 8₃), a central section of the 120-cell that bisects the 3-sphere the way an equator bisects an ordinary sphere.{{Efn|The 120-cell's curved 3-space surface is a honeycomb of 120 dodecahedron cells. In this 3-space a dodecahedron cell lies inside at the center of each section 8₃ rhombicosidodecahedron, face-bonded to 12 other dodecahedron cells which surround it, also inside the rhombicosidodecahedron. We find the opposite pentagon faces of those 12 surrounding cells on the surface of the section 8₃ rhombicosidodecahedron. These twelve dodecahedra surrounding one dodecahedron partially fill the volume of the rhombicosidodecahedron, leaving 30 concavities in its surface at the rectangle faces, and 12 deeper concavities between them at the triangle faces. 30 more dodecahedra fit into the rectangle concavities, lying half inside and half outside the rhombicosidodecahedron. The diagonal of each rectangle face is a long diameter of a dodecahedron cell. 12 more dodecahedra fit into the triangle face concavities, lying ....|name=dodecahedral cells in the section 8 rhombicosidodecahedron}} Such a central polyhedron is the dimensional analog of an equatorial great circle polygon. Its 60 vertices lie in the same 3-dimensional hyperplane, a flat 3-dimensional section sliced through the center of the 120-cell. There are 60 distinct stacks of 15 parallel section ''n''₃ hyperplanes in the 120-cell, one stack spindled on each axis that connects a dodecahedron cell-center to its antipodal dodecahedron cell-center. Each central section 8₃ has ''two'' disjoint sets of smaller sections nested within it, that lie in opposite directions from the 120-cell's center along its 4th dimension axis. The largest-radius central slice lies in the center of the stack, and the smaller non-central section hyperplanes occur in parallel pairs on either side of the central slice. The 120-cell therefore contains 120 instances of each kind of non-central section 1₃ through 7₃, and 60 instances of the central section 8₃.{{Efn|A central section is concave on its inside and also on its outside: it has two insides. It may be helpful to imagine the central 60-point section as two mirror-image 60-point polyhedra whose points are coincident, but which are convex in opposite directions: the inside of one is the outside of the other. Each has seven smaller polyhedra nested within itself, but their two volumes are disjoint.}} [[File:Tensegrity Icosahedron.png|thumb|[[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|Tensegrity icosahedron]] structure.{{Sfn|Burkhardt|1994}} First built by [[W:Kenneth Snelson|Kenneth Snelson]] in 1949. Geometrically a [[w:Jessen's_icosahedron|Jessen's icosahedron]] with 6 reflex ''long'' edge struts, and 24 ''short'' edge tension cables around 8 equilateral triangle faces. 3 pairs of parallel struts lie in 3 orthogonal central planes.]] We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that the 60 Moxness's Hull #8 rhombicosidodecahedra are real polyhedra which the abstract hemi-icosahedra represent in some manner, but we have not yet identified 11 real face-bonded cells, at 11 distinct locations in the 120-cell, as an 11-cell. The abstract hemi-icosahedron's 10 faces correspond to actual 5-cell faces inscribed in real rhombicosidodecahedra, and its 15 edges correspond to 5-cell edges (of length {{radic|5}} in a {{radic|2}}-radius 120-cell) that occur as chords lurking under the surface of the rhombicosidodecahedra. [[File:Buckminster-Fuller-holding-a-geodesic-tensegrity-sphere.png|thumb|200px|Buckminster Fuller holding a 3-dimensional geodesic tensegrity 2-sphere, an infinitesimally mobile rigid polytope consisting of tension cable edges and disjoint compression strut chords.<ref>{{Cite journal|last=Álvarez Elipe|first=Dolores|title=Ensegrities and Tensioned Structures|journal=Journal of Architectural Environment & Structural Engineering Research|date=July 2020|volume=3|issue=3|url=https://www.researchgate.net/publication/343652287_Ensegrities_and_Tensioned_Structures}}</ref>]] A rhombicosidodecahedron is constructed from a regular icosahedron by truncating its vertices, making them into pentagon faces. The regular icosahedron frames all the regular and semi-regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter & Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Sfn|Coxeter, du Val, Flather & Petrie|1938|p=4|ps=; "Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "[[W:Golden section|golden section]]". Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''."}} Before we can move on to locating the 11 discrete hemi-icosahedral cells of the 11-cell in the 120-cell, it is important that we take notice of one more icosahedral symmetry of the hidden {{radic|5}} chords lurking below the surface of Moxness's Hull #8 rhombicosidodecahedron. The 12 little pentagon faces (120-cell faces) are connected to each other in parallel pairs, by 10 sets of six disjoint {{radic|5}} chords (5-cell edges). Each six-chord set is the six reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], which is to say that the six disjoint chords are the parallel-orthogonal strut chords of a [[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|tensegrity icosahedron]]. The six chords of each set are disjoint (they don't touch or form 5-cell faces), and they are symmetrically arranged as 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal {12} central planes.{{Efn|The Jessen's icosahedron has 8 equilateral triangle faces, which are not rhombicosidodecahedron triangle faces or 5-cell triangle faces, they are 24-cell triangle faces. Each 120-cell pentagon face lies at one end of 20 5-cell edges, from 20 distinct Jessen's icosahedra and five disjoint 5-cells: four at each pentagon vertex from each 5-cell.}} Five disjoint instances of the Jessen's icosahedron may be inscribed in each Moxness's Hull #8 rhombicosidodecahedron, their struts propping the rhombicosidodecahedron and the 120-cell itself open like a tensegrity structure.{{Efn|Moxness's Hull #8 rhombicosidodecahedron is a compound of five disjoint Jessen's icosahedra, because the 60 {{radic|5}} chords meet two-at-a-vertex and form 10 distinct Jessen's icosahedra: five disjoint Jessen's, in two different ways. The dimensionally analogous construction is the [[120-cell#Compound of five 600-cells|120-cell as a compound of five disjoint 600-cells]], in two different ways.}} But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, and work our way patiently outwards, before in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we approach that rare child Bucky Fuller's orthogonal 12-point tensegrity icosahedron, an [[WikiJournal Preprints/Kinematics of the cuboctahedron|in-folded cuboctahedron]], the unique pyritohedral fish swimming deep in the 3-sphere ocean. == Eleven == Each pair of rhombicosidodecahedra that are not completely orthogonal intersect in a central plane containing an irregular {12} dodecagon. Ten irregular great dodecagons occur in each 60-point (central section 8₃) rhombicosidodecahedron, with 2 dodecagons crossing orthogonally at each vertex. Each rhombicosidodecahedron shares a {12} central plane with ten other rhombicosidodecahedra. ''Groups of 11 rhombicosidodecahedra share central planes pairwise.'' Here, at last, we find eleven of something, a group which must comprise an 11-cell. There are eleven {12} central planes in the group, with one of the eleven absent from each rhombicosidodecahedron. {|class="wikitable floatright" width=450 !colspan=2|Perspective views{{Efn|1=These images are ''non-orthogonal'' orthographic projections of the chords described in the caption. Those chords do not lie in a plane parallel to the projection plane, so they appear foreshortened.{{Efn|name=orthogonal triacontagram projections}} Consecutive chords of the helical Petrie polygon slant toward and away from the viewer. Any three consecutive chords, but no four, are edges of the same cell, in the 4-polytope whose edges are the chord.{{Efn|name=Petrie polygon of a honeycomb}}}} of a compound of six disjoint 5-cells in dual position |- ![[W:Triacontagon#Triacontagram|{30/12}{{=}}6{5/2} compound]] ![[W:Triacontagon#Triacontagram|{30/8}{{=}}2{15/4} compound]]{{Efn|name=orthogonal triacontagram projections|1=The {30/''n''} triacontagrams can each be seen as an ''orthogonal projection'' of the 120-cell showing all instances of the {30/''n''} chord. Each chord lies orthogonal to the line of sight, in a plane parallel to the projection plane. The diameter of the image is the diameter of the 120-cell. For example, the {30/8}=2{15/4} triacontagram is an orthogonal projection showing the 120-cell's 1200 {30/8} chords, the edges of 120 5-cells. Each edge of the triacontagram covers 40 5-cell edges, and each vertex covers 20 120-cell vertices. This projection can also be viewed as a compound of six 5-cells and their 30 unique vertices. But viewed that way, only 30 of the 60 5-cell edges are visible. Two edges meet at each vertex, but the other two are invisible. They are visible in the orthogonal view, the {30/4}=2{15} projection.}} |- valign=top |[[File:Regular_star_figure_6(5,2).svg|240px]]<BR>The 6{5/2} compound of six 5-cells. The six disjoint pentagrams in this view are six disjoint 5-cells.{{Efn|name=5-cell edges do not intersect is S³}} The 120-cell, with 120 disjoint 5-cells, is a compound of 20 of these compounds. All edges are 5-cell edges, but only five of each 5-cell's ten edges are shown. The other five edges, connecting the points of the six 5-cell pentagrams, are shown in the 6{5} projection below, the orthogonal view:<BR>[[File:Regular_star_figure_6(5,1).svg|240px]]These two views look straight down the orthogonal axes of a [[w:Duocylinder|duocylinder]], from inside the curved 3-dimensional space of the 120-cell's surface. They are like looking down a column of 5-cells stacked on top of one another in curved 3-space, but the column is actually circular: it is bent into a torus in the fourth dimension. |[[File:Regular_star_figure_2(15,4).svg|240px]]<BR>The 2{15/4} rotation circuits of the 5-cell isoclinic rotation. In this view, all edges are 75.5° chords of length {{radic|3}}, the 180° complement chord of the 5-cell edges of length {{radic|5}}.{{Efn|These are not 15-gons of 5-cell edges. There are no skew {15} polygons of 5-cell edges in the 120-cell. The 120 5-cells are completely disjoint, so the largest circuit along 5-cell edges is a skew {5}. Each vertex in the 120-cell is {{radic|5}} away from four and only four other vertices. No {{radic|5}} chords connect disjoint 5-cells; they are connected by several other chords. The skew {15} polygons are the discrete continuous spiral paths of moving vertices during an isoclinic rotation, and their edges are {{radic|3}} chords connecting 5-cells, not 5-cell edges.}} Each skew {15} polygon is the spiral chord-path of half the 30 vertices during the isoclinic rotation. The twined vertex orbits lie skew in 4-space; they form a circular double helix of two 15-gon spiral isoclines, winding through all four dimensions. These two completely orthogonal views look straight down an axis of a double helix cylinder, from inside the curved 3-dimensional space of the 120-cell's surface. Since the duocylinder is bent into a [[w:Clifford_torus|Clifford torus]] in the fourth dimension, the sightline axis in curved 3-space is a geodesic great circle in 4-space.<BR>[[File:Regular_star_figure_2(15,2).svg|240px]] |- ![[W:Triacontagon#Triacontagram|{30/6}{{=}}6{5} compound]] ![[W:Triacontagon#Triacontagram|{30/4}{{=}}2{15/2} compound]] |- |colspan=2|Images by Tom Ruen in [[W:Triacontagon#Triacontagram|Triacontagram compounds and stars]].{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}} |} Each shared {12} central plane contains six disjoint 5-cell edges, from six completely disjoint 5-cells. Each rhombicosidodecahedron contains 60 5-cell edges, which form 20 disjoint 5-cell faces within the rhombicosidodecahedron, under and parallel to its own 20 smaller triangle faces. Four 5-cell edges meet at each vertex at the 5-cell's tetrahedral vertex figure. Two 5-cell edges of a face within the rhombicosidodecahedron meet two edges belonging to other faces of the 5-cell: edges and faces outside the rhombicosidodecahedron, in some neighboring rhombicosidodecahedron.{{Efn|name=orthogonal triacontagram projections}} Each 5-cell face is shared by two tetrahedral cells of one 5-cell. It has its three 104.5° {{radic|5}} edges in three distinct {12} central planes, and is parallel to a fourth {12} central plane. In each rhombicosidodecahedron there are ten sets of five parallel planes: a {12} central plane, a pair of 5-cell faces on either side of it (from disjoint 5-cells), and a pair of rhombicosidodecahedron triangle faces. Each rhombicosidodecahedron is sliced into five parallel planes, ten distinct ways. There is no face sharing between 5-cells: the 120 5-cells in the 120-cell are completely disjoint. 5-cells never share any elements, but they are related to each other positionally, in groups of six, in the '''characteristic rotation of the regular 5-cell'''. That rigid isoclinic rotation takes the six 5-cells within each group to each other's positions, and back to their original positions, in a circuit of 15 rotational displacements.{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439|ps=; in symmetry group 𝛢₄ the operation [15]𝑹_q3,q3 is the 15 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 5-cell; in symmetry group 𝛨₄ the operation [1200]𝑹_q3,q13 is the 1200 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 120-cell.}} Each displacement takes every 104.5° 5-cell edge of length {{radic|5}} to an edge 75.5° and {{radic|3}} away in another 5-cell in the group of six 5-cells. The 30 vertices of the six 5-cells rotate along 15-chord helical-circular isocline paths from 5-cell to 5-cell, before closing their circuits and returning the moving 5-cells to their original locations and orientations.{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. The orthogonal distance equals half the total Pythagorean distance. For example, when the {{radic|2}}-radius 5-cell rotates isoclinically 104.5° in the invariant central planes of its 104.5° edges of length {{radic|5}}, each vertex is displaced to another vertex 75.5° and {{radic|3}} away, moving {{radic|3/4}} in four orthogonal directions at once.|name=isoclinic 4-dimensional diagonal}} The six rotationally related 5-cells form a stellated compound, a non-convex 4-polytope with 30 star points.{{Efn|name=compound of six 5-cells}} The star compound, and the rotation of the 5-cell within it, are here illustrated by orthogonal projections from four different perspective viewpoints. To help us visualize the 4-polytopes within the 120-cell, we can examine 2-dimensional orthographic projections from various points of view. Such images filtered to include only chords of a single length are especially revealing, because they pick out the edges of a particular 4-polytope, or the isocline chords of its rotational orbits, the chords which link 4-polytopes together. No view of a single chord from a single point of view is sufficient by itself, but if we visualize various chords from various perspectives, we may imagine the 4-dimensional rotational geometry of interrelated objects within the 120-cell. The star compound as a whole has ten {12} central planes, like a rhombicosidodecahedron. Each {12} central plane contains one edge from each of the six 5-cells. Each {12} central plane is shared by two rhombicosidodecahedra in the group of eleven, and by six 5-cells in the group of six. == The eleventh chord == [[File:Major chord 11 of 135.5° in the 120-cell.png|thumb|The 120-cell contains 200 irregular {12} central planes containing 1200 135.5° {30/11} chords, six in each plane (shown in blue). They lie parallel to six 104.5° {30/8} chords (the 5-cell edges, shown in red), to which they are joined by 15.5° {30/1} 120-cell edges, and by 120° {30/10} great triangle edges (only one of the four great triangles is shown, in green).]] In addition to six 104.5° {30/8} 5-cell edge chords of length {{radic|5}}, the {12} central plane contains six 135.5° {30/11} chords of length <math>\phi^2</math>, parallel to the {{radic|5}} chords. The {30/11} chord spans an arc of five shorter chords: * 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} = 135.5° {30/11) * 15.5° {30/1} + 104.5° (30/8) + 15.5° {30/1} = 135.5° {30/11) * 15.5° {30/1} + 120° (30/10) = 135.5° {30/11) and its chord length is the linear sum of five shorter chords: * 1/𝜙^2 {30/1} + 1/𝜙^2 {30/1} + 1/𝜙 {30/2} + 1/𝜙 {30/2} + 1/𝜙 {30/2} = 𝜙^2 {30/11) Two distinct chords are always related to each other in two different ways: by their degrees-of-arc-difference, and by their linear difference chord. The 135.5° {30/11) chord is ''two'' 15.5° (30/1) 120-cell edge-arcs longer than the 104.5° (30/8) 5-cell edge chord. But the <math>\phi^2</math> {30/11} chord ''length'' is just ''one'' {30/1} 120-cell edge chord length longer than the {{radic|5}} {30/8} 5-cell edge chord.{{Efn|In a <small><math>\sqrt{2}</math></small>-radius 120-cell, the 15.5° {30/1} 120-cell edge chord has length <small><math>\phi^{-2}</math></small>. The 25.2° {30/2} pentagon face diagonal chord of length <small><math>\phi^{-1}</math></small> is <small><math>\phi</math></small> times the {30/1} edge length. The 41.1° 5-cell isocline chord of length <small><math>\sqrt{1}</math></small> is <small><math>\phi^2</math></small> times the {30/1} edge length. The 69.8° chord of length <small><math>\phi</math></small> is <small><math>\phi^3</math></small> times the {30/1} edge length. The 135.5° {30/11} 11-cell edge chord of length <small><math>\phi^2</math></small> is <small><math>\phi^4</math></small> times the {30/1} edge length.}} The {30/11} chord can be bisected into two shorter 120-cell chords in three different ways: * 15.5° {30/1} 120-cell edge + 104.5° {30/8} 5-cell edge = {30/11} chord * 25.2° {30/2} 120-cell pentagon face diagonal + 90° {30/15} 16-cell edge = {30/11} chord * 41.4° {30/1}+{30/2} chord + 69.8° {30/2}+{30/1}+{30/2} chord = {30/11} chord [[File:Regular_star_polygon_30-11.svg|thumb|The [[W:Triacontagon#Triacontagram|{30/11} regular triacontagram]] of the 11-cell rotation.{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}} In this 2-dimensional projection of a 30-edge 4-dimensional helix ring, the 30 chords pictured lie in 30 distinct central planes, and no two planes are orthogonal.]] The last of those bisections trisects the {30/11} chord into three distinct shorter chords: * 15.5° {30/1} + 25.2° {30/2} + 44.5° {30/4} chord = 135.5° {30/11} chord The {30/11} chords do not form triangle faces within the rhombicosidodecahedron the way the {30/8} chords do, but they do meet at a tetrahedral vertex figure. Groups of 11 rhombicosidodecahedra (an 11-cell) share central planes pairwise, including all the chords in the {12} central plane. When 11 things, all pairwise-adjacent to each other, are arranged in any circuit of 30 positions, there exists another pairwise circuit of 30 positions through every eleventh position, whether the things are 11 vertices, 11 rhombicosidodecahedra, or 11 [[w:Aardvark|aardvarks]] (although it might be unwieldy in practice to so arrange 11 live aardvarks, e.g. by tying them together pairwise with cords in both circuits). This intrinsic property of the [[w:Rational_number|rational number]] 30/11 is responsible for the existence of the {30/11} regular triacontagram (see illustration). The 11 rhombicosidodecahedra of the 11-cell are linked by a regular {30/11} triacontagram of 30 chords which runs through them. Each successive chord of the 30 in the triacontagram is shared by a distinct pair of rhombicosidodecahedra in the 11-cell group. An isoclinic rotation characteristic of the 11-cell takes the rhombicosidodecahedra in each 11-cell to each other's positions, pair by pair, in a circuit of 30 rotational displacements. It takes every {12} central plane to a Clifford parallel {12} central plane that is 44.5° away in two completely orthogonal angles. One 135.5° {30/11} chord separates each of the 12 vertex pairs. In this '''characteristic rotation of the 11-cell''' in its edge planes, the invariant planes are {12} central planes, the edges of the 11-cell are {30/11} chords, and the isocline chords of the vertex orbits are also {30/11} 11-cell edges, because the triacontagram is regular.{{Efn|In the 120-cell there are three ''regular isoclinic rotations'' in which the rotation edge and the isocline chord are the same chord. These rotations are each described by a [[W:Triacontagon#Triacontagram|regular triacontagram]]: the {30/7} rotation characteristic of the 16-cell in great square invariant planes, the {30/11} rotation characteristic of the 11-cell, and the {30/13} rotation.}} The 44.5° {30/4} chord of length <small><math>\sqrt{3}/\phi</math></small>, the 180° complement of the {30/11} chord, is the orthogonal distance between nearest parallel {30/11} chords.{{Efn|In its characteristic isoclinic rotation, a 4-polytope rotates an equal arc distance in each invariant {12} edge plane in each rotational displacement. In the 11-cell, every invariant plane rotates 44.5° (like a wheel), and tilts sideways 44.5° (like a coin flipping) in the completely orthogonal invariant plane, to occupy another invariant plane in the group of eleven. Each pair of original and destination {12} central planes are Clifford parallel and intersect only at one point (the center of the 4-polytope), but six other {12} central planes intersect them both. Two parallel {30/11} chords in each of the six spanning {12} central planes separate two vertex pairs in the original and destination planes, and these are the isocline chords over which the two vertices move in the rotation. None of the six spanning {12} central planes are contained in either the original or destination rhombicosidodecahedron. A total of ten {12} central planes span each original and destination rhombicosidodecahedron; they comprise a third rhombicosidodecahedron which does not belong to the group of eleven. The edges of an 11-cell and the isocline chords of an 11-cell are disjoint sets of {30/11} chords.}} The 60 vertices of each rhombicosidodecahedron rotate in parallel, on non-intersecting 30-chord spiral orbital paths, from rhombicosidodecahedron to rhombicosidodecahedron, before closing their circuits and returning the moving rhombicosidodecahedron to its original location and orientation. In this isoclinic rotation of a rigid 120-cell, the 60 rhombicosidodecahedra do this concurrently. Each of the 600 vertices moves on a 4-dimensionally-curved helical isocline, over a skew regular polygram of 30 {30/11} chords, in which a {30/11} chord connects every eleventh vertex of a {30} triacontagram. In the course of a complete revolution (the 30 rotational displacements of this isoclinic rotation), an 11-cell visits the positions of three 11-cells (including itself) 10 times each (in 10 different orientations), and returns to its original position and orientation.{{Sfn|Coxeter|1984|loc=§9. Eleven disjoint decagons}} At each step it occupies the same distinct group of 11 rhombicosidodecahedra sharing planes pairwise, and its 11 vertex positions are those of a distinct 11-cell in the group of eleven 11-cells. A group of 4-polytopes related by an isoclinic rotation is contained in a larger compound 4-polytope which subsumes them. This group of eleven 11-cells related by an isoclinic rotation is not a compound of eleven disjoint 11-cells (since they share vertices), but it is a compound of eleven non-disjoint 11-cells, in the same sense that a 24-cell is a compound of three non-disjoint 8-cell tesseracts. Consider the incidence of these 30-chord {30/11} triacontagram rotation paths, and their intersections. Each rhombicosidodecahedron has 60 vertices and 60 {30/11} chords, which rotate concurrently on Clifford parallel triacontagrams. The 120-cell has only 600 vertices and 1200 {30/11} chords, so at most 20 triacontagrams can be disjoint; some must intersect. But the 11 vertices of an individual 11-cell must be linked by disjoint 30-position {30/11} triacontagram helices, such that their rotation paths never intersect.{{Efn|The isoclines on which a 4-polytope's vertices rotate in parallel never intersect. Isoclinic rotation is a concurrent motion of Clifford parallel (disjoint) elements over Clifford parallel (non-intersecting) circles.}} Each 11-cell has two disjoint triacontagram helicies, its left and right isoclinic rotations, in each of its four discrete fibrations. The 120-cell has 60 distinct {30/11} triacontagram helices, which are 11 disjoint {30/11} triacontagram helices in 11 distinct ways. {{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} == Compounds in the 120-cell == [[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 for 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 other 5 regular convex 4-polytopes]]. {{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}} === How many building blocks, how many ways === The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy. The 24-point also compounds by <math>5^2</math> non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. [[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|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}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]] The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times <math>5^2</math> (75) disjoint instances of itself into the 600-point, which then magically contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point, and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the original 8-point. They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[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''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}} 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]], the final regular [[W:Stellation|stellation]] of the 120-cell, 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:Stellation core|stellation core]] deep inside. The compound of 120 regular 5-cells can be seen to be equivalent to the compound of 5 disjoint 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. === Building the building blocks themselves === We have built every regular 4-polytope except the 5-cell out of 16-cells, but we haven't made the 16-cell (or the 5-cell) out of anything. So far, we have just accepted them both a priori, like [[W:Euclid's postulates|Euclid's postulates]], and proceeded to build with them. But it turns out that while they are the two atomic regular 4-polytopes, they are not indivisible, and can be built up as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct [[w:Symmetry_group|symmetry group]]. Every regular convex ''n''-polytope can be subdivided into instances of its characteristic [[W:Orthoscheme|Schläfli orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-[[w:Orthoplex|orthoplex]]!) is an ''irregular'' ''n''-[[w:Simplex_(geometry)|simplex]] with faces that are various right triangles instead of congruent equilateral triangles. A characteristic ''n''-orthoscheme possesses the complete symmetry of its ''n''-polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements. It is the gene for the polytope, which can be replicated to construct the polytope.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope because 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 partitioned radially by its planes of symmetry (Coxeter's "reflecting circles") into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme and its chiral mirror image can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its <math>A_4</math> planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into <small><math>120\times 120 = 14400</math></small> of these 4-orthoschemes, so that is the symmetry order of the 120-cell. The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its <math>B_4</math> planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order <small><math>75\times 384 / 2 = 14400</math></small>. The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its <math>F_4</math> planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order <small><math>25\times 1152 / 2 = 14400</math></small>. The regular 120-point (600-cell) is subdivided into 14400 instances of its [[600-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its <math>H_4</math> planes of symmetry at once intersecting at its center, so its symmetry is of order 14400. The regular 600-point (120-cell) is the convex hull of the regular compound of 5 disjoint 600-cells (which have 5-fold reflective symmetry), so its symmetry is of order <small><math>5 \times 14400 / 5 = 14400</math></small>. === Building with sticks === [[File:15 major chords.png|thumb|300px|The 15 major chords {30/1} ... {30/15} join vertex pairs which are 1 to 15 edges apart on a skew {30} [[w:Petrie_polygon|Petrie polygon]] of the 120-cell.{{Efn|Drawing the fan of major chords with #1 and #11 at a different origin than all the others was an artistic choice, since all the chords are incident at every vertex. We could just as well have fanned all the chords from the same origin vertex, but this arrangement notices the important parallel relationship between #8 and #11, and calls attention to the 11-cell's maverick edge chord.|name=fan of 15 major chords}} The 15 minor chords (not shown) fall between two major chords, and their length is the sum of two other major chords; e.g. the 41.4° minor chord of length {30/1}+{30/2} falls between the 36° {30/3} and 44.5° {30/4} chords.]] We have seen how all the regular convex 4-polytopes except the 5-cell, including the largest one on the cover of the box, can be built from a box containing 675 16-cell building blocks, provided we can arrange the blocks on top of one another in 4-space, as interpenetrating objects. An alternate box, containing 120 regular 5-cell building blocks, builds the great grand stellated 120-cell (the picture on ''its'' cover), by the same method. In these boxes, the atomic building part is one of the two smallest regular 4-polytopes (5-cell or 16-cell), each generated by its characteristic isoclinic rotation as an expression of its symmetry group (<math>A_4</math> or <math>B_4</math>). All the regular convex 4-polytopes, including the largest one on the cover of the box, can also be built from a box containing a certain number of building sticks and rubber joints, provided we can connect the sticks together in 4-space with the rubber joints. In this box, the atomic building parts are 1-dimensional edges and chords of just 15 distinct arc-lengths. The regular 4-polytopes do not contain a vast variety of stick lengths, but only 30 of them: only 15 unique pairs of 180° complementary chords. The 15 ''major chords'' {30/1} ... {30/15} suffice to construct all the regular 4-polytopes. The 15 ''minor chords'' occur only in the 120-cell, not in any smaller regular 4-polytope; they emerge as a consequence of building the largest 4-polytope on the cover of the box from major chords. In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the unit-radius sphere of a distinct length <math>l</math>, and a distinct rational number <math>h</math>, a unique flavor. If the polytope is regular, it is a noteworthy distinctive flavor. The chord's length <math>l</math> is a square root, related to the rational number <math>h = k/d</math> and to the polygon <small><math>\{k/d\}</math></small> it represents, by a formula discovered by Steinbach.{{Sfn|Steinbach|1997|loc=''Golden Fields''; §1. The Diagonal Product Formula|pp=22-24|ps=; The product of two diagonals is a sum of a sequence of diagonals (in the fan, every other one) centered on the longer of the two, for all regular polygons. We may express products and quotients of diagonals <math>d_k</math> of an <math>n</math>-gon (with edge <math>d_0=1</math>) as linear combinations of diagonals.}} The chord length <math>l</math> is related to the number of sides of the regular polygon <small><math>\{k\}</math></small>, and to the winding number or density of the polygram (its denominator <math>d</math>).{{Sfn|Kappraff & Adamson|2004}} The largest <math>k</math> of any major chord in the 120-cell is 30, and the polygrams <small><math>\{30/d\}</math></small> represent all the skew Petrie polygons and characteristic isoclinic rotations of the regular 4-polytopes. == Concentric 120-cells == The 8-point 16-cell, not the 5-point 5-cell, is the smallest regular 4-polytope which compounds to every larger regular 4-polytope. The 5-point 5-cell is also an atomic building block, but one that compounds to nothing else regular except the leviathan 120-cell polytope: the picture on the cover of the box, that is built from everything in the box. In the [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|sequence of 4-polytope compounds]], we actually start with the 16-cell at the small end, and the 5-cell emerges only at the large end. To build with the 16-cell blocks, we simply put them on top of each other as interpenetrating compounds. We can build every other regular 4-polytope from them by that method, except the individual regular 5-cell. We can also try to build with the 5-cell that way, as when we tried to build a 4-polytope of 11 hemi-icosahedral cells from 11 5-cells, but that was rather hard going. We somehow found 5-cell edges and faces lurking inside hemi-icosahedral rhombicosidodecahedra, and 11 rhombicosidodecahedra sharing central planes pairwise, and even the edges and characteristic rotation of the 11-cell, but we didn't quite get all the way to a discrete 11-cell 4-polytope made from 11 5-cells. That's because ''compounding'' isn't the easiest method for building with the 5-cell. The 5-cell is the last building block hierarchically, not the first, and the most natural way to build with it is in reverse, by ''subdividing'' it, to find all the parts inscribed inside it. When we've taken the 5-cell apart, all the ways we possibly can, into certain ''irregular'' 4-polytopes found within it, we will have a new set of irregular 4-polytope building blocks, which compound to the 5-cells and everything else, including the 11-cells. Subdividing a polytope is done by a geometric operation called ''[[w:Truncation_(geometry)|truncation]]''. There are myriad ways to truncate a 5-cell, each corresponding to a distinct ''depth'' of truncation at a particular point on an edge, or a line on a face, or a face on a cell, where a piece of the 5-cell is cut off. The simplest truncations, such as [[w:Rectification_(geometry)|cutting off each vertex at the midedge of each incident edge]], have been very well-studied; but how should we proceed? Let us see what happens when we [[w:Truncated_5-cell|truncate the 5-cells]] found in the 120-cell, by the simplest kinds of truncation. These three semi-regular 10-cells are closely related truncations of the regular 5-cell: * The 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of six 5-cells. * The 20-point 10-cell [[w:Truncated_5-cell|truncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of four 5-cells. * The 10-point 10-cell [[w:Rectified_5-cell|rectified 5-cell]] is the convex hull, and the convex common core, of a stellated compound of two 5-cells. In the following sections, we explore the effect of performing these truncations on the 120-cell's 120 5-cells. We begin by identifying some promising truncation points on the 120-cell's 5-cell edge chords at which to cut. If we cut off the 120-cell's 600 vertices at some point on its 1200 5-cell edges, we create new vertices on the edges of the 120 5-cells, which lie on a smaller 3-sphere than the 120-cell. How many vertices does the smaller 4-polytope thus created have? That is, how many distinct 5-cell edge truncation points occur in the 120-cell? As many as 1200, the number of 5-cell edges, or perhaps 2400, if each edge is truncated at both ends. But also perhaps fewer; for example, if the 120-cell contains pairs of 5-cells with intersecting edges, and the edges intersect at the point on each edge where we make our cut. [[File:Great_(12)_chords_of_radius_√2.png|thumb|400px|Chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {{radic|2}} chords form two regular {6} hexagons (black).{{Efn|name=compound of 5 cuboctahedra}} The 120-cell edges form two irregular {6} hexagons (red truncated triangles) with the {{radic|5}} chords. The {6} intersection points (black) of the {{radic|5}} chords form a smaller red regular hexagon of radius {{radic|1}} (inscribed in the red circle).]]In the irregular {12} central plane chord diagram, we see six truncation points on the six 104.5° 5-cell edges of length {{Radic|5}}, where two co-planar 5-cell edges intersect, directly under the midpoint of a 44.5° chord (and under the intersection point of two 60° chords). The six truncation points lie on a red circle that is a circumference of the smaller 4-polytope created by this truncation. They form a red regular hexagon inscribed in the red circle. The edge length of this regular hexagon is {{radic|1}}. The two intersection points on the {{Radic|5}} chord divide it into its golden sections. The center section of the chord is <small><math>1</math></small>. The center section plus either of the smaller sections is <small><math>\phi = \tfrac{\sqrt{5} + 1}{2} \approx 1.618</math></small>, the larger golden section. Each of the two smaller sections is <small><math>\Phi = \phi - 1 = \tfrac{1}{\phi} \approx 0.618</math></small>, the smaller golden section.{{Efn|The bitruncated {30/8} chord of the 120-cell of radius <small><math>2</math></small> provides a geometric derivation of the golden ratio formulas. First consider a 120-cell of radius <small><math>2\sqrt{2}</math></small> in which the {30/8} chord is <small><math>2\sqrt{5}</math></small> and the center section of the chord is <small><math>2</math></small>. Divide results by <small><math>2</math></small> to get a radius <small><math>\sqrt{2}</math></small> result. The left section of the chord is: :<small><math>\tfrac{\sqrt{5} - 1}{2} \approx 0.618</math></small> The center section plus the right section is: :<small><math>\tfrac{1 + \sqrt{5}}{2} \approx 1.618</math></small> The sum of these two golden sections is <small><math>\sqrt{5} \approx 2.236</math></small>, the chord length.}} The smaller golden sections <small><math>\Phi \approx 0.618</math></small> of the 5-cell edge are the same length as the 120-cell's 25.2° pentagon face diagonal chords. No 25.2° chords appear in the {12} central plane diagram, because they do not lie in {12} central planes. Each 104.5° 5-cell edge chord of length {{Radic|5}} has ''two'' points of intersection with other 5-cell edges, exactly 60° apart, the ''arc'' of a 24-cell edge chord, but with ''length'' {{radic|1}}. The center segment of the 5-cell edge, between the two intersection points, is a 24-cell edge in the smaller 4-polytope, and the red hexagon is a [[24-cell#Great hexagons|24-cell's great hexagon]] in the smaller 4-polytope. Nine other of its great hexagons, in other planes, each intersect with an antipodal pair of these {6} vertices. The dihedral angles between hexagon planes in a 24-cell are 60°, and four great hexagons intersect at each vertex. The 1200 5-cell edges, with two intersection points each, are reduced to 600 distinct vertices, so the smaller 4-polytope is a smaller 120-cell. The larger 120-cell, of radius {{radic|2}}, is concentric to a smaller instance of itself, of radius {{radic|1}}. Each 120-cell contains 225 distinct (25 disjoint) inscribed 24-cells. The smaller 24-cells are the [[w:Inscribed_sphere|insphere]] duals of the larger 24-cells. The vertices of the smaller 120-cell are located at the octahedral cell centers of the 24-cells in the larger 120-cell. Four 5-cell edges meet in 600 tetrahedral vertex figures. Four orthogonally intersecting 5-cell edges of the larger 120-cell meet in cubic vertex figures of 24-cells in the smaller 120-cell. Two disjoint 5-cell tetrahedral vertex figures are inscribed in alternate positions in each 24-cell cubic vertex figure. The 24-cell edges of the smaller 120-cell are the 5-cell edges of the larger 120-cell, truncated at both ends. The distance between the two points of intersection on a {{radic|5}} chord is {{radic|1}}, the same length as the 41.4° chord. But the actual 41.4° chords of the 120-cell do not appear in this diagram at all, because they do not lie in the 200 irregular {12} dodecagon central planes. === Bitruncating the 5-cells === The smaller concentric 120-cell can be built from 5-cell building blocks, by applying a specific kind of truncation operation to the blocks of the larger 120-cell called [[w:Bitruncation|''bitruncation'']]. This reveals a smaller irregular 4-polytope inside each 5-cell called the [[w:Bitruncated_5-cell|bitruncated 5-cell]]. The smaller unit-radius 120-cell is the convex hull of a compound of 20 disjoint (and 60 distinct) bitruncated 5-cells, bitruncated from the 120 disjoint 5-cells of the larger {{Radic|2}}-radius 120-cell. Bitruncation of the 120 disjoint 5-cells is the same truncation of the 120-cell described in the previous section, at the two golden section truncation points on each 104.5° 5-cell edge where two co-planar 5-cell edges intersect. [[File:Truncatedtetrahedron.gif|thumb|A 12-point [[w:Truncated_tetrahedron|truncated tetrahedron]] cell of the 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]].{{Sfn|Cyp: Truncated tetrahedron|2005}} Its edges are 41.4° chords of length 1 in a {{radic|2}}-radius 120-cell (or length {{radic|1/2}} in a unit-radius 120-cell). The 120-cell contains 20 disjoint (60 distinct) bitruncated 5-cells, containing 600 distinct truncated tetrahedra.]] The bitruncated 5-cell is a 30-vertex convex 4-polytope with 10 [[W:Truncated tetrahedron|truncated tetrahedron]] cells that have faces of two kinds: 4 triangle faces opposite 4 hexagon faces. The bitruncated 5-cell has 60 edges of the same length, 20 triangle faces, and 20 hexagon faces. Its 20 hexagon face planes are not [[24-cell#Great hexagons|24-cell central plane hexagons]]; they intersect each other at their edges, not at their long diameters. Its edges are not 60° 24-cell edge chords (the {{radic|2}} or 1 radius chords), but shorter 41.4° chords (of length 1 or {{radic|1/2}}), which do not appear at all in the diagram above, because they do not lie in the {12} central planes. The long diameter of the hexagon faces is not a 180° 120-cell long diameter chord (of length 2{{radic|2}} or 2) but a 90° 16-cell edge chord (of length 2 or {{radic|2}}). Consequently, three 16-cell tetrahedron cells (from three disjoint 16-cells) are inscribed in each truncated tetrahedron, at the three vertices of each face triangle. The truncated tetrahedron cell is a truncation of a tetrahedron of the same size as the tetrahedral cells of the 120-cell's 5-cells. The four smaller tetrahedra truncated from the corners of the larger tetrahedron have edges which are 25.2° chords (of length 1/𝜙 or {{radic|0.19}}). The truncated tetrahedron edges (of length 1 or {{radic|1/2}}) are equal in length to the 41.4° center sections of the 104.5° 5-cell edge chords (of length {{radic|5}} or {{radic|5/2}}). The shorter diagonal of the hexagon faces is the 75.5° chord (of length {{radic|3}} or {{radic|1.5}}), which is the 180° complement of the 104.5° 5-cell edge chord. The dimensions of the truncated tetrahedron cell suggest that it was cut directly from a 5-cell tetrahedron cell, simply by cutting off the tetrahedron corners, but remarkably, that is not the case. The edges of the bitruncated 5-cell are not actually center sections of 5-cell edges, although they are exactly that length, because the edges of the bitruncated 5-cell do not lie in the same {12} central planes as the 5-cell edges. They are not colinear with 5-cell edges in any way, and only intersect 5-cell edges at vertices (the 5-cell edges' intersection points). Bitruncation of the 5-cells does ''not'' simply truncate each tetrahedron cell in place. By creating new edges which connect the intersection points of 5-cell edges, bitruncation does create 600 truncated tetrahedron cells perfectly sized to fit within the 600 original tetrahedron cells, but at new locations, not centered on an original 5-cell tetrahedron cell. These new locations lie on a smaller 3-sphere than the original locations. [[File:Bitruncated_5-cell_net.png|thumb|Net of the bitruncated 5-cell honeycomb. 10 truncated tetrahedron cells alternately colored red and yellow.{{Sfn|Ruen: Net of the bitruncated 5-cell|2007}}]] The 3-dimensional surface of each bitruncated 5-cell is a honeycomb of 10 truncated tetrahedron cells. The truncated tetrahedra are joined face-to-face in a 3-sphere-filling honeycomb (like the cells of any 4-polytope), at both their hexagon and triangle faces. Each hexagonal face of a cell is joined in complementary orientation to the neighboring cell. Three cells meet at each edge, which is shared by two hexagons and one triangle. Four cells meet at each vertex in a [[w:Tetragonal_disphenoid|tetragonal disphenoid]] vertex figure. The 30-point bitruncated 5-cell is the convex common core (spatial [[w:Intersection|intersection]]) of six 5-point 5-cells in dual position. These six 5-cells are completely disjoint: they share no vertices, but their edges intersect orthogonally, at two points on each edge. Four 5-cell edges, from four of the six 5-cells, cross orthogonally in 30 places, the two intersection points on 60 5-cell edges: the 30 vertices of a bitruncated 5-cell. The six 5-cells are three dual pairs (in two different ways) of the self-dual 5-cell: six pairs of duals reciprocated at their common midsphere. Each dual pair intersects at just one of the two intersection points on each edge.{{Sfn|Klitzing|2025|loc=''sted'' (Stellated Decachoron)|ps=; [https://bendwavy.org/klitzing/incmats/sted.htm ''sted''] is the compound of two [https://bendwavy.org/klitzing/incmats/pen.htm ''pen'' (Pentachoron)] in dual position. Their intersection core ("Admiral of the fleet") is [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)].}} We have seen these six 5-cells before, illustrated in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Eleven|§Eleven]]'' above; they are the compound of six completely disjoint 5-cells visited during each 5-cell's characteristic isoclinic rotation of period 15.{{Efn|1=The 5-cell edges of the six disjoint pentagrams in the {30/12}=6{5/2} triacontagram illustration do not appear to intersect, as the 5-cell edge chords of the bitruncated 5-cell compound are said to intersect. The {30/12}=6{5/2} projection is a perspective view from inside the curved 3-dimensional space of the 120-cell's surface, looking straight down a cylindrical column of six stacked 5-cells. None of the 5-cell edges intersect in that curved 3-space, except where they meet at the 30 120-cell vertices. The 60 5-cell edges do intersect orthogonally in 4-space, in groups of four, at 30 points which lie on a smaller 3-sphere than the 120-cell. None of those 4-space intersections are visible in these projections of points and lines on the 120-cell's 3-sphere surface.|name=5-cell edges do not intersect is S³}} The six 5-cell compound is a stellated 4-polytope with 30 star-points, inscribed in the 120-cell.{{Efn|The stellated compound of six 5-cells in dual position is three pairs of 5-cells reciprocated at their common midsphere. It is composed of dual pairs of the [[W:Compound of five tetrahedra|compound of five tetrahedra]], which form the [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]; its 30 tetrahedral cells are three such dual pairs. In the compound of five tetrahedra the edges of the tetrahedra do not intersect. In the compound of ten tetrahedra they intersect orthogonally, but not at their midpoints. Each edge has two points of intersection on it. The compound of ten tetrahedra is five pairs of dual tetrahedra reciprocated at their common midsphere. It is inscribed in a dodecahedron (its convex hull). Its ''stellation core'' is an icosahedron, but its ''common core'' where the tetrahedron edges intersect is a dodecahedron, the tetrahedrons' convex spatial intersection. The stellated compound of six 5-cells has the analogous property: it is inscribed in a bitruncated 5-cell (its convex hull), and its common core is a smaller bitruncated 5-cell. (Its stellation core is a [[W:Truncated 5-cell#Disphenoidal 30-cell|disphenoidal 30-cell]], its dual polytope.)|name=compound of six 5-cells}} It is 1/20th of the 600-point [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#How many building blocks, how many ways|great grand stellated 120-cell]], the compound of 120 5-cells. The convex hull of its 30 star-points is a bitruncated 5-cell. In this stellated compound of six 5-cells in dual position, the bitruncated 5-cell occurs in two places and two sizes: as both the convex hull, and the convex common core, of the six 5-cells. Inscribed in the larger 120-cell of radius {{radic|2}}, the convex hull of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length 1. The convex common core of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length {{radic|1/2}}, inscribed in the smaller 120-cell of radius 1. In the 120-cell, 120 disjoint 5-cell building blocks combine in dual position groups of six related by the 5-cell's isoclinic rotation, to make 60 bitruncated 5-cells inscribed in the self-dual 5-cells' midsphere (at their edge intersections), and also 60 larger bitruncated 5-cells inscribed in the 120-cell, with each of the 600 vertices shared by three bitruncated 5-cells. The 120-cell is the convex hull of a compound of 20 disjoint (60 distinct) 30-point bitruncated 5-cells, generated by the characteristic rotation of its 120 completely disjoint 5-cells.{{Sfn|Klitzing|2025|loc= ''teppix'' (tripesic hexacosachoron)|ps=; ''[https://bendwavy.org/klitzing/incmats/teppix.htm teppix]'' is a compound of 60 [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)] with 3 ''deca'' sharing each vertex.}}{{Efn|In the 120-cell, 600 tetrahedron cells of 120 completely disjoint 5-cells intersect at two truncation points on each edge. Those 2400 truncation points are the vertices of 200 disjoint (and 600 distinct) truncated tetrahedra, which are the cells of 20 disjoint (and 60 distinct) bitruncated 5-cells. The 60 bitruncated 5-cells share vertices, but not edges, faces or cells. Each bitruncated 5-cell finds its 30 vertices at the 30 intersection points of 4 orthogonal 5-cell edges, belonging to 6 disjoint 5-cells, in the original 120-cell. Each bitruncated 5-cell vertex lies on an edge of 4 disjoint original 5-cells. Each bitruncated 5-cell edge touches intersection points on all 6 disjoint original 5-cells, and is shared by 3 truncated tetrahedra of just one bitruncated 5-cell.}} In [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Concentric 120-cells|the previous section]] we saw that the six 5-cell edges in each central plane intersect at the {6} vertices of the red hexagon, a great hexagon of a 24-cell. Each 5-cell edge, truncated at both ends at those intersection points, is a 24-cell edge of one of the 24-cells inscribed in a smaller 120-cell: the 600 intersection points. In this section we have seen how that truncation of 5-cell edges at both ends is the bitruncation of the 5-cell, and those 5-cell edges, truncated at both ends, are the same length as edges of bitruncated 5-cells inscribed in the original 120-cell. Bitruncating the {{radic|2}}-radius 120-cell's 120 5-cells reveals a smaller unit-radius 120-cell. The 24-cell edges of the smaller 120-cell are 5-cell edges of a larger-radius-by-{{radic|2}} 120-cell, truncated at both ends. Both 120-cells have 24-point 24-cells and 30-point bitruncated 5-cells inscribed in them. The 60° edge length of the 24-cells equals the radius; it is {{radic|2}} times the 41.4° edge length of the bitruncated 5-cells. The 60° 24-cell edges lie in the {12} central planes with the 5-cell edges and the 120-cell edges; but the 41.4° bitruncated 5-cell edges do not. The 120-cell contains 25 disjoint (225 distinct) 24-cells, and 20 disjoint (60 distinct) bitruncated 5-cells. Although regular 5-cells do not combine to form any regular 4-polytope smaller than the 120-cell, the 5-cells do combine to form semi-regular bitruncated 5-cells which are subsumed in the 120-cell.{{Efn|Although only major chords occur in regular 4-polytopes smaller than the 120-cell, minor chords do occur in semi-regular 4-polytopes smaller than the 120-cell. Truncating the 5-cell creates minor chords, such as the 41.1° edges of the bitruncated 5-cell.}} The 41.4° edge of the 30-point bitruncated 5-cell is also the triangle face edge we found in the 60-point central [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The real hemi-icosahedron|section 8₃ (Moxness's Hull #8) rhombicosidodecahedron]]. There are 60 distinct section 8₃ rhombicosidodecahedra and 600 distinct truncated tetrahedron cells of 60 distinct (20 disjoint) bitruncated 5-cells, and they share triangle faces, but little else. The truncated tetrahedron cells cannot be inscribed in the rhombicosidodecahedra, and the only chords they share are the 41.4° triangle edge and the 75.5° chord (the 180° complement of the 104.5° 5-cell edge chord). The section 8₃ rhombicosidodecahedron's 20 triangle faces lie over the centers of 20 larger-by-√2 5-cell faces, parallel to them and to a {12} central plane. The 5-cell faces are inscribed in the rhombicosidodecahedron, but are not edge-bound to each other; the 20 faces belong to 10 completely disjoint 5-cells. The 5-cell edges (but not the 5-cell faces) lie in {12} central planes; the 5-cell faces, the bitruncated 5-cell edges and their triangle and hexagon faces do not. Each section 8₃ rhombicosidodecahedron is the intersection of ten {12} central planes, shared pairwise with ten other rhombicosidodecahedra; 11 rhombicosidodecahedra share ten {12} central planes pairwise, as cells of a 4-polytope share face planes pairwise. Each truncated tetrahedron cell of a bitruncated 5-cell shares none of the {12} central planes; it is the intersection of 6 great rectangles, with two parallel 41.1° edges lying in each, alternating with two parallel 138.6° chords (its hexagon face diameters). Each bitruncated 5-cell is the intersection of 30 great rectangle {4} central planes. A truncated tetrahedron is face-bonded to the outside of each triangle face of a rhombicosidodecahedron. Three of its hexagon faces stand on the long edge of a rectangle face, perpendicular to the rectangle. We find the 25.2° chord as the edge of the non-central section 6₃ (Moxness's Hull #6) rhombicosidodecahedron. Those 120 semi-regular rhombicosidodecahedra have only that single edge (of length 1/𝜙 in a {{radic|2}}-radius 120-cell, or 1/𝜙{{radic|2}} in a unit-radius 120-cell). This edge length is in the golden ratio to the 41.4° edge of the 30-point bitruncated 5-cells, which is also the triangle face edge of the central section 8₃ (Moxness's Hull #8) rhombicosidodecahedron. The 120 semi-regular section 6₃ rhombicosidodecahedra share their smaller edges with 720 pentagonal prisms, 1200 hexagonal prisms and 600 truncated tetrahedron cells, in a semi-regular honeycomb of the 120-cell discovered by Alicia Boole Stott and described in her 1910 paper.{{Sfn|Boole Stott|1910|loc=Table of Polytopes in S₄|ps=; <math>e_2e_3C_{120}\ RID\ P_5\ P_6\ tT</math>}} These truncated tetrahedra are 1/𝜙 smaller than the 600 cells of the bitruncated 5-cells. The 60 distinct section 8₃ rhombicosidodecahedra (Moxness's Hull #8) share pentagon faces. Each of the 120 dodecahedron cells lies just inside 12 distinct rhombicosidodecahedra which share its volume. Each rhombicosidodecahedron includes a ball of 13 dodecahedron cells, 12 around one at the center of the rhombicosidodecahedron, within its volume. The remainder of the rhombicosidodecahedron is filled by 30 dodecahedron cell fragments that fit into the concavities of the 13 cell ball of dodecahedra. These fragments have triangle and rectangle faces. === Rectifying the 16-cells === Bitruncation is not the only way to truncate a regular polytope, or even the simplest way. The simplest method of truncation is [[w:Rectification_(geometry)|''rectification'']], complete truncation at the midpoint of each edge. Moreover, the 5-cell is not the only 120-cell building block we can truncate. We saw how bitruncation of the {{radic|2}}-radius 120-cell's 5-cells reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 20 disjoint (60 distinct) bitruncated 5-cells. In the next paragraph we describe how rectification of the {{radic|2}}-radius 120-cell's 16-cells also reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 25 disjoint (225 distinct) 24-cells. Those two operations on the 120-cell are equivalent. They are the same truncation of the 120-cell, which bitruncates 5-cells into bitruncated 5-cells, and also rectifies 16-cells into 24-cells. This single truncation of the 120-cell captures the distant relationship of 5-cell building blocks to 16-cell building blocks. Rectifying a {{radic|2}}-radius 16-cell of edge 2 creates a unit-radius 24-cell of unit edge, which is the compound of three unit-radius 16-cells. Rectifying one of those inscribed unit-radius 16-cells of edge {{radic|2}} creates a smaller 24-cell of radius and edge {{radic|1/2}}, which is the [[24-cell#Relationships among interior polytopes|common core (intersection]]) of the unit 24-cell and its three inscribed 16-cells. Like the 120-cell itself, the 24-cell is concentric to a smaller instance of itself of {{radic|1/2}} its radius. The common core of each of the 24-cells inscribed in the 120-cell is the corresponding 24-cell in the smaller 120-cell. === Rectifying the 5-cells === In the previous section we bitruncated the 5-cells and rectified the 16-cells, as one combined truncation operation that yields a smaller 120-cell of {{radic|1/2}} the radius. We can also rectify the 5-cells; but that is another distinct truncation operation, that yields a smaller 4-polytope of {{radic|3/8}} the radius. [[File:Great (12) chords of rectified 5-cell.png|thumb|400px|5-cell edge chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {6} bitruncation points (two on each of the 104.5° {{radic|5}} 5-cell edges) lie on a smaller 120-cell of radius 1 (the red circle); they are bitruncated 5-cell vertices. The {6} rectification points (at the midpoints of the 5-cell edges) lie on a still smaller 1200-point 4-polytope of radius {{radic|0.75}} ≈ 0.866 (the magenta circle); they are rectified 5-cell vertices.]] Rectifying the 5-cell creates the 10-point 10-cell semi-regular [[W:Rectified 5-cell|rectified 5-cell]], with 5 tetrahedral cells and 5 octahedral cells. It has 30 edges and 30 equilateral triangle faces. The 3-dimensional surface of the rectified 5-cell is an alternating [[W:Tetrahedral-octahedral honeycomb|tetrahedral-octahedral honeycomb]] of just 5 tetrahedra and 5 octahedra, tessellating the 3-sphere. Its vertex figure is the cuboctahedron. The rectified 5-cell is a [[w:Blind_polytope|Blind polytope]], because it is convex with only regular facets. It is a bistratic lace tower which has exactly three vertex layers with the same Coxeter symmetry, aligned on top of each other.{{Sfn|Klitzing|2025|loc=''[https://bendwavy.org/klitzing/incmats/rap.htm rap (rectified pentachoron)]''}} If the 120 5-cells in a radius {{radic|2}} 120-cell are rectified, the rectified 5-cells lie on a smaller 4-polytope of radius {{radic|3/4}} (the magenta circle in the diagram), inscribed at the 1200 midedges of the 5-cells.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the ''unit-radius'' 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the unit-radius 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of the [[24-cell#Characteristic orthoscheme|24-cell's 4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[24-cell#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the ''area'' of the equilateral triangle face of the unit-edge, unit-radius 24-cell.|name=root 3/4}} This smaller 4-polytope is not a smaller 120-cell; it is the convex hull of a 1200-point compound of two 120-cells. The rectified 5-cell does not occur inscribed in the 120-cell; it only occurs in this compound of two 120-cells, 240 regular 5-cells, and 120 rectified 5-cells. The rectified 5-cell with its 80.4° edge chord does not occur anywhere in a single 120-cell, so the rectified 5-cell's edges are not the edges of any polytope found in the 120-cell. The rectified 5-cell's significance to the 120-cell is well-hidden, but we shall see that it has an indirect role as a building block of the 11-cells in the 120-cell. Each 10-point rectified 5-cell is the convex hull of a stellated compound of two completely orthogonal 5-point 5-cells: five pairs of antipodal vertices. Their edges intersect at the midedge, and they are ''not'' in dual position (not reciprocated at their common 3-sphere). In this stellated compound of two completely orthogonal 5-cells (which does not occur in the 120-cell), the rectified 5-cell occurs in two places and two sizes: as both the convex hull of the vertices, and the convex common core of the midedge intersections. The edge length of the rectified 5-cells in the smaller 1200-point 4-polytope of radius {{radic|3/4}} is {{radic|5/4}}. The edge length of a unit-radius rectified 5-cell is {{radic|5/3}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|3}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}. In the 120-cell of radius {{radic|2}}, the 104.5° {{radic|5}} chord is the 5-cell edge, and the 75.5° {{radic|3}} chord is the distance between two parallel 5-cell edges (belonging to two disjoint 5-cells). The 104.5° and 75.5° chords are 180° complements, so they form great rectangles in the {12} central planes of the 120-cell (the red rectangles in the diagram). In the 1200-point compound of two 120-cells of radius {{radic|3}} where 120 rectified 5-cells occur, the {{radic|3}} chord is the ''radius'' (not the 75.5° chord), and the {{radic|5}} chord is the ''rectified'' 5-cell edge of arc 80.4° (not the 104.5° regular 5-cell edge). === Truncating the 5-cells === [[File:Great (12) chords of unit thirds radius.png|thumb|400px|Truncating the 120-cell's 5-cells at ''one-third'' of their edge length produces a smaller 120-cell of ''one-half'' the radius, with vertices at {6} one-third intersection points of the 120° {{Radic|6}} chords (''not'' of the 104.5° {{Radic|5}} 5-cell edge chords). The green {6} hexagon is a 24-cell great hexagon in the resulting smaller-by-one-half 1200-point 4-polytopes. Because there are {12} such intersection points in each {12} central plane, there are two chiral ways to perform this truncation, which produce disjoint 1200-point 4-polytopes.]] A third simple way to truncate the 5-cell is at one-third of its edge length. This truncation of the 5-cell creates a 20-point, 10-cell semi-regular 4-polytope, known somewhat ambiguously as ''the'' [[w:Truncated_5-cell|truncated 5-cell]], with 5 truncated tetrahedron cells (like the bitruncated 5-cell's), and 5 regular tetrahedron cells (like the rectified 5-cell's). The 3-dimensional surface of the truncated 5-cell is an alternating honeycomb of 5 truncated tetrahedra and 5 regular tetrahedra. It resembles the smaller rectified 5-cell with truncated tetrahedra instead of octahedra, or the larger bitruncated 5-cell with half its truncated tetrahedra replaced by regular tetrahedra. When the regular 5-cell is truncated at ''one-third'' of its edge length, the radius and edge length of the the resulting truncated 5-cell are ''one-half'' the regular 5-cell's radius and edge length. When the 120 5-cells in a 120-cell of radius 2 are truncated at one-third of their edge length, the truncated 5-cells lie on a smaller 120-cell of radius 1. The edge length of the unit-radius truncated 5-cell is {{radic|5/8}}, one-half the unit-radius 5-cell's edge length of {{radic|5/2}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|8}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}, and the rectified 5-cell is characterized by the ratio {{radic|5}} to {{radic|3}}. The 20-point truncated 5-cell is the convex common core of a stellated compound of four 5-cells (the four 5-cells' spatial intersection). The convex common core has half the radius of the convex hull of the compound. The four 5-cells are orthogonal (aligned on the four orthogonal axes), but none of their 20 vertices are antipodal. The 5-cells are ''not'' in dual position (not reciprocated at their common 3-sphere). The 5-cell edges do ''not'' intersect, but truncating the 120-cell's 5-cell edge chords at their one-third points truncates the 120-cell's other chords similarly. It is the 120-cell's 120° chords (of length {{Radic|6}} in a {{Radic|2}}-radius 120-cell, or {{Radic|3}} in a unit-radius 120-cell) which intersect each other at their one-third points. Four edges (one from each 5-cell) intersect orthogonally at just ''one'' of the two one-third intersection points on each of the 2400 120° chords that join vertices of two disjoint 5-cells. There are two chiral ways to perform this truncation of the 120-cell; they use the alternate intersection points on each edge, and produce disjoint 600-point 120-cells. The 52.25° edge chord of the truncated 5-cell (one-half the 5-cell's 104.5° edge chord) is not among the [[120-cell#Chords|chords of the 120-cell]], so the truncated 5-cell does not occur inscribed in the 120-cell; it occurs only in a compound of four 120-cells, and 480 regular 5-cells, and 120 truncated 5-cells. In the stellated compound of four orthogonal 5-cells (which does not occur in the 120-cell), the truncated 5-cell occurs in two places and two sizes: as both the convex hull of the 20 vertices, and the convex common core (of half the radius of the convex hull) of the 20 intersection points of four orthogonal 120° chords. == The perfection of Fuller's cyclic design == [[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 orthogonal axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]] This section is not an historical digression, but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]''}} After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session<ref>{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}</ref> where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.<ref>{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}</ref>|name=Snelson and Fuller}} A tensegrity icosahedron is an icosahedral geodesic sphere whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables. The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 10 of in Moxness's Hull #8 rhombicosidodecahedron, the real cell of the 11-cell. The Jessen's was named by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a [[W:Cubist|cubist]] shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point ''vector equilibrium'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point vector equilibrium, and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposite equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (a hexad), then backing away again while still rotating in the same directions. All this was overlaid with Fuller's own deep commentary, in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing "vector edge cube" (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}} Earlier, we noticed 10 Jessen's inscribed in each 60-point rhombicosidodecahedron central section of the 120-cell (each real hemi-icosahedron). Each rhombicosidodecahedron is a compound of 5 disjoint Jessen's, in two different ways, just the way the 120-cell is a compound of 5 disjoint 600-cells, in two different ways. In the rhombicosidodecahedron each regular icosahedron vertex has been replaced by the five vertices of a little pentagon face (a 120-cell face), and the regular icosahedron has been replaced by 5 disjoint (10 distinct) Jessen's icosahedra.{{Efn|name=compound of 5 cuboctahedra}} The 3 pairs of parallel 5-cell edges in each Jessen's lie a bit uncertainly, infinitesimally mobile and [[Kinematics of the cuboctahedron#Elastic-edge transformation|behaving like the struts of a tensegrity icosahedron]], so we can push any parallel pair of them apart or together infinitesimally, making each Jessen's icosahedron expand or contract infinitesimally. All 600 Jessen's, all 60 rhombicosidodecahedra, and the 120-cell itself expand or contract infinitesimally, together.{{Efn|name=tensegrity 120-cell}} Expansion and contraction are Boole Stott's operators of dimensional analogy, and that infinitesimal mobility is the infinite calculus of an inter-dimensional symmetry. The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three rectangular chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables. [[File:Jessen's √2 radius dimensions.png|thumb|400px|Moxness's 60-point section 8₃ rhombicosidodecahedron is a compound of 5 of this 12-point Jessen's icosahedron, shown here in a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed {{radic|1.5}} green cube, and its 8 equilateral triangle faces are 24-cell faces. This is a ''vertex figure'' of the 120-cell. The center point is also a vertex of the 120-cell.]] As a matter of convenience in this paper, we have used {{radic|2}}-radius metrics for 3-sphere polytopes, so e.g. the 5-cell edge is {{radic|5}}, where in unit-radius coordinates it would be {{Radic|5/2}}. Here we give two illustrations of the Jessen's using two different metrics: the 2-sphere Jessen's has a {{radic|5}} diameter, and the 3-sphere Jessen's has a {{radic|2}} radius. This reveals a curiously cyclic way in which our 2-sphere and 3-sphere metrics correspond. In the embedding into 4-space the characteristic root factors of the Jessen's seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord. We might have expected to find the 6-point hemi-icosahedron's 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the Jessen's are the 5-cell edges. A 5-cell face has its three {{radic|5}} edges in three different Jessen's icosahedra. The Jessen's is not a cell, but one of the 120-cell's vertex figures, like the [[600-cell#Icosahedra|120 regular icosahedron vertex figures in the 600-cell]]. That is why we find 600 Jessen's, of course. The center point in this Jessen's illustration is another ''vertex'' of the 120-cell, not the empty center of a cell.{{Efn|The 13 vertices of the illustration which include its center point lie in the curved 3-space of the 3-sphere, on the 120-cell's surface. In 4-space, this object is an [[W:Icosahedral pyramid|icosahedral pyramid]] with a Jessen's icosahedron as its base, and the apical center vertex as its apex. The center point in the illustration is a vertex of the 120-cell, and the center of the curved Jessen's, and the apex of the icosahedral pyramid, but it is not the center point in 4-space of a flat 3-dimensional Jessen's icosahedron. The center point of the base Jessen's icosahedron is a point inside the 120-cell, not a 120-cell vertex on its surface. It lies in the same 3-dimensional flat-slice hyperplane as the 12 vertices of the base Jessen's icosahedron, directly below the 13th 120-cell vertex.}} Each Jessen's includes the central apex vertex, {{radic|2}} radii, {{radic|2}} edges and {{radic|5}} chords of a vertex figure around the 120-cell vertex at its center. The {{radic|2}} face edges are 24-cell edges (also tesseract edges), and the inscribed green cube is the 24-cell's cube vertex figure. The 8 {{radic|2}} face triangles occur in 8 distinct 24-cells that meet at the apex vertex.{{Efn|Eight 24-cells meet at each vertex of a [[24-cell#Radially equilateral honeycomb|honeycomb of 24-cells]]: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell.{{Efn|In the 600-cell, which contains [[600-cell#Twenty-five 24-cells|25 24-cells]], 5 24-cells meet at each vertex. Each pair of 24-cells at the vertex meets at one of 200 distinct great hexagon central planes. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 8 other 24-cells. In the 120-cell, which contains 10 600-cells (5 disjoint 600-cells two different ways) and 225 24-cells (25 disjoint 24-cells), 8 24-cells meet at each vertex. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 208 other 24-cells. But since in the 120-cell the great hexagons lie in pairs in one of 200 {12} central planes (containing 400 great hexagons), each 24-cell shares one of its {12} central ''planes'' with .. other 24-cells.}}}} This Jessen's vertex figure includes 5-cell edges and 24-cell edges (which are also tesseract edges), so it is descriptive of the relationship between those regular 4-polytopes, but it does not include any 120-cell edges or 600-cell edges, so it has nothing to say, by itself, about the <math>H_4</math> polytopes. It is only a tiny fraction of the 120-cell's full vertex figure, which is a staggeringly complex star: 600 chords of 30 distinct lengths meet at each of the 600 vertices. The {{radic|5}} chords are 5-cell edges, connecting vertices in different 24-cells. The 3 pairs of parallel 5-cell edges in each Jessen's lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th pair of parallel 5-cell edges orthogonal to all of them, in fact three more orthogonal pairs, since 6 orthogonal planes (not just 4) intersect at a point in 4-space. The Jessen's situation is that it lies completely orthogonal to another Jessen's, the vertex figure of the antipodal vertex, and its 3 orthogonal planes (xy, yz, zx) lie completely orthogonal to its antipodal Jessen's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} These 6 pairs of parallel 5-cell edges form a 24-point 4-polytope, composed of two completely orthogonal 12-point Jessen's, inscribed in two completely orthogonal rhombicosidodecahedra. This 24-point 4-polytope is not a 24-cell: the 24-cell is not a compound of two 12-point Jessen's. But it turns out that two completely orthogonal 12-point Jessen's indirectly define a 24-point 24-cell. We shall see that their 4-space intersection is a 24-cell. This finding, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, brings Fuller's study of [[w:Tesseract#Radial_equilateral_symmetry|radially equilateral]] vector equilibrium polytopes to its completion in the 24-cell. Fuller began with the hexagon, the 6-point vector equilibrium in 2 dimensions, the only polygon with its radius equal to its edge length. He studied the cuboctahedron, the 12-point vector equilibrium in 3 dimensions, the only polyhedron with its radius equal to its edge length, in all its flexible guises. He discovered its stable equilibrium as the the Jessen's shadfish, with its cube of 6 open mouths and 90° dihedral angles between all its faces, the geometric center of [[WikiJournal Preprints/Kinematics of the cuboctahedron|the cuboctahedron's kinematic transformation]] through the regular polyhedra: tetrahedron, octahedron, Jessen's, regular icosahedron, and cuboctahedron. Fuller's study of kinematic Euclidean geometry did not reach the 4-polytopes, and the ultimate 24-point vector equilibrium in 4 dimensions, the 24-cell, the unique <math>F_4</math> symmetry found only in 4 dimensions. But Fuller led us up to it, through the kinematics of infinitesimal mobility, and that route to it is our clue to the infinite calculus of dimensional expansion and contraction. We observe this geometry, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, only in the 120-cell. The 600-cell contains 12-point Jessen's, but no completely orthogonal pairs of them. The 24-cell individually, and the 25 24-cells in the 600-cell, are not occupied by a pair of 12-point Jessen's. The 24-point 24-cell is not, in fact, a compound of two 12-point Jessen's. While the 120-cell's ratio of disjoint 12-point Jessen's to disjoint 24-point 24-cells is <math>50/25 = 2/1</math>, the ratio of distinct 12-point Jessen's to distinct 24-point 24-cells is <math>600/225 = 8/3 </math>. We observe another geometry, of 24-cells in dual positions, only in the 120-cell. No two 24-cells in the 600-cell are in dual positions, but in the 120-cell with 225 distinct 24-cells (25 disjoint 24-cells), every 24-cell is in dual position to other 24-cells. The 24-cell is self-dual, and when two 24-cells of the same radius are in dual position, they are completely disjoint with respect to vertices, but they intersect at the midpoints of their 96 orthogonal edges. Since four orthogonal lines intersect at a point in 4-space, in addition to the midedge radius and the two intersecting edges there is a third intersecting edge through each point of contact: ''three'' 24-cells lie in dual positions to each other, with their orthogonal edges intersecting. Three ''pairs'' of 24-cells lie in orthogonal dual positions to each other, sharing no vertices, but the same 96 midedge points. We also observe this geometry, of 24-cells in dual positions, in the irregular {12} dodecagon central planes, which have two inscribed great {6} hexagons, offset from each other irregularly by a 15.5° arc on one side (a 120-cell edge chord) and a 44.5° arc on the other side. The 600-cell and the 24-cell contain only great {6} hexagon planes. The two inscribed great {6} hexagons in each {12} central plane belong to a pair of 24-cells in dual position. We observe inscribed 5-cells only in the 120-cell. The 600-cell has <math>5^2 = 25</math> distinct 24-cells inscribed in 120 vertices, and is a regular compound of <math>5</math> disjoint 24-cells in 10 different ways, but it has no inscribed 5-point 5-cells joining corresponding vertices of 5 of its 25 24-cells.{{Efn|The 600-cell does have inscribed 5-point great pentagons joining corresponding vertices of 5 of its 25 24-cells. The 600-cell has 2-dimensional pentads, but only the 120-cell has 4-dimensional pentads.}} The 120-cell has <math>5^2 \times 3^2 = 225</math> distinct 24-cells inscribed in 600 vertices, and is a regular compound of <math>5^2 = 25</math> disjoint 24-point 24-cells in 10 different ways, and it has 120 inscribed 5-cells joining corresponding vertices of 5 of its 225 24-cells. [[File:Great 5-cell √5 digons rectangle.png|thumb|400px|Three {{radic|5}} x {{radic|3}} rectangles (red) are found in 200 central planes of the radius {{radic|2}} 120-cell, and in its 600 Jessen's icosahedra, where 3 orthogonal rectangles comprise each 12-point Jessen's. Each central plane intersects {12} vertices in an irregular great dodecagon. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges (solid red), which form two opposing ''irregular'' great hexagons (truncated triangles) with the {{radic|5}} chords. The {12} central planes also contain four {{radic|6}} great triangles (green), inscribed in two {{radic|2}} ''regular'' great hexagons. 1200 smaller {{radic|5}} 5-cell ''face'' triangles (blue) occupy 600 other, non-central planes.]] The Jessen's eight {{radic|6}} triangle faces lie in eight great {6} hexagons in eight {12} central planes of the 120-cell. The Jessen's {{radic|5}} chords lie in great {4} rectangles ({{radic|5}} by {{radic|3}}) in orthogonal central planes of the Jessen's. These are ''also'' {12} central planes of the 120-cell. We can pick out the {{radic|5}} by {{radic|3}} rectangles in the {12} central plane chord diagrams (bounded by red dashed lines). The Jessen's vertex figure is bounded by eight {12} face planes, and divided by six orthogonal {12} central planes, and all 14 planes are {12} central planes of the 120-cell. The 5-cells' ''face'' planes are ''not'' central planes of the 120-cell. Recall that 10 distinct Jessen's are inscribed in each rhombicosidodecahedron, as two chiral sets of 5 completely disjoint Jessen's, such that two {{radic|5}} 5-cell edges meet at each vertex of the rhombicosidodecahedron. These are two of the four 5-cell edges that meet at each vertex of the 5-cell: edges of a 5-cell face, 20 of which are disjointly inscribed in each rhombicosidodecahedron. In each Jessen's the 6 {{radic|5}} reflex edges are disjoint, and in each rhombicosidodecahedron only two edges meet at each vertex, but in the 120-cell each {{radic|5}} chord meets three others, that lie in three other Jessen's. Each 5-cell face triangle has each edge in a distinct Jessen's, but the face triangle lies in just one rhombicosidodecahedron. The 1200 5-cell face triangles lie in opposing pairs, in one of 600 ''non-central'' hexagon ''face'' planes. Each of the 60 rhombicosidodecahedra is a compound of 10 Jessen's (5 disjoint Jessen's in two different ways), just the way the 120-cell is a compound of 10 600-cells (5 disjoint 600-cells in two different ways), and the 120-cell's dodecahedron cell is a compound of 10 600-cell tetrahedron cells (5 disjoint tetrahedra in two different ways). The 600 Jessen's in the 120-cell occur in bundles of 8 disjoint Jessen's, in 4 completely orthogonal pairs, each pair aligned with one of the four axes of the Cartesian coordinate system. Collectively they comprise 3 disjoint 24-cells in orthogonal dual position. They are [[24-cell#Clifford parallel polytopes|Clifford parallel 4-polytopes]], 3 completely disjoint 24-cells 90° apart, and two sets of 4 completely disjoint Jessen's 15.5° apart. Opposite triangle faces in a Jessen's occupy opposing positions in opposite great hexagons. In contrast, the two completely orthogonal Jessen's are completely disjoint, with completely orthogonal bounding planes that intersect only at one point, the center of the 120-cell. The corresponding {{radic|6}} triangle faces of two completely orthogonal Jessen's occupy completely orthogonal {12} central planes that share no vertices. If we look again at a single Jessen's, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) Jessen's lies in 4-space, it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (vertex figure) Jessen's is part of a 16-point (8-cell) tesseract containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already found it was part of a 24-point (24-cell) 4-polytope, which contains 3 16-point (8-cell) tesseracts. Each 12-point (6 {{radic|5}} reflex edge) Jessen's is one of 10 concentric Jessen's in a rhombicosidodecahedron, two sets of 5 disjoint Jessen's rotated with respect to each other isoclinically by 12° x 12° = 15.5°, with a total of 60 disjoint {{radic|5}} edges. Each 12-point (24 {{radic|6}} edge) Jessen's is one of 8 concentric Jessen's in two 24-cells in dual positions, rotated with respect to each other isoclinically by 41.4° x 41.4° = 90°, with a total of 192 {{radic|6}} edges.{{Efn|There are 96 {{radic|6}} chords in each 24-cell, linking every other vertex under its 96 {{radic|2}} edges.}} The 24-point 24-cell has 4 Hopf fibrations of 4 hexagonal great circle fibers, so it is a complex of 16 great hexagons, generally not orthogonal to each other, but containing 3 sets of 4 orthogonal great hexagons. Three Borromean link great rectangles are inscribed in each great hexagon, and three tesseracts are inscribed in each 24-cell. Four of the 6 orthogonal [[w:Borromean_rings|Borromean link]] great rectangles in each completely orthogonal pair of Jessen's are inscribed in each tesseract. == Conclusion == Thus we see what the 11-cell really is: an unexpected seventh regular convex 4-polytope falling between the 600-cell and 120-cell, a quasi-regular compound of 600-cell and 5-cell (an icosahedron-tetrahedron analogue), as the 24-cell is an unexpected sixth regular convex polytope falling between the 8-cell and 600-cell, a quasi-regular compound of 8-cell and 16-cell (a cube-octahedron analogue). Like the 5-cell, the 11-cell is a far-side 4-polytope with its long edges spanning the near and far halves of the 3-sphere. Unlike the 5-cell, the 11-cell's left and right rotational instances are not the same object: they have distinct cell polyhedra, which are duals. The 11-cell is a real regular convex 4-polytope, not just an [[W:abstract polytope|abstract 4-polytope]], but not just a singleton regular convex 4-polytope, and not just a single kind of cell honeycomb on the 3-sphere.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} Though it is all those things singly, it never occurs singly, but its multiple instances in the 120-cell compound to all those things, and significantly more. The 11-cell (singular) is the 11-vertex (17 cell) non-uniform Blind 4-polytope, with 11 non-uniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. The abstract regular 11-point (11-cell) has a realization in Euclidean 4-space as this convex 4-polytope, with regular facets and regular triangle faces. The 11-cell (plural) is subsumed in the 120-cell, as all the regular convex 4-polytopes are. The compound of eleven 11-cells (the ..-cell) and Schoute's compound of five 24-cells (the 600-cell) is the quasi-regular 137-point (..-cell) 4-polytope, an object of further study. The 11-cells' realization in the 120-cell as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (4-simplex and 4-orthoplex), which are both inscribed in the 11-point (17-cell), 137-point (..-cell) and 600-point (120-cell), but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationship between the regular ''n''-polytopes of different ''n''. The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies. == Build with the blocks == <blockquote>"The best of truths is of no use unless it has become one's most personal inner experience."{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: "The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery."}}}}</blockquote> <blockquote>"Even the very wise cannot see all ends."{{Sfn|Tolkien|1954|loc=Gandalf}}</blockquote> No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to show how I came by my understanding of these objects, since I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, in my imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am. {{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=2}} {{Regular convex 4-polytopes|columns=7|wiki=W:|radius=1}} == Acknowledgements == ... == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite web| url = https://www.youtube.com/watch?v=9sM44p385Ws| title = Vector Equilibrium | first = R. 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Gregory|editor-link4=W:User:Jgmoxness|year=2024|ref={{SfnRef|Ruen & Goucher et al. eds. 600-cell|2024}}}} * {{Citation|title=120-cell|title-link=120-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|ref={{SfnRef|Ruen & Goucher et al. eds. 120-cell|2024}}}} * {{Cite book|last=Sandperl|first=Ira|author-link=W:Ira Sandperl|title=A Little Kinder|year=1974|publisher=Science and Behavior Books|place=Palo Alto, CA|isbn=0-8314-0035-8|lccn=73-93870|url=https://www.allinoneboat.org/a-little-kinder-an-old-friend-moves-on/|ref={{SfnRef|Sandperl|1974}}}} * {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|orig-date=1954|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|date=1967|publisher=Houghton Mifflin|place=Boston|author-link=W:J.R.R.Tolkien|title-link=W:The Lord of the Rings|ref={{SfnRef|Tolkien|1954}}}} {{Refend}} syuv3s49satf4o2ww09b5u90tp46nrs 2816268 2816265 2026-06-19T01:21:29Z Dc.samizdat 2856930 /* Concentric 120-cells */ 2816268 wikitext text/x-wiki = A symmetrical arrangement of eleven 11-cells = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|March 2024 - June 2026}} <blockquote>[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]] ₅{3,5,3}₅, a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]] {3,5}₅, a hexad non-orientable polyhedron. The 11-cell is described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, we find that the 11-cell has a realization in Euclidean 4-space inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the convex regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra and 120 regular 5-cells. We find that the 120-cell also contains: a non-uniform icosahedral polyhedron that contains the realization of the abstract hemi-icosahedron; real 11-point 11-cells made from 11 of it; and a compound of eleven real 11-cells. We also find a quasi-regular compound of the compound of eleven 11-cells and [[w:Schoute|Schoute]]'s compound of five 24-cells (the 600-cell). We describe the real 11-point 11-cell 4-polytope; its compound of eleven 11-cells; the quasi-regular compound; and their relation to the regular polytopes.</blockquote> == Introduction == [[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hemi-icosahedral hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them. [[File:120-cell.gif|thumb|360px|The picture on the cover of the box of 4-dimensional building blocks.{{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]]."}} Only the 120-cell's own edges are shown. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and 11-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]] The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-hypercube, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the 600-point [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). The 5-cell has a fundamental relationship to all the other 4-polytopes, but not one as simple as compounding, so it is not immediately useful to children trying to learn to build with 4-dimensional building blocks. But the 16-cell is our very starting point, and the most frequently used tool in the box. Nevertheless, to build the 11-cell, we start with the 5-cell. The 5-cell and 11-cell are both self-reciprocal (their own duals). They are the only 4-polytopes where every cell shares a face with every other cell. The 5-cell is a tetrahedron surrounded by 4 other tetrahedra, in five different ways. The 11-cell is a hemi-icosahedron surrounded by 10 other hemi-icosahedra, in eleven different ways. The 5-cell has 5 vertices that form 5 tetrahedral cells, and a total of 10 triangular faces and 10 edges. The 11-cell has 11 vertices that form 11 hemi-icosahedral cells, each with 6 verticies 10 triangular faces and 15 edges, and a total of 55 triangular faces and 55 edges. == 5-cells and hemi-icosahedra in the 11-cell == [[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its <math>2^5</math> elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]] [[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two or more real elements found in different places in a concrete realization of the 11-cell.{{Sfn|Ruen: Hemi-icosahedron|2007}}]] The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we find a facet congruence between a 4-polytope and a 3-polytope we suspect a dimensional analogy. In the exceptional case of 5-cell and icosahedron, which share the same symmetry group <math>A_5</math>, we fully expect a dimensional analogy.{{Efn|There is an exceptional inter-dimensional duality between the regular icosahedron and the 5-cell because they share <math>A_5</math> symmetry. See this question asked on [https://math.stackexchange.com/questions/4235783/the-rotational-symmetry-groups-of-the-5-cell-and-the-icosahedron-are-isomorphi math.stackexchange.com 2021].}} Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization as the 6-point 5-simplex. Yet another real hexad is the 6-point 3-orthoplex; thus as a hexad the hemi-icosahedron is related by dimensional analogy to the 4-simplex (5-cell) from above, and to the 4-orthoplex (16-cell) from below, while those two simplest 4-polytope building blocks are only related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The cell of the 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, the 4-simplex (5-cell) and 4-orthoplex (16-cell), who are famously stand-offish with each other. Interesting! The cell of the 11-cell is an abstract hexad hemi-icosahedron with 5 central planes, most handsomely illustrated by Séquin.{{Sfn|Séquin|2012|loc=A 10-Dimensional Jewel}}{{Sfn|Séquin & Lanier|2007|p=3|loc=Figure 4: (b,c) two views of the hemi-icosahedron projected into 3D space|ps=; Séquin et. al. have a lovely colored illustration of the hemi-icosahedron, subdivided into 10 triangular faces by 5 central planes of its icosahedral symmetry, revealing rings of polytopes nestled in its interior. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}}{{Sfn|Séquin & Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point [[W:Hemi-isosahedron|hemi-isosahedron]] is the vertex figure of the 11-cell's dual 4-polytope the 57-point [[W:57-cell|57-cell]].}} The 11 hemi-icosahedral cells have 10 triangle faces each, and each cell is face-bonded to the other 10 cells. The 5-cell's 5 tetrahedral cells have 10 faces and 10 edges altogether, and each cell is face-bonded to the other 4 cells. If 11-cell faces correspond to 5-cell faces, then 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, that does not belong to the same hemi-icosahedral cell as its opposing face. He found that the 10 edges opposite each hemi-icosahedron's 10 faces are the 10 edges of a single 5-cell, which does not share any vertices, edges or faces with the hemi-icosahedron. For each cell of the 11-point 11-cell, there is exactly one 5-point 5-cell that is completely disjoint from the 6-point hemi-icosahedron cell.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; "We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary <math>a_4</math> [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron."}} There are 11 disjoint 5-cell 4-polytopes inscribed in each 11-cell, which also contains 11 hemi-icosahedral cells, 55 faces, 55 edges and 11 vertices. The real 11-cell is more complex than the abstract 11-cell representing it, because the real hemi-icosahedron is more complex and harder to find than the abstract hemi-icosahedron. Seeing the real 11-cell will be easier once we have identified the real hemi-icosahedron, and seen exactly where the 11-cell's real elements reside in the other 4-polytopes within the 120-cell with which the 11-cell intermingles. The 5-cell has 10 faces, and the 11-cell has 10 faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron is face-bonded to the other 10 hemi-icosahedra, and to 10 of the 11 5-cells, and there is exactly one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point 5-cell is the other 5 vertices of the 11-point 11-cell that are not vertices of this 6-point hemi-icosahedron: the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces which belong to 10 distinct hemi-icosahedra of the 11-cell, and there is just one hemi-icosahedron with which it does not share a face. In the abstract 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 completely disjoint 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction.{{Efn|We shall see that this small eliminated distance is in fact the length of a 120-cell edge, the shortest chordal distance found in the 120-cell.}} Seemingly adjacent hemi-icosahedron faces do not actually meet at an edge; there is a polygon separating them, which has been abstracted to an edge. The 10 hemi-icosahedron faces are 5-cell faces from 10 distinct 5-cells, and they do not actually touch each other: the 120 5-cells in the 120-cell are completely disjoint. In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of tetrahedral cells together, because each 11-cell face represents two actual 5-cell faces in different places. Each duplex 11-cell face bonds tetrahedra in two 5-cells in different places, without binding the 5-cells together (they are completely disjoint). One actual 5-cell face is one half of a duplex 11-cell face, so 110 5-cell faces are 55 duplex 11-cell faces. The 11-cell's 11 abstract vertices represent all 55 distinct vertices of the 11 disjoint 5-cells, so they must be abstract conflations of at least 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; 11-cells must be sharing vertices, not disjoint as the 5-cells are. == The real hemi-icosahedron == [[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right| Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; "Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, & 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 & 5 are each pairs of Truncated icosahedrons. Hulls 6 & 8 are Rhombicosidodecahedrons."}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes starting with a dodecahedron cell. Hull #8 with 60 vertices (lower right) is a central section of the 120-cell, the 8th and largest section starting with a cell.{{Efn|1=Although the 8 hulls are illustrated as the same size, in the 120-cell they have increasing size as numbered, and occur nested inside each other like Russian dolls. Only Hull #8 is a central section of the same radius as the 120-cell itself, analogous to the equator. Sections 1-7 occur in pairs on opposite sides of the central section, and are analogous to lines of latitude. Section 1 is simply a dodecahedral cell. The "Combined hulls" is for illustrative purposes only; no such compound polyhedron exists in the 120-cell.}}]] We shall see in subsequent sections that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the realization of the abstract hemi-icosahedron, as an actual polyhedron that occurs in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''² = ''j''² = ''k''² = ''ijk'' = −1}} as the [[W:Tesseract|16-point (8-cell) tesseract]] 4-polytope. He did not realize then that he had discovered the 4-hypercube polytope and [[W:Tesseractic honeycomb|its Euclidean honeycomb]], the (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because [[w:Quaternions_and_spatial_rotation|quaternions make rotations]] and projections in 3D or 4D space as simple as matrix multiplications.{{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 quaternions are 4-hypercube building blocks, analogous to the 3-hypercube wooden blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the [[24-cell]], but we digress). Moxness used his software to render illustrations of polyhedra inside the 120-cell, some of which he published. Notice his "Hull # = 8 with 60 vertices", lower right in his illustration of the 120-cell sections starting with a cell. It is a real icosahedron that occurs in the 120-cell, and we shall see that the abstract hemi-icosahedron represents it. Moxness's 60-point Hull #8 is a concrete realization of the 6-point hemi-icosahedron in spherical 3-space <math>S^3</math>, embedded in Euclidean 4-space <math>\mathbb{R}^4</math>. Its 12 little pentagon faces are 120-cell faces. It also has 20 triangle faces like any icosahedron, separated from each other by rectangles, but beware: those triangles are not the 5-cell faces. They are smaller equilateral triangles, of edge length <math>1</math> in a {{radic|2}}-radius 120-cell, where the 5-cell face triangles have edge length {{radic|5}}.{{Efn|The 41.4° chord of edge length 1 in a {{radic|2}}-radius 120-cell occurs only in the 120-cell; it is not the edge of any smaller regular 4-polytope inscribed in the 120-cell. The equilateral triangle faces of Moxness's Hull #8 rhombicosidodecahedron are not the 5-cell faces of edge length <small><math>\sqrt{5} \approx 2.236</math> </small>(104.5°), not the 16-cell faces of edge length <small><math>2</math></small> (90°), not the 24-cell faces of edge length <small><math>\sqrt{2} \approx 1.414</math></small> (60°), and not the 600-cell faces of edge length <small><math>\sqrt{2}/\phi \approx 0.874</math></small> (36°).|name=Moxness 60-point triangle faces}} [[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges, in 200 irregular {12} dodecagon central planes. The 5-cell ''faces'' do not lie in central planes.]] Edges of the larger 5-cell face triangles of length {{radic|5}} can also be found in Hull #8, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that six 120-cell edges (little pentagon edges) lie on a great circle, alternating with six rectangle diagonals. Also lying on this irregular {12} great circle are six 5-cell edges, invisible chords joining every other 120-cell edge and running under the 120-cell edge between them. The six long chords and six short edges form two opposing irregular {6} great hexagons (truncated triangles) of alternating 5-cell edges and 120-cell edges, as illustrated. The irregular great {12} lies on a great circle of Moxness's Hull #8, and also on a great circle of the 120-cell, because Hull #8 is the ''central'' cell-first section of the 120-cell.{{Efn|The cell-first central section of the 600-cell (and of the 24-cell) is a cuboctahedron with 24-cell edges. The 120-cell is the regular compound of 5 600-cells (and of 25 24-cells), so Moxness's Hull #8, as the cell-first central section of the 120-cell, is the regular compound of 5 cuboctahedra. Their 24-cell edges, like the 5-cell edges, are invisible chords of Hull #8 that lie below its surface, on the same irregular {12} great circles. Each 24-cell edge chord spans one 120-cell edge chord (one little pentagon edge) and one rectangle face diagonal chord. Six 24-cell edge chords form a regular great {6} hexagon, inscribed in the irregular great {12} dodecagon.|name=compound of 5 cuboctahedra}} There are 10 great dodecagon central planes and 60 5-cell edges in Moxness's Hull #8, and 200 great dodecagon central planes and 1200 5-cell edges in the 120-cell. [[File:Central cell-first section of the 120-cell with 5-cell face triangle.png|thumb|Orthogonal projection of the cell-first central section of the 120-cell, Hull #8 rendered by Moxness, with one of 20 inscribed 5-cell faces (black chords) drawn under portions of three of its ten great circle {12} dodecagons (green).{{Efn|The point of view in this rendering is not quite right to best illustrate that a rhombicosidodecahedron triangle face lies over the center of a 5-cell face parallel to it, such that it would be perfectly inscribed in the center of the larger black triangle in an orthogonal view.}}]] But the 5-cell ''faces'' do not lie in those central planes. We can locate them in the 60-point polyhedron where they lie parallel to and under each small face triangle of edge length <math>1</math>. Truncating at a triangle face of Moxness's Hull #8 exposes a deeper 5-cell triangle face.{{Efn|Each face triangle of edge length <math>1</math> is surrounded by 3 rectangles, and beyond each rectangle by another face triangle. The distant vertices of those 3 surrounding triangles form a {{radic|5}} triangle, a 5-cell face.}} There are 20 such 5-cell faces inscribed in the Hull #8 polyhedron, all completely disjoint. We find 60 vertices, 60 edges and 20 faces of various 5-cells in each Hull #8 polyhedron, but no whole tetrahedral cells of the 5-cells.{{Efn|The fourth vertex of each 5-cell tetrahedron lies opposite the small face triangle of edge length <math>1</math> that lies over the 5-cell face. Since Moxness's Hull #8 polyhedron has opposing triangle faces (like any icosahedron), the fourth vertex of the 5-cell tetrahedron lies over the center of the opposing face, outside the Hull #8 polyhedron. This is a vertex of some other Hull #8 polyhedron in the 120-cell. Each tetrahedral cell of a 5-cell spans four Hull #8 polyhedra, with one face inscribed in each, and one vertex outside of each.}} [[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point Hull #8 is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation of the icosahedron with smaller red pentagons and narrower rhombs. Rhombicosidodecahedra are also made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the unique 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 60 of Moxness's Hull #8 rhombicosidodecahedron. Each occupies a central hyperplane, and so is analogous to an equator dividing the sphere in half.]] Moxness's Hull #8 is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point [[W:icosidodecahedron|icosidodecahedron]], which has the same relationship to Moxness's Hull #8 that the 6-point hemi-icosahedron does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The regular 5-point 5-cell can be another abstraction of Moxness's 60-point Hull #8, 12-vertices-into-1. None of these contractions of Moxness's Hull #8 is an instance of her operation actually described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length, but she did explicitly describe contractions of the semi-regular Archimedean rhomibicosidodecahedron.}} Moxness was not the first person to find rhombicosidodecahedra in the 120-cell. Alicia Boole Stott identified the 6th section of the 120-cell beginning with a cell as the semi-regular rhombicosidodecahedron that is her ''e₂ expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e₂ expansion|p=7}} But that 6th section rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8, it is the semi-regular Archimedean solid (Moxness's Hull #6), with a single edge length and square faces. Moxness's Hull #8, with its two distinct edge lengths and rectangular faces, is Coxeter's 8₃, the 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; "Alicia Boole Stott (1860-1940) ... also constructed the sections i₃ of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1₃-7₃; but 8₃ is missing. Incidentally, Diagram XIII (our 6₃) is a rhombicosidodecahedron, the Archimedean solid."}} Coxeter was the first to describe the central section 8₃, and he gave its coordinates, but he did not identify it as an irregular rhombicosidodecahedron. His table entry for its description is empty (characteristically, since it is not a regular or semi-regular polyhedron), so he gives us no indication that he actually visualized it. Although Moxness was not the first to compute the 60-point 8₃ section, he may have been the first person to ''see'' it. The 30-point icosidodecahedron is the quasi-regular product of 5-point pentagon and 6-point hexagon, recalling Coxeter's original discovery of the 11-cell in pentads and hexads, and also the two child's building blocks: one so useless the 5-point (pentad) 5-cell, and the other so useful the 8-point 16-cell with its four orthogonal 6-point (hexad) octahedron central sections, which can be compounded into everything larger. Some children building with the 30-point icosidodecahedron notice that it occurs as the central section 4₀ of the 120-point 600-cell. It is less often noticed that Moxness's Hull #8 rhombicosidodecahedron is the central section 8₃ of the 600-point 120-cell. It occupies a flat 3-dimensional hyperplane that bisects the 120-cell, and since there are 120 dodecahedral cells, there are 60 such central hyperplanes, each perpendicular to an axis that connects the centers of two antipodal cells. The 60 central hyperplanes, each containing an instance of Moxness's Hull #8, are rotated with respect to each other. They intersect, with 6 rhombicosidodecahedra sharing each vertex and 3 sharing each edge, but each little pentagon face (120-cell face) belongs to just one rhombicosidodecahedron. The 60 central sections lie in isoclinic hyperplanes, that is, the rhombicosidodecahedra are rotated symmetrically with respect to each other, by two equal angles.{{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.)"}} Each pair of rhombicosidodecahedra intersect in a central plane containing an irregular {12} dodecagon, unless they are completely orthogonal and intersect only at the center of the 4-polytope. Each of the 120 dodecahedral cells lies in the closed, curved 3-dimensional space of the 3-sphere as the 1st and smallest section beginning with a cell (section 1₃), the innermost of a series of concentric polyhedral hulls of increasing size, which nest like Russian dolls around it. Moxness's Hull #8 rhombicosidodecahedron is the 8th and largest concentric hull beginning with a cell (section 8₃), a central section of the 120-cell that bisects the 3-sphere the way an equator bisects an ordinary sphere.{{Efn|The 120-cell's curved 3-space surface is a honeycomb of 120 dodecahedron cells. In this 3-space a dodecahedron cell lies inside at the center of each section 8₃ rhombicosidodecahedron, face-bonded to 12 other dodecahedron cells which surround it, also inside the rhombicosidodecahedron. We find the opposite pentagon faces of those 12 surrounding cells on the surface of the section 8₃ rhombicosidodecahedron. These twelve dodecahedra surrounding one dodecahedron partially fill the volume of the rhombicosidodecahedron, leaving 30 concavities in its surface at the rectangle faces, and 12 deeper concavities between them at the triangle faces. 30 more dodecahedra fit into the rectangle concavities, lying half inside and half outside the rhombicosidodecahedron. The diagonal of each rectangle face is a long diameter of a dodecahedron cell. 12 more dodecahedra fit into the triangle face concavities, lying ....|name=dodecahedral cells in the section 8 rhombicosidodecahedron}} Such a central polyhedron is the dimensional analog of an equatorial great circle polygon. Its 60 vertices lie in the same 3-dimensional hyperplane, a flat 3-dimensional section sliced through the center of the 120-cell. There are 60 distinct stacks of 15 parallel section ''n''₃ hyperplanes in the 120-cell, one stack spindled on each axis that connects a dodecahedron cell-center to its antipodal dodecahedron cell-center. Each central section 8₃ has ''two'' disjoint sets of smaller sections nested within it, that lie in opposite directions from the 120-cell's center along its 4th dimension axis. The largest-radius central slice lies in the center of the stack, and the smaller non-central section hyperplanes occur in parallel pairs on either side of the central slice. The 120-cell therefore contains 120 instances of each kind of non-central section 1₃ through 7₃, and 60 instances of the central section 8₃.{{Efn|A central section is concave on its inside and also on its outside: it has two insides. It may be helpful to imagine the central 60-point section as two mirror-image 60-point polyhedra whose points are coincident, but which are convex in opposite directions: the inside of one is the outside of the other. Each has seven smaller polyhedra nested within itself, but their two volumes are disjoint.}} [[File:Tensegrity Icosahedron.png|thumb|[[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|Tensegrity icosahedron]] structure.{{Sfn|Burkhardt|1994}} First built by [[W:Kenneth Snelson|Kenneth Snelson]] in 1949. Geometrically a [[w:Jessen's_icosahedron|Jessen's icosahedron]] with 6 reflex ''long'' edge struts, and 24 ''short'' edge tension cables around 8 equilateral triangle faces. 3 pairs of parallel struts lie in 3 orthogonal central planes.]] We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that the 60 Moxness's Hull #8 rhombicosidodecahedra are real polyhedra which the abstract hemi-icosahedra represent in some manner, but we have not yet identified 11 real face-bonded cells, at 11 distinct locations in the 120-cell, as an 11-cell. The abstract hemi-icosahedron's 10 faces correspond to actual 5-cell faces inscribed in real rhombicosidodecahedra, and its 15 edges correspond to 5-cell edges (of length {{radic|5}} in a {{radic|2}}-radius 120-cell) that occur as chords lurking under the surface of the rhombicosidodecahedra. [[File:Buckminster-Fuller-holding-a-geodesic-tensegrity-sphere.png|thumb|200px|Buckminster Fuller holding a 3-dimensional geodesic tensegrity 2-sphere, an infinitesimally mobile rigid polytope consisting of tension cable edges and disjoint compression strut chords.<ref>{{Cite journal|last=Álvarez Elipe|first=Dolores|title=Ensegrities and Tensioned Structures|journal=Journal of Architectural Environment & Structural Engineering Research|date=July 2020|volume=3|issue=3|url=https://www.researchgate.net/publication/343652287_Ensegrities_and_Tensioned_Structures}}</ref>]] A rhombicosidodecahedron is constructed from a regular icosahedron by truncating its vertices, making them into pentagon faces. The regular icosahedron frames all the regular and semi-regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter & Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Sfn|Coxeter, du Val, Flather & Petrie|1938|p=4|ps=; "Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "[[W:Golden section|golden section]]". Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''."}} Before we can move on to locating the 11 discrete hemi-icosahedral cells of the 11-cell in the 120-cell, it is important that we take notice of one more icosahedral symmetry of the hidden {{radic|5}} chords lurking below the surface of Moxness's Hull #8 rhombicosidodecahedron. The 12 little pentagon faces (120-cell faces) are connected to each other in parallel pairs, by 10 sets of six disjoint {{radic|5}} chords (5-cell edges). Each six-chord set is the six reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], which is to say that the six disjoint chords are the parallel-orthogonal strut chords of a [[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|tensegrity icosahedron]]. The six chords of each set are disjoint (they don't touch or form 5-cell faces), and they are symmetrically arranged as 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal {12} central planes.{{Efn|The Jessen's icosahedron has 8 equilateral triangle faces, which are not rhombicosidodecahedron triangle faces or 5-cell triangle faces, they are 24-cell triangle faces. Each 120-cell pentagon face lies at one end of 20 5-cell edges, from 20 distinct Jessen's icosahedra and five disjoint 5-cells: four at each pentagon vertex from each 5-cell.}} Five disjoint instances of the Jessen's icosahedron may be inscribed in each Moxness's Hull #8 rhombicosidodecahedron, their struts propping the rhombicosidodecahedron and the 120-cell itself open like a tensegrity structure.{{Efn|Moxness's Hull #8 rhombicosidodecahedron is a compound of five disjoint Jessen's icosahedra, because the 60 {{radic|5}} chords meet two-at-a-vertex and form 10 distinct Jessen's icosahedra: five disjoint Jessen's, in two different ways. The dimensionally analogous construction is the [[120-cell#Compound of five 600-cells|120-cell as a compound of five disjoint 600-cells]], in two different ways.}} But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, and work our way patiently outwards, before in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we approach that rare child Bucky Fuller's orthogonal 12-point tensegrity icosahedron, an [[WikiJournal Preprints/Kinematics of the cuboctahedron|in-folded cuboctahedron]], the unique pyritohedral fish swimming deep in the 3-sphere ocean. == Eleven == Each pair of rhombicosidodecahedra that are not completely orthogonal intersect in a central plane containing an irregular {12} dodecagon. Ten irregular great dodecagons occur in each 60-point (central section 8₃) rhombicosidodecahedron, with 2 dodecagons crossing orthogonally at each vertex. Each rhombicosidodecahedron shares a {12} central plane with ten other rhombicosidodecahedra. ''Groups of 11 rhombicosidodecahedra share central planes pairwise.'' Here, at last, we find eleven of something, a group which must comprise an 11-cell. There are eleven {12} central planes in the group, with one of the eleven absent from each rhombicosidodecahedron. {|class="wikitable floatright" width=450 !colspan=2|Perspective views{{Efn|1=These images are ''non-orthogonal'' orthographic projections of the chords described in the caption. Those chords do not lie in a plane parallel to the projection plane, so they appear foreshortened.{{Efn|name=orthogonal triacontagram projections}} Consecutive chords of the helical Petrie polygon slant toward and away from the viewer. Any three consecutive chords, but no four, are edges of the same cell, in the 4-polytope whose edges are the chord.{{Efn|name=Petrie polygon of a honeycomb}}}} of a compound of six disjoint 5-cells in dual position |- ![[W:Triacontagon#Triacontagram|{30/12}{{=}}6{5/2} compound]] ![[W:Triacontagon#Triacontagram|{30/8}{{=}}2{15/4} compound]]{{Efn|name=orthogonal triacontagram projections|1=The {30/''n''} triacontagrams can each be seen as an ''orthogonal projection'' of the 120-cell showing all instances of the {30/''n''} chord. Each chord lies orthogonal to the line of sight, in a plane parallel to the projection plane. The diameter of the image is the diameter of the 120-cell. For example, the {30/8}=2{15/4} triacontagram is an orthogonal projection showing the 120-cell's 1200 {30/8} chords, the edges of 120 5-cells. Each edge of the triacontagram covers 40 5-cell edges, and each vertex covers 20 120-cell vertices. This projection can also be viewed as a compound of six 5-cells and their 30 unique vertices. But viewed that way, only 30 of the 60 5-cell edges are visible. Two edges meet at each vertex, but the other two are invisible. They are visible in the orthogonal view, the {30/4}=2{15} projection.}} |- valign=top |[[File:Regular_star_figure_6(5,2).svg|240px]]<BR>The 6{5/2} compound of six 5-cells. The six disjoint pentagrams in this view are six disjoint 5-cells.{{Efn|name=5-cell edges do not intersect is S³}} The 120-cell, with 120 disjoint 5-cells, is a compound of 20 of these compounds. All edges are 5-cell edges, but only five of each 5-cell's ten edges are shown. The other five edges, connecting the points of the six 5-cell pentagrams, are shown in the 6{5} projection below, the orthogonal view:<BR>[[File:Regular_star_figure_6(5,1).svg|240px]]These two views look straight down the orthogonal axes of a [[w:Duocylinder|duocylinder]], from inside the curved 3-dimensional space of the 120-cell's surface. They are like looking down a column of 5-cells stacked on top of one another in curved 3-space, but the column is actually circular: it is bent into a torus in the fourth dimension. |[[File:Regular_star_figure_2(15,4).svg|240px]]<BR>The 2{15/4} rotation circuits of the 5-cell isoclinic rotation. In this view, all edges are 75.5° chords of length {{radic|3}}, the 180° complement chord of the 5-cell edges of length {{radic|5}}.{{Efn|These are not 15-gons of 5-cell edges. There are no skew {15} polygons of 5-cell edges in the 120-cell. The 120 5-cells are completely disjoint, so the largest circuit along 5-cell edges is a skew {5}. Each vertex in the 120-cell is {{radic|5}} away from four and only four other vertices. No {{radic|5}} chords connect disjoint 5-cells; they are connected by several other chords. The skew {15} polygons are the discrete continuous spiral paths of moving vertices during an isoclinic rotation, and their edges are {{radic|3}} chords connecting 5-cells, not 5-cell edges.}} Each skew {15} polygon is the spiral chord-path of half the 30 vertices during the isoclinic rotation. The twined vertex orbits lie skew in 4-space; they form a circular double helix of two 15-gon spiral isoclines, winding through all four dimensions. These two completely orthogonal views look straight down an axis of a double helix cylinder, from inside the curved 3-dimensional space of the 120-cell's surface. Since the duocylinder is bent into a [[w:Clifford_torus|Clifford torus]] in the fourth dimension, the sightline axis in curved 3-space is a geodesic great circle in 4-space.<BR>[[File:Regular_star_figure_2(15,2).svg|240px]] |- ![[W:Triacontagon#Triacontagram|{30/6}{{=}}6{5} compound]] ![[W:Triacontagon#Triacontagram|{30/4}{{=}}2{15/2} compound]] |- |colspan=2|Images by Tom Ruen in [[W:Triacontagon#Triacontagram|Triacontagram compounds and stars]].{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}} |} Each shared {12} central plane contains six disjoint 5-cell edges, from six completely disjoint 5-cells. Each rhombicosidodecahedron contains 60 5-cell edges, which form 20 disjoint 5-cell faces within the rhombicosidodecahedron, under and parallel to its own 20 smaller triangle faces. Four 5-cell edges meet at each vertex at the 5-cell's tetrahedral vertex figure. Two 5-cell edges of a face within the rhombicosidodecahedron meet two edges belonging to other faces of the 5-cell: edges and faces outside the rhombicosidodecahedron, in some neighboring rhombicosidodecahedron.{{Efn|name=orthogonal triacontagram projections}} Each 5-cell face is shared by two tetrahedral cells of one 5-cell. It has its three 104.5° {{radic|5}} edges in three distinct {12} central planes, and is parallel to a fourth {12} central plane. In each rhombicosidodecahedron there are ten sets of five parallel planes: a {12} central plane, a pair of 5-cell faces on either side of it (from disjoint 5-cells), and a pair of rhombicosidodecahedron triangle faces. Each rhombicosidodecahedron is sliced into five parallel planes, ten distinct ways. There is no face sharing between 5-cells: the 120 5-cells in the 120-cell are completely disjoint. 5-cells never share any elements, but they are related to each other positionally, in groups of six, in the '''characteristic rotation of the regular 5-cell'''. That rigid isoclinic rotation takes the six 5-cells within each group to each other's positions, and back to their original positions, in a circuit of 15 rotational displacements.{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439|ps=; in symmetry group 𝛢₄ the operation [15]𝑹_q3,q3 is the 15 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 5-cell; in symmetry group 𝛨₄ the operation [1200]𝑹_q3,q13 is the 1200 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 120-cell.}} Each displacement takes every 104.5° 5-cell edge of length {{radic|5}} to an edge 75.5° and {{radic|3}} away in another 5-cell in the group of six 5-cells. The 30 vertices of the six 5-cells rotate along 15-chord helical-circular isocline paths from 5-cell to 5-cell, before closing their circuits and returning the moving 5-cells to their original locations and orientations.{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. The orthogonal distance equals half the total Pythagorean distance. For example, when the {{radic|2}}-radius 5-cell rotates isoclinically 104.5° in the invariant central planes of its 104.5° edges of length {{radic|5}}, each vertex is displaced to another vertex 75.5° and {{radic|3}} away, moving {{radic|3/4}} in four orthogonal directions at once.|name=isoclinic 4-dimensional diagonal}} The six rotationally related 5-cells form a stellated compound, a non-convex 4-polytope with 30 star points.{{Efn|name=compound of six 5-cells}} The star compound, and the rotation of the 5-cell within it, are here illustrated by orthogonal projections from four different perspective viewpoints. To help us visualize the 4-polytopes within the 120-cell, we can examine 2-dimensional orthographic projections from various points of view. Such images filtered to include only chords of a single length are especially revealing, because they pick out the edges of a particular 4-polytope, or the isocline chords of its rotational orbits, the chords which link 4-polytopes together. No view of a single chord from a single point of view is sufficient by itself, but if we visualize various chords from various perspectives, we may imagine the 4-dimensional rotational geometry of interrelated objects within the 120-cell. The star compound as a whole has ten {12} central planes, like a rhombicosidodecahedron. Each {12} central plane contains one edge from each of the six 5-cells. Each {12} central plane is shared by two rhombicosidodecahedra in the group of eleven, and by six 5-cells in the group of six. == The eleventh chord == [[File:Major chord 11 of 135.5° in the 120-cell.png|thumb|The 120-cell contains 200 irregular {12} central planes containing 1200 135.5° {30/11} chords, six in each plane (shown in blue). They lie parallel to six 104.5° {30/8} chords (the 5-cell edges, shown in red), to which they are joined by 15.5° {30/1} 120-cell edges, and by 120° {30/10} great triangle edges (only one of the four great triangles is shown, in green).]] In addition to six 104.5° {30/8} 5-cell edge chords of length {{radic|5}}, the {12} central plane contains six 135.5° {30/11} chords of length <math>\phi^2</math>, parallel to the {{radic|5}} chords. The {30/11} chord spans an arc of five shorter chords: * 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} = 135.5° {30/11) * 15.5° {30/1} + 104.5° (30/8) + 15.5° {30/1} = 135.5° {30/11) * 15.5° {30/1} + 120° (30/10) = 135.5° {30/11) and its chord length is the linear sum of five shorter chords: * 1/𝜙^2 {30/1} + 1/𝜙^2 {30/1} + 1/𝜙 {30/2} + 1/𝜙 {30/2} + 1/𝜙 {30/2} = 𝜙^2 {30/11) Two distinct chords are always related to each other in two different ways: by their degrees-of-arc-difference, and by their linear difference chord. The 135.5° {30/11) chord is ''two'' 15.5° (30/1) 120-cell edge-arcs longer than the 104.5° (30/8) 5-cell edge chord. But the <math>\phi^2</math> {30/11} chord ''length'' is just ''one'' {30/1} 120-cell edge chord length longer than the {{radic|5}} {30/8} 5-cell edge chord.{{Efn|In a <small><math>\sqrt{2}</math></small>-radius 120-cell, the 15.5° {30/1} 120-cell edge chord has length <small><math>\phi^{-2}</math></small>. The 25.2° {30/2} pentagon face diagonal chord of length <small><math>\phi^{-1}</math></small> is <small><math>\phi</math></small> times the {30/1} edge length. The 41.1° 5-cell isocline chord of length <small><math>\sqrt{1}</math></small> is <small><math>\phi^2</math></small> times the {30/1} edge length. The 69.8° chord of length <small><math>\phi</math></small> is <small><math>\phi^3</math></small> times the {30/1} edge length. The 135.5° {30/11} 11-cell edge chord of length <small><math>\phi^2</math></small> is <small><math>\phi^4</math></small> times the {30/1} edge length.}} The {30/11} chord can be bisected into two shorter 120-cell chords in three different ways: * 15.5° {30/1} 120-cell edge + 104.5° {30/8} 5-cell edge = {30/11} chord * 25.2° {30/2} 120-cell pentagon face diagonal + 90° {30/15} 16-cell edge = {30/11} chord * 41.4° {30/1}+{30/2} chord + 69.8° {30/2}+{30/1}+{30/2} chord = {30/11} chord [[File:Regular_star_polygon_30-11.svg|thumb|The [[W:Triacontagon#Triacontagram|{30/11} regular triacontagram]] of the 11-cell rotation.{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}} In this 2-dimensional projection of a 30-edge 4-dimensional helix ring, the 30 chords pictured lie in 30 distinct central planes, and no two planes are orthogonal.]] The last of those bisections trisects the {30/11} chord into three distinct shorter chords: * 15.5° {30/1} + 25.2° {30/2} + 44.5° {30/4} chord = 135.5° {30/11} chord The {30/11} chords do not form triangle faces within the rhombicosidodecahedron the way the {30/8} chords do, but they do meet at a tetrahedral vertex figure. Groups of 11 rhombicosidodecahedra (an 11-cell) share central planes pairwise, including all the chords in the {12} central plane. When 11 things, all pairwise-adjacent to each other, are arranged in any circuit of 30 positions, there exists another pairwise circuit of 30 positions through every eleventh position, whether the things are 11 vertices, 11 rhombicosidodecahedra, or 11 [[w:Aardvark|aardvarks]] (although it might be unwieldy in practice to so arrange 11 live aardvarks, e.g. by tying them together pairwise with cords in both circuits). This intrinsic property of the [[w:Rational_number|rational number]] 30/11 is responsible for the existence of the {30/11} regular triacontagram (see illustration). The 11 rhombicosidodecahedra of the 11-cell are linked by a regular {30/11} triacontagram of 30 chords which runs through them. Each successive chord of the 30 in the triacontagram is shared by a distinct pair of rhombicosidodecahedra in the 11-cell group. An isoclinic rotation characteristic of the 11-cell takes the rhombicosidodecahedra in each 11-cell to each other's positions, pair by pair, in a circuit of 30 rotational displacements. It takes every {12} central plane to a Clifford parallel {12} central plane that is 44.5° away in two completely orthogonal angles. One 135.5° {30/11} chord separates each of the 12 vertex pairs. In this '''characteristic rotation of the 11-cell''' in its edge planes, the invariant planes are {12} central planes, the edges of the 11-cell are {30/11} chords, and the isocline chords of the vertex orbits are also {30/11} 11-cell edges, because the triacontagram is regular.{{Efn|In the 120-cell there are three ''regular isoclinic rotations'' in which the rotation edge and the isocline chord are the same chord. These rotations are each described by a [[W:Triacontagon#Triacontagram|regular triacontagram]]: the {30/7} rotation characteristic of the 16-cell in great square invariant planes, the {30/11} rotation characteristic of the 11-cell, and the {30/13} rotation.}} The 44.5° {30/4} chord of length <small><math>\sqrt{3}/\phi</math></small>, the 180° complement of the {30/11} chord, is the orthogonal distance between nearest parallel {30/11} chords.{{Efn|In its characteristic isoclinic rotation, a 4-polytope rotates an equal arc distance in each invariant {12} edge plane in each rotational displacement. In the 11-cell, every invariant plane rotates 44.5° (like a wheel), and tilts sideways 44.5° (like a coin flipping) in the completely orthogonal invariant plane, to occupy another invariant plane in the group of eleven. Each pair of original and destination {12} central planes are Clifford parallel and intersect only at one point (the center of the 4-polytope), but six other {12} central planes intersect them both. Two parallel {30/11} chords in each of the six spanning {12} central planes separate two vertex pairs in the original and destination planes, and these are the isocline chords over which the two vertices move in the rotation. None of the six spanning {12} central planes are contained in either the original or destination rhombicosidodecahedron. A total of ten {12} central planes span each original and destination rhombicosidodecahedron; they comprise a third rhombicosidodecahedron which does not belong to the group of eleven. The edges of an 11-cell and the isocline chords of an 11-cell are disjoint sets of {30/11} chords.}} The 60 vertices of each rhombicosidodecahedron rotate in parallel, on non-intersecting 30-chord spiral orbital paths, from rhombicosidodecahedron to rhombicosidodecahedron, before closing their circuits and returning the moving rhombicosidodecahedron to its original location and orientation. In this isoclinic rotation of a rigid 120-cell, the 60 rhombicosidodecahedra do this concurrently. Each of the 600 vertices moves on a 4-dimensionally-curved helical isocline, over a skew regular polygram of 30 {30/11} chords, in which a {30/11} chord connects every eleventh vertex of a {30} triacontagram. In the course of a complete revolution (the 30 rotational displacements of this isoclinic rotation), an 11-cell visits the positions of three 11-cells (including itself) 10 times each (in 10 different orientations), and returns to its original position and orientation.{{Sfn|Coxeter|1984|loc=§9. Eleven disjoint decagons}} At each step it occupies the same distinct group of 11 rhombicosidodecahedra sharing planes pairwise, and its 11 vertex positions are those of a distinct 11-cell in the group of eleven 11-cells. A group of 4-polytopes related by an isoclinic rotation is contained in a larger compound 4-polytope which subsumes them. This group of eleven 11-cells related by an isoclinic rotation is not a compound of eleven disjoint 11-cells (since they share vertices), but it is a compound of eleven non-disjoint 11-cells, in the same sense that a 24-cell is a compound of three non-disjoint 8-cell tesseracts. Consider the incidence of these 30-chord {30/11} triacontagram rotation paths, and their intersections. Each rhombicosidodecahedron has 60 vertices and 60 {30/11} chords, which rotate concurrently on Clifford parallel triacontagrams. The 120-cell has only 600 vertices and 1200 {30/11} chords, so at most 20 triacontagrams can be disjoint; some must intersect. But the 11 vertices of an individual 11-cell must be linked by disjoint 30-position {30/11} triacontagram helices, such that their rotation paths never intersect.{{Efn|The isoclines on which a 4-polytope's vertices rotate in parallel never intersect. Isoclinic rotation is a concurrent motion of Clifford parallel (disjoint) elements over Clifford parallel (non-intersecting) circles.}} Each 11-cell has two disjoint triacontagram helicies, its left and right isoclinic rotations, in each of its four discrete fibrations. The 120-cell has 60 distinct {30/11} triacontagram helices, which are 11 disjoint {30/11} triacontagram helices in 11 distinct ways. {{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} == Compounds in the 120-cell == [[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 for 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 other 5 regular convex 4-polytopes]]. {{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}} === How many building blocks, how many ways === The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy. The 24-point also compounds by <math>5^2</math> non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. [[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|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}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]] The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times <math>5^2</math> (75) disjoint instances of itself into the 600-point, which then magically contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point, and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the original 8-point. They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[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''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}} 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]], the final regular [[W:Stellation|stellation]] of the 120-cell, 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:Stellation core|stellation core]] deep inside. The compound of 120 regular 5-cells can be seen to be equivalent to the compound of 5 disjoint 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. === Building the building blocks themselves === We have built every regular 4-polytope except the 5-cell out of 16-cells, but we haven't made the 16-cell (or the 5-cell) out of anything. So far, we have just accepted them both a priori, like [[W:Euclid's postulates|Euclid's postulates]], and proceeded to build with them. But it turns out that while they are the two atomic regular 4-polytopes, they are not indivisible, and can be built up as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct [[w:Symmetry_group|symmetry group]]. Every regular convex ''n''-polytope can be subdivided into instances of its characteristic [[W:Orthoscheme|Schläfli orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-[[w:Orthoplex|orthoplex]]!) is an ''irregular'' ''n''-[[w:Simplex_(geometry)|simplex]] with faces that are various right triangles instead of congruent equilateral triangles. A characteristic ''n''-orthoscheme possesses the complete symmetry of its ''n''-polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements. It is the gene for the polytope, which can be replicated to construct the polytope.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope because 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 partitioned radially by its planes of symmetry (Coxeter's "reflecting circles") into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme and its chiral mirror image can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its <math>A_4</math> planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into <small><math>120\times 120 = 14400</math></small> of these 4-orthoschemes, so that is the symmetry order of the 120-cell. The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its <math>B_4</math> planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order <small><math>75\times 384 / 2 = 14400</math></small>. The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its <math>F_4</math> planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order <small><math>25\times 1152 / 2 = 14400</math></small>. The regular 120-point (600-cell) is subdivided into 14400 instances of its [[600-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its <math>H_4</math> planes of symmetry at once intersecting at its center, so its symmetry is of order 14400. The regular 600-point (120-cell) is the convex hull of the regular compound of 5 disjoint 600-cells (which have 5-fold reflective symmetry), so its symmetry is of order <small><math>5 \times 14400 / 5 = 14400</math></small>. === Building with sticks === [[File:15 major chords.png|thumb|300px|The 15 major chords {30/1} ... {30/15} join vertex pairs which are 1 to 15 edges apart on a skew {30} [[w:Petrie_polygon|Petrie polygon]] of the 120-cell.{{Efn|Drawing the fan of major chords with #1 and #11 at a different origin than all the others was an artistic choice, since all the chords are incident at every vertex. We could just as well have fanned all the chords from the same origin vertex, but this arrangement notices the important parallel relationship between #8 and #11, and calls attention to the 11-cell's maverick edge chord.|name=fan of 15 major chords}} The 15 minor chords (not shown) fall between two major chords, and their length is the sum of two other major chords; e.g. the 41.4° minor chord of length {30/1}+{30/2} falls between the 36° {30/3} and 44.5° {30/4} chords.]] We have seen how all the regular convex 4-polytopes except the 5-cell, including the largest one on the cover of the box, can be built from a box containing 675 16-cell building blocks, provided we can arrange the blocks on top of one another in 4-space, as interpenetrating objects. An alternate box, containing 120 regular 5-cell building blocks, builds the great grand stellated 120-cell (the picture on ''its'' cover), by the same method. In these boxes, the atomic building part is one of the two smallest regular 4-polytopes (5-cell or 16-cell), each generated by its characteristic isoclinic rotation as an expression of its symmetry group (<math>A_4</math> or <math>B_4</math>). All the regular convex 4-polytopes, including the largest one on the cover of the box, can also be built from a box containing a certain number of building sticks and rubber joints, provided we can connect the sticks together in 4-space with the rubber joints. In this box, the atomic building parts are 1-dimensional edges and chords of just 15 distinct arc-lengths. The regular 4-polytopes do not contain a vast variety of stick lengths, but only 30 of them: only 15 unique pairs of 180° complementary chords. The 15 ''major chords'' {30/1} ... {30/15} suffice to construct all the regular 4-polytopes. The 15 ''minor chords'' occur only in the 120-cell, not in any smaller regular 4-polytope; they emerge as a consequence of building the largest 4-polytope on the cover of the box from major chords. In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the unit-radius sphere of a distinct length <math>l</math>, and a distinct rational number <math>h</math>, a unique flavor. If the polytope is regular, it is a noteworthy distinctive flavor. The chord's length <math>l</math> is a square root, related to the rational number <math>h = k/d</math> and to the polygon <small><math>\{k/d\}</math></small> it represents, by a formula discovered by Steinbach.{{Sfn|Steinbach|1997|loc=''Golden Fields''; §1. The Diagonal Product Formula|pp=22-24|ps=; The product of two diagonals is a sum of a sequence of diagonals (in the fan, every other one) centered on the longer of the two, for all regular polygons. We may express products and quotients of diagonals <math>d_k</math> of an <math>n</math>-gon (with edge <math>d_0=1</math>) as linear combinations of diagonals.}} The chord length <math>l</math> is related to the number of sides of the regular polygon <small><math>\{k\}</math></small>, and to the winding number or density of the polygram (its denominator <math>d</math>).{{Sfn|Kappraff & Adamson|2004}} The largest <math>k</math> of any major chord in the 120-cell is 30, and the polygrams <small><math>\{30/d\}</math></small> represent all the skew Petrie polygons and characteristic isoclinic rotations of the regular 4-polytopes. == Concentric 120-cells == The 8-point 16-cell, not the 5-point 5-cell, is the smallest regular 4-polytope which compounds to every larger regular 4-polytope. The 5-point 5-cell is also an atomic building block, but one that compounds to nothing else regular except the leviathan 120-cell polytope: the picture on the cover of the box, that is built from everything in the box. In the [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|sequence of 4-polytope compounds]], we actually start with the 16-cell at the small end, and the 5-cell emerges only at the large end. To build with the 16-cell blocks, we simply put them on top of each other as interpenetrating compounds. We can build every other regular 4-polytope from them by that method, except the individual regular 5-cell. We can also try to build with the 5-cell that way, as when we tried to build a 4-polytope of 11 hemi-icosahedral cells from 11 5-cells, but that was rather hard going. We somehow found 5-cell edges and faces lurking inside hemi-icosahedral rhombicosidodecahedra, and 11 rhombicosidodecahedra sharing central planes pairwise, and even the edges and characteristic rotation of the 11-cell, but we didn't quite get all the way to a discrete 11-cell 4-polytope made from 11 5-cells. That's because ''compounding'' isn't the easiest method for building with the 5-cell. The 5-cell is the last building block hierarchically, not the first, and the most natural way to build with it is in reverse, by ''subdividing'' it, to find all the parts inscribed inside it. When we've taken the 5-cell apart, all the ways we possibly can, into certain ''irregular'' 4-polytopes found within it, we will have a new set of irregular 4-polytope building blocks, which compound to the 5-cells and everything else, including the 11-cells. Subdividing a polytope is done by a geometric operation called ''[[w:Truncation_(geometry)|truncation]]''. There are myriad ways to truncate a 5-cell, each corresponding to a distinct ''depth'' of truncation at a particular point on an edge, or a line on a face, or a face on a cell, where a piece of the 5-cell is cut off. The simplest truncations, such as [[w:Rectification_(geometry)|cutting off each vertex at the midedge of each incident edge]], have been very well-studied; but how should we proceed? Let us see what happens when we [[w:Truncated_5-cell|truncate the 5-cells]] found in the 120-cell, by the simplest kinds of truncation. These three semi-regular 10-cells are closely related truncations of the regular 5-cell: * The 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of six 5-cells. * The 20-point 10-cell [[w:Truncated_5-cell|truncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of four 5-cells. * The 10-point 10-cell [[w:Rectified_5-cell|rectified 5-cell]] is the convex hull, and the convex common core, of a stellated compound of two 5-cells. In the following sections, we explore the effect of performing these truncations on the 120-cell's 120 5-cells. We begin by identifying some promising truncation points on the 120-cell's 5-cell edge chords at which to cut. If we cut off the 120-cell's 600 vertices at some point on its 1200 5-cell edges, we create new vertices on the edges of the 120 5-cells, which lie on a smaller 3-sphere than the 120-cell. How many vertices does the smaller 4-polytope thus created have? That is, how many distinct 5-cell edge truncation points occur in the 120-cell? As many as 1200, the number of 5-cell edges, or perhaps 2400, if each edge is truncated at both ends. But also perhaps fewer; for example, if the 120-cell contains pairs of 5-cells with intersecting edges, and the edges intersect at the point on each edge where we make our cut. [[File:Great_(12)_chords_of_radius_√2.png|thumb|400px|Chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {{radic|2}} chords form two regular {6} hexagons (black).{{Efn|name=compound of 5 cuboctahedra}} The 120-cell edges form two irregular {6} hexagons (red truncated triangles) with the {{radic|5}} chords. The {6} intersection points (black) of the {{radic|5}} chords form a smaller red regular hexagon of radius {{radic|1}} (inscribed in the red circle).]]In the irregular {12} central plane chord diagram, we see six truncation points on the six 104.5° 5-cell edges of length {{Radic|5}}, where two co-planar 5-cell edges intersect, directly under the midpoint of a 44.5° chord (and under the intersection point of two 60° chords). The six truncation points lie on a red circle that is a circumference of the smaller 4-polytope created by this truncation. They form a red regular hexagon inscribed in the red circle. The edge length of this regular hexagon is {{radic|1}}. The two intersection points on the {{Radic|5}} chord divide it into its golden sections. The center section of the chord is <small><math>1</math></small>. The center section plus either of the smaller sections is <small><math>\phi = \tfrac{\sqrt{5} + 1}{2} \approx 1.618</math></small>, the larger golden section. Each of the two smaller sections is <small><math>\Phi = \phi - 1 = \tfrac{1}{\phi} \approx 0.618</math></small>, the smaller golden section.{{Efn|The bitruncated {30/8} chord of the 120-cell of radius <small><math>\sqrt{2}</math></small> provides a geometric derivation of the golden ratio formulas. First consider a 120-cell of radius <small><math>2\sqrt{2}</math></small> in which the {30/8} chord is <small><math>2\sqrt{5}</math></small> and the center section of the chord is <small><math>2</math></small>. Divide results by <small><math>2</math></small> to get a radius <small><math>\sqrt{2}</math></small> result. The left section of the chord is: :<small><math>\tfrac{\sqrt{5} - 1}{2} \approx 0.618</math></small> The center section plus the right section is: :<small><math>\tfrac{1 + \sqrt{5}}{2} \approx 1.618</math></small> The sum of these two golden sections is <small><math>\sqrt{5} \approx 2.236</math></small>, the chord length.}} The smaller golden sections <small><math>\Phi \approx 0.618</math></small> of the 5-cell edge are the same length as the 120-cell's 25.2° pentagon face diagonal chords. No 25.2° chords appear in the {12} central plane diagram, because they do not lie in {12} central planes. Each 104.5° 5-cell edge chord of length {{Radic|5}} has ''two'' points of intersection with other 5-cell edges, exactly 60° apart, the ''arc'' of a 24-cell edge chord, but with ''length'' {{radic|1}}. The center segment of the 5-cell edge, between the two intersection points, is a 24-cell edge in the smaller 4-polytope, and the red hexagon is a [[24-cell#Great hexagons|24-cell's great hexagon]] in the smaller 4-polytope. Nine other of its great hexagons, in other planes, each intersect with an antipodal pair of these {6} vertices. The dihedral angles between hexagon planes in a 24-cell are 60°, and four great hexagons intersect at each vertex. The 1200 5-cell edges, with two intersection points each, are reduced to 600 distinct vertices, so the smaller 4-polytope is a smaller 120-cell. The larger 120-cell, of radius {{radic|2}}, is concentric to a smaller instance of itself, of radius {{radic|1}}. Each 120-cell contains 225 distinct (25 disjoint) inscribed 24-cells. The smaller 24-cells are the [[w:Inscribed_sphere|insphere]] duals of the larger 24-cells. The vertices of the smaller 120-cell are located at the octahedral cell centers of the 24-cells in the larger 120-cell. Four 5-cell edges meet in 600 tetrahedral vertex figures. Four orthogonally intersecting 5-cell edges of the larger 120-cell meet in cubic vertex figures of 24-cells in the smaller 120-cell. Two disjoint 5-cell tetrahedral vertex figures are inscribed in alternate positions in each 24-cell cubic vertex figure. The 24-cell edges of the smaller 120-cell are the 5-cell edges of the larger 120-cell, truncated at both ends. The distance between the two points of intersection on a {{radic|5}} chord is {{radic|1}}, the same length as the 41.4° chord. But the actual 41.4° chords of the 120-cell do not appear in this diagram at all, because they do not lie in the 200 irregular {12} dodecagon central planes. === Bitruncating the 5-cells === The smaller concentric 120-cell can be built from 5-cell building blocks, by applying a specific kind of truncation operation to the blocks of the larger 120-cell called [[w:Bitruncation|''bitruncation'']]. This reveals a smaller irregular 4-polytope inside each 5-cell called the [[w:Bitruncated_5-cell|bitruncated 5-cell]]. The smaller unit-radius 120-cell is the convex hull of a compound of 20 disjoint (and 60 distinct) bitruncated 5-cells, bitruncated from the 120 disjoint 5-cells of the larger {{Radic|2}}-radius 120-cell. Bitruncation of the 120 disjoint 5-cells is the same truncation of the 120-cell described in the previous section, at the two golden section truncation points on each 104.5° 5-cell edge where two co-planar 5-cell edges intersect. [[File:Truncatedtetrahedron.gif|thumb|A 12-point [[w:Truncated_tetrahedron|truncated tetrahedron]] cell of the 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]].{{Sfn|Cyp: Truncated tetrahedron|2005}} Its edges are 41.4° chords of length 1 in a {{radic|2}}-radius 120-cell (or length {{radic|1/2}} in a unit-radius 120-cell). The 120-cell contains 20 disjoint (60 distinct) bitruncated 5-cells, containing 600 distinct truncated tetrahedra.]] The bitruncated 5-cell is a 30-vertex convex 4-polytope with 10 [[W:Truncated tetrahedron|truncated tetrahedron]] cells that have faces of two kinds: 4 triangle faces opposite 4 hexagon faces. The bitruncated 5-cell has 60 edges of the same length, 20 triangle faces, and 20 hexagon faces. Its 20 hexagon face planes are not [[24-cell#Great hexagons|24-cell central plane hexagons]]; they intersect each other at their edges, not at their long diameters. Its edges are not 60° 24-cell edge chords (the {{radic|2}} or 1 radius chords), but shorter 41.4° chords (of length 1 or {{radic|1/2}}), which do not appear at all in the diagram above, because they do not lie in the {12} central planes. The long diameter of the hexagon faces is not a 180° 120-cell long diameter chord (of length 2{{radic|2}} or 2) but a 90° 16-cell edge chord (of length 2 or {{radic|2}}). Consequently, three 16-cell tetrahedron cells (from three disjoint 16-cells) are inscribed in each truncated tetrahedron, at the three vertices of each face triangle. The truncated tetrahedron cell is a truncation of a tetrahedron of the same size as the tetrahedral cells of the 120-cell's 5-cells. The four smaller tetrahedra truncated from the corners of the larger tetrahedron have edges which are 25.2° chords (of length 1/𝜙 or {{radic|0.19}}). The truncated tetrahedron edges (of length 1 or {{radic|1/2}}) are equal in length to the 41.4° center sections of the 104.5° 5-cell edge chords (of length {{radic|5}} or {{radic|5/2}}). The shorter diagonal of the hexagon faces is the 75.5° chord (of length {{radic|3}} or {{radic|1.5}}), which is the 180° complement of the 104.5° 5-cell edge chord. The dimensions of the truncated tetrahedron cell suggest that it was cut directly from a 5-cell tetrahedron cell, simply by cutting off the tetrahedron corners, but remarkably, that is not the case. The edges of the bitruncated 5-cell are not actually center sections of 5-cell edges, although they are exactly that length, because the edges of the bitruncated 5-cell do not lie in the same {12} central planes as the 5-cell edges. They are not colinear with 5-cell edges in any way, and only intersect 5-cell edges at vertices (the 5-cell edges' intersection points). Bitruncation of the 5-cells does ''not'' simply truncate each tetrahedron cell in place. By creating new edges which connect the intersection points of 5-cell edges, bitruncation does create 600 truncated tetrahedron cells perfectly sized to fit within the 600 original tetrahedron cells, but at new locations, not centered on an original 5-cell tetrahedron cell. These new locations lie on a smaller 3-sphere than the original locations. [[File:Bitruncated_5-cell_net.png|thumb|Net of the bitruncated 5-cell honeycomb. 10 truncated tetrahedron cells alternately colored red and yellow.{{Sfn|Ruen: Net of the bitruncated 5-cell|2007}}]] The 3-dimensional surface of each bitruncated 5-cell is a honeycomb of 10 truncated tetrahedron cells. The truncated tetrahedra are joined face-to-face in a 3-sphere-filling honeycomb (like the cells of any 4-polytope), at both their hexagon and triangle faces. Each hexagonal face of a cell is joined in complementary orientation to the neighboring cell. Three cells meet at each edge, which is shared by two hexagons and one triangle. Four cells meet at each vertex in a [[w:Tetragonal_disphenoid|tetragonal disphenoid]] vertex figure. The 30-point bitruncated 5-cell is the convex common core (spatial [[w:Intersection|intersection]]) of six 5-point 5-cells in dual position. These six 5-cells are completely disjoint: they share no vertices, but their edges intersect orthogonally, at two points on each edge. Four 5-cell edges, from four of the six 5-cells, cross orthogonally in 30 places, the two intersection points on 60 5-cell edges: the 30 vertices of a bitruncated 5-cell. The six 5-cells are three dual pairs (in two different ways) of the self-dual 5-cell: six pairs of duals reciprocated at their common midsphere. Each dual pair intersects at just one of the two intersection points on each edge.{{Sfn|Klitzing|2025|loc=''sted'' (Stellated Decachoron)|ps=; [https://bendwavy.org/klitzing/incmats/sted.htm ''sted''] is the compound of two [https://bendwavy.org/klitzing/incmats/pen.htm ''pen'' (Pentachoron)] in dual position. Their intersection core ("Admiral of the fleet") is [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)].}} We have seen these six 5-cells before, illustrated in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Eleven|§Eleven]]'' above; they are the compound of six completely disjoint 5-cells visited during each 5-cell's characteristic isoclinic rotation of period 15.{{Efn|1=The 5-cell edges of the six disjoint pentagrams in the {30/12}=6{5/2} triacontagram illustration do not appear to intersect, as the 5-cell edge chords of the bitruncated 5-cell compound are said to intersect. The {30/12}=6{5/2} projection is a perspective view from inside the curved 3-dimensional space of the 120-cell's surface, looking straight down a cylindrical column of six stacked 5-cells. None of the 5-cell edges intersect in that curved 3-space, except where they meet at the 30 120-cell vertices. The 60 5-cell edges do intersect orthogonally in 4-space, in groups of four, at 30 points which lie on a smaller 3-sphere than the 120-cell. None of those 4-space intersections are visible in these projections of points and lines on the 120-cell's 3-sphere surface.|name=5-cell edges do not intersect is S³}} The six 5-cell compound is a stellated 4-polytope with 30 star-points, inscribed in the 120-cell.{{Efn|The stellated compound of six 5-cells in dual position is three pairs of 5-cells reciprocated at their common midsphere. It is composed of dual pairs of the [[W:Compound of five tetrahedra|compound of five tetrahedra]], which form the [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]; its 30 tetrahedral cells are three such dual pairs. In the compound of five tetrahedra the edges of the tetrahedra do not intersect. In the compound of ten tetrahedra they intersect orthogonally, but not at their midpoints. Each edge has two points of intersection on it. The compound of ten tetrahedra is five pairs of dual tetrahedra reciprocated at their common midsphere. It is inscribed in a dodecahedron (its convex hull). Its ''stellation core'' is an icosahedron, but its ''common core'' where the tetrahedron edges intersect is a dodecahedron, the tetrahedrons' convex spatial intersection. The stellated compound of six 5-cells has the analogous property: it is inscribed in a bitruncated 5-cell (its convex hull), and its common core is a smaller bitruncated 5-cell. (Its stellation core is a [[W:Truncated 5-cell#Disphenoidal 30-cell|disphenoidal 30-cell]], its dual polytope.)|name=compound of six 5-cells}} It is 1/20th of the 600-point [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#How many building blocks, how many ways|great grand stellated 120-cell]], the compound of 120 5-cells. The convex hull of its 30 star-points is a bitruncated 5-cell. In this stellated compound of six 5-cells in dual position, the bitruncated 5-cell occurs in two places and two sizes: as both the convex hull, and the convex common core, of the six 5-cells. Inscribed in the larger 120-cell of radius {{radic|2}}, the convex hull of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length 1. The convex common core of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length {{radic|1/2}}, inscribed in the smaller 120-cell of radius 1. In the 120-cell, 120 disjoint 5-cell building blocks combine in dual position groups of six related by the 5-cell's isoclinic rotation, to make 60 bitruncated 5-cells inscribed in the self-dual 5-cells' midsphere (at their edge intersections), and also 60 larger bitruncated 5-cells inscribed in the 120-cell, with each of the 600 vertices shared by three bitruncated 5-cells. The 120-cell is the convex hull of a compound of 20 disjoint (60 distinct) 30-point bitruncated 5-cells, generated by the characteristic rotation of its 120 completely disjoint 5-cells.{{Sfn|Klitzing|2025|loc= ''teppix'' (tripesic hexacosachoron)|ps=; ''[https://bendwavy.org/klitzing/incmats/teppix.htm teppix]'' is a compound of 60 [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)] with 3 ''deca'' sharing each vertex.}}{{Efn|In the 120-cell, 600 tetrahedron cells of 120 completely disjoint 5-cells intersect at two truncation points on each edge. Those 2400 truncation points are the vertices of 200 disjoint (and 600 distinct) truncated tetrahedra, which are the cells of 20 disjoint (and 60 distinct) bitruncated 5-cells. The 60 bitruncated 5-cells share vertices, but not edges, faces or cells. Each bitruncated 5-cell finds its 30 vertices at the 30 intersection points of 4 orthogonal 5-cell edges, belonging to 6 disjoint 5-cells, in the original 120-cell. Each bitruncated 5-cell vertex lies on an edge of 4 disjoint original 5-cells. Each bitruncated 5-cell edge touches intersection points on all 6 disjoint original 5-cells, and is shared by 3 truncated tetrahedra of just one bitruncated 5-cell.}} In [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Concentric 120-cells|the previous section]] we saw that the six 5-cell edges in each central plane intersect at the {6} vertices of the red hexagon, a great hexagon of a 24-cell. Each 5-cell edge, truncated at both ends at those intersection points, is a 24-cell edge of one of the 24-cells inscribed in a smaller 120-cell: the 600 intersection points. In this section we have seen how that truncation of 5-cell edges at both ends is the bitruncation of the 5-cell, and those 5-cell edges, truncated at both ends, are the same length as edges of bitruncated 5-cells inscribed in the original 120-cell. Bitruncating the {{radic|2}}-radius 120-cell's 120 5-cells reveals a smaller unit-radius 120-cell. The 24-cell edges of the smaller 120-cell are 5-cell edges of a larger-radius-by-{{radic|2}} 120-cell, truncated at both ends. Both 120-cells have 24-point 24-cells and 30-point bitruncated 5-cells inscribed in them. The 60° edge length of the 24-cells equals the radius; it is {{radic|2}} times the 41.4° edge length of the bitruncated 5-cells. The 60° 24-cell edges lie in the {12} central planes with the 5-cell edges and the 120-cell edges; but the 41.4° bitruncated 5-cell edges do not. The 120-cell contains 25 disjoint (225 distinct) 24-cells, and 20 disjoint (60 distinct) bitruncated 5-cells. Although regular 5-cells do not combine to form any regular 4-polytope smaller than the 120-cell, the 5-cells do combine to form semi-regular bitruncated 5-cells which are subsumed in the 120-cell.{{Efn|Although only major chords occur in regular 4-polytopes smaller than the 120-cell, minor chords do occur in semi-regular 4-polytopes smaller than the 120-cell. Truncating the 5-cell creates minor chords, such as the 41.1° edges of the bitruncated 5-cell.}} The 41.4° edge of the 30-point bitruncated 5-cell is also the triangle face edge we found in the 60-point central [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The real hemi-icosahedron|section 8₃ (Moxness's Hull #8) rhombicosidodecahedron]]. There are 60 distinct section 8₃ rhombicosidodecahedra and 600 distinct truncated tetrahedron cells of 60 distinct (20 disjoint) bitruncated 5-cells, and they share triangle faces, but little else. The truncated tetrahedron cells cannot be inscribed in the rhombicosidodecahedra, and the only chords they share are the 41.4° triangle edge and the 75.5° chord (the 180° complement of the 104.5° 5-cell edge chord). The section 8₃ rhombicosidodecahedron's 20 triangle faces lie over the centers of 20 larger-by-√2 5-cell faces, parallel to them and to a {12} central plane. The 5-cell faces are inscribed in the rhombicosidodecahedron, but are not edge-bound to each other; the 20 faces belong to 10 completely disjoint 5-cells. The 5-cell edges (but not the 5-cell faces) lie in {12} central planes; the 5-cell faces, the bitruncated 5-cell edges and their triangle and hexagon faces do not. Each section 8₃ rhombicosidodecahedron is the intersection of ten {12} central planes, shared pairwise with ten other rhombicosidodecahedra; 11 rhombicosidodecahedra share ten {12} central planes pairwise, as cells of a 4-polytope share face planes pairwise. Each truncated tetrahedron cell of a bitruncated 5-cell shares none of the {12} central planes; it is the intersection of 6 great rectangles, with two parallel 41.1° edges lying in each, alternating with two parallel 138.6° chords (its hexagon face diameters). Each bitruncated 5-cell is the intersection of 30 great rectangle {4} central planes. A truncated tetrahedron is face-bonded to the outside of each triangle face of a rhombicosidodecahedron. Three of its hexagon faces stand on the long edge of a rectangle face, perpendicular to the rectangle. We find the 25.2° chord as the edge of the non-central section 6₃ (Moxness's Hull #6) rhombicosidodecahedron. Those 120 semi-regular rhombicosidodecahedra have only that single edge (of length 1/𝜙 in a {{radic|2}}-radius 120-cell, or 1/𝜙{{radic|2}} in a unit-radius 120-cell). This edge length is in the golden ratio to the 41.4° edge of the 30-point bitruncated 5-cells, which is also the triangle face edge of the central section 8₃ (Moxness's Hull #8) rhombicosidodecahedron. The 120 semi-regular section 6₃ rhombicosidodecahedra share their smaller edges with 720 pentagonal prisms, 1200 hexagonal prisms and 600 truncated tetrahedron cells, in a semi-regular honeycomb of the 120-cell discovered by Alicia Boole Stott and described in her 1910 paper.{{Sfn|Boole Stott|1910|loc=Table of Polytopes in S₄|ps=; <math>e_2e_3C_{120}\ RID\ P_5\ P_6\ tT</math>}} These truncated tetrahedra are 1/𝜙 smaller than the 600 cells of the bitruncated 5-cells. The 60 distinct section 8₃ rhombicosidodecahedra (Moxness's Hull #8) share pentagon faces. Each of the 120 dodecahedron cells lies just inside 12 distinct rhombicosidodecahedra which share its volume. Each rhombicosidodecahedron includes a ball of 13 dodecahedron cells, 12 around one at the center of the rhombicosidodecahedron, within its volume. The remainder of the rhombicosidodecahedron is filled by 30 dodecahedron cell fragments that fit into the concavities of the 13 cell ball of dodecahedra. These fragments have triangle and rectangle faces. === Rectifying the 16-cells === Bitruncation is not the only way to truncate a regular polytope, or even the simplest way. The simplest method of truncation is [[w:Rectification_(geometry)|''rectification'']], complete truncation at the midpoint of each edge. Moreover, the 5-cell is not the only 120-cell building block we can truncate. We saw how bitruncation of the {{radic|2}}-radius 120-cell's 5-cells reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 20 disjoint (60 distinct) bitruncated 5-cells. In the next paragraph we describe how rectification of the {{radic|2}}-radius 120-cell's 16-cells also reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 25 disjoint (225 distinct) 24-cells. Those two operations on the 120-cell are equivalent. They are the same truncation of the 120-cell, which bitruncates 5-cells into bitruncated 5-cells, and also rectifies 16-cells into 24-cells. This single truncation of the 120-cell captures the distant relationship of 5-cell building blocks to 16-cell building blocks. Rectifying a {{radic|2}}-radius 16-cell of edge 2 creates a unit-radius 24-cell of unit edge, which is the compound of three unit-radius 16-cells. Rectifying one of those inscribed unit-radius 16-cells of edge {{radic|2}} creates a smaller 24-cell of radius and edge {{radic|1/2}}, which is the [[24-cell#Relationships among interior polytopes|common core (intersection]]) of the unit 24-cell and its three inscribed 16-cells. Like the 120-cell itself, the 24-cell is concentric to a smaller instance of itself of {{radic|1/2}} its radius. The common core of each of the 24-cells inscribed in the 120-cell is the corresponding 24-cell in the smaller 120-cell. === Rectifying the 5-cells === In the previous section we bitruncated the 5-cells and rectified the 16-cells, as one combined truncation operation that yields a smaller 120-cell of {{radic|1/2}} the radius. We can also rectify the 5-cells; but that is another distinct truncation operation, that yields a smaller 4-polytope of {{radic|3/8}} the radius. [[File:Great (12) chords of rectified 5-cell.png|thumb|400px|5-cell edge chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {6} bitruncation points (two on each of the 104.5° {{radic|5}} 5-cell edges) lie on a smaller 120-cell of radius 1 (the red circle); they are bitruncated 5-cell vertices. The {6} rectification points (at the midpoints of the 5-cell edges) lie on a still smaller 1200-point 4-polytope of radius {{radic|0.75}} ≈ 0.866 (the magenta circle); they are rectified 5-cell vertices.]] Rectifying the 5-cell creates the 10-point 10-cell semi-regular [[W:Rectified 5-cell|rectified 5-cell]], with 5 tetrahedral cells and 5 octahedral cells. It has 30 edges and 30 equilateral triangle faces. The 3-dimensional surface of the rectified 5-cell is an alternating [[W:Tetrahedral-octahedral honeycomb|tetrahedral-octahedral honeycomb]] of just 5 tetrahedra and 5 octahedra, tessellating the 3-sphere. Its vertex figure is the cuboctahedron. The rectified 5-cell is a [[w:Blind_polytope|Blind polytope]], because it is convex with only regular facets. It is a bistratic lace tower which has exactly three vertex layers with the same Coxeter symmetry, aligned on top of each other.{{Sfn|Klitzing|2025|loc=''[https://bendwavy.org/klitzing/incmats/rap.htm rap (rectified pentachoron)]''}} If the 120 5-cells in a radius {{radic|2}} 120-cell are rectified, the rectified 5-cells lie on a smaller 4-polytope of radius {{radic|3/4}} (the magenta circle in the diagram), inscribed at the 1200 midedges of the 5-cells.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the ''unit-radius'' 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the unit-radius 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of the [[24-cell#Characteristic orthoscheme|24-cell's 4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[24-cell#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the ''area'' of the equilateral triangle face of the unit-edge, unit-radius 24-cell.|name=root 3/4}} This smaller 4-polytope is not a smaller 120-cell; it is the convex hull of a 1200-point compound of two 120-cells. The rectified 5-cell does not occur inscribed in the 120-cell; it only occurs in this compound of two 120-cells, 240 regular 5-cells, and 120 rectified 5-cells. The rectified 5-cell with its 80.4° edge chord does not occur anywhere in a single 120-cell, so the rectified 5-cell's edges are not the edges of any polytope found in the 120-cell. The rectified 5-cell's significance to the 120-cell is well-hidden, but we shall see that it has an indirect role as a building block of the 11-cells in the 120-cell. Each 10-point rectified 5-cell is the convex hull of a stellated compound of two completely orthogonal 5-point 5-cells: five pairs of antipodal vertices. Their edges intersect at the midedge, and they are ''not'' in dual position (not reciprocated at their common 3-sphere). In this stellated compound of two completely orthogonal 5-cells (which does not occur in the 120-cell), the rectified 5-cell occurs in two places and two sizes: as both the convex hull of the vertices, and the convex common core of the midedge intersections. The edge length of the rectified 5-cells in the smaller 1200-point 4-polytope of radius {{radic|3/4}} is {{radic|5/4}}. The edge length of a unit-radius rectified 5-cell is {{radic|5/3}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|3}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}. In the 120-cell of radius {{radic|2}}, the 104.5° {{radic|5}} chord is the 5-cell edge, and the 75.5° {{radic|3}} chord is the distance between two parallel 5-cell edges (belonging to two disjoint 5-cells). The 104.5° and 75.5° chords are 180° complements, so they form great rectangles in the {12} central planes of the 120-cell (the red rectangles in the diagram). In the 1200-point compound of two 120-cells of radius {{radic|3}} where 120 rectified 5-cells occur, the {{radic|3}} chord is the ''radius'' (not the 75.5° chord), and the {{radic|5}} chord is the ''rectified'' 5-cell edge of arc 80.4° (not the 104.5° regular 5-cell edge). === Truncating the 5-cells === [[File:Great (12) chords of unit thirds radius.png|thumb|400px|Truncating the 120-cell's 5-cells at ''one-third'' of their edge length produces a smaller 120-cell of ''one-half'' the radius, with vertices at {6} one-third intersection points of the 120° {{Radic|6}} chords (''not'' of the 104.5° {{Radic|5}} 5-cell edge chords). The green {6} hexagon is a 24-cell great hexagon in the resulting smaller-by-one-half 1200-point 4-polytopes. Because there are {12} such intersection points in each {12} central plane, there are two chiral ways to perform this truncation, which produce disjoint 1200-point 4-polytopes.]] A third simple way to truncate the 5-cell is at one-third of its edge length. This truncation of the 5-cell creates a 20-point, 10-cell semi-regular 4-polytope, known somewhat ambiguously as ''the'' [[w:Truncated_5-cell|truncated 5-cell]], with 5 truncated tetrahedron cells (like the bitruncated 5-cell's), and 5 regular tetrahedron cells (like the rectified 5-cell's). The 3-dimensional surface of the truncated 5-cell is an alternating honeycomb of 5 truncated tetrahedra and 5 regular tetrahedra. It resembles the smaller rectified 5-cell with truncated tetrahedra instead of octahedra, or the larger bitruncated 5-cell with half its truncated tetrahedra replaced by regular tetrahedra. When the regular 5-cell is truncated at ''one-third'' of its edge length, the radius and edge length of the the resulting truncated 5-cell are ''one-half'' the regular 5-cell's radius and edge length. When the 120 5-cells in a 120-cell of radius 2 are truncated at one-third of their edge length, the truncated 5-cells lie on a smaller 120-cell of radius 1. The edge length of the unit-radius truncated 5-cell is {{radic|5/8}}, one-half the unit-radius 5-cell's edge length of {{radic|5/2}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|8}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}, and the rectified 5-cell is characterized by the ratio {{radic|5}} to {{radic|3}}. The 20-point truncated 5-cell is the convex common core of a stellated compound of four 5-cells (the four 5-cells' spatial intersection). The convex common core has half the radius of the convex hull of the compound. The four 5-cells are orthogonal (aligned on the four orthogonal axes), but none of their 20 vertices are antipodal. The 5-cells are ''not'' in dual position (not reciprocated at their common 3-sphere). The 5-cell edges do ''not'' intersect, but truncating the 120-cell's 5-cell edge chords at their one-third points truncates the 120-cell's other chords similarly. It is the 120-cell's 120° chords (of length {{Radic|6}} in a {{Radic|2}}-radius 120-cell, or {{Radic|3}} in a unit-radius 120-cell) which intersect each other at their one-third points. Four edges (one from each 5-cell) intersect orthogonally at just ''one'' of the two one-third intersection points on each of the 2400 120° chords that join vertices of two disjoint 5-cells. There are two chiral ways to perform this truncation of the 120-cell; they use the alternate intersection points on each edge, and produce disjoint 600-point 120-cells. The 52.25° edge chord of the truncated 5-cell (one-half the 5-cell's 104.5° edge chord) is not among the [[120-cell#Chords|chords of the 120-cell]], so the truncated 5-cell does not occur inscribed in the 120-cell; it occurs only in a compound of four 120-cells, and 480 regular 5-cells, and 120 truncated 5-cells. In the stellated compound of four orthogonal 5-cells (which does not occur in the 120-cell), the truncated 5-cell occurs in two places and two sizes: as both the convex hull of the 20 vertices, and the convex common core (of half the radius of the convex hull) of the 20 intersection points of four orthogonal 120° chords. == The perfection of Fuller's cyclic design == [[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 orthogonal axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]] This section is not an historical digression, but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]''}} After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session<ref>{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}</ref> where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.<ref>{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}</ref>|name=Snelson and Fuller}} A tensegrity icosahedron is an icosahedral geodesic sphere whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables. The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 10 of in Moxness's Hull #8 rhombicosidodecahedron, the real cell of the 11-cell. The Jessen's was named by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a [[W:Cubist|cubist]] shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point ''vector equilibrium'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point vector equilibrium, and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposite equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (a hexad), then backing away again while still rotating in the same directions. All this was overlaid with Fuller's own deep commentary, in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing "vector edge cube" (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}} Earlier, we noticed 10 Jessen's inscribed in each 60-point rhombicosidodecahedron central section of the 120-cell (each real hemi-icosahedron). Each rhombicosidodecahedron is a compound of 5 disjoint Jessen's, in two different ways, just the way the 120-cell is a compound of 5 disjoint 600-cells, in two different ways. In the rhombicosidodecahedron each regular icosahedron vertex has been replaced by the five vertices of a little pentagon face (a 120-cell face), and the regular icosahedron has been replaced by 5 disjoint (10 distinct) Jessen's icosahedra.{{Efn|name=compound of 5 cuboctahedra}} The 3 pairs of parallel 5-cell edges in each Jessen's lie a bit uncertainly, infinitesimally mobile and [[Kinematics of the cuboctahedron#Elastic-edge transformation|behaving like the struts of a tensegrity icosahedron]], so we can push any parallel pair of them apart or together infinitesimally, making each Jessen's icosahedron expand or contract infinitesimally. All 600 Jessen's, all 60 rhombicosidodecahedra, and the 120-cell itself expand or contract infinitesimally, together.{{Efn|name=tensegrity 120-cell}} Expansion and contraction are Boole Stott's operators of dimensional analogy, and that infinitesimal mobility is the infinite calculus of an inter-dimensional symmetry. The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three rectangular chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables. [[File:Jessen's √2 radius dimensions.png|thumb|400px|Moxness's 60-point section 8₃ rhombicosidodecahedron is a compound of 5 of this 12-point Jessen's icosahedron, shown here in a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed {{radic|1.5}} green cube, and its 8 equilateral triangle faces are 24-cell faces. This is a ''vertex figure'' of the 120-cell. The center point is also a vertex of the 120-cell.]] As a matter of convenience in this paper, we have used {{radic|2}}-radius metrics for 3-sphere polytopes, so e.g. the 5-cell edge is {{radic|5}}, where in unit-radius coordinates it would be {{Radic|5/2}}. Here we give two illustrations of the Jessen's using two different metrics: the 2-sphere Jessen's has a {{radic|5}} diameter, and the 3-sphere Jessen's has a {{radic|2}} radius. This reveals a curiously cyclic way in which our 2-sphere and 3-sphere metrics correspond. In the embedding into 4-space the characteristic root factors of the Jessen's seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord. We might have expected to find the 6-point hemi-icosahedron's 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the Jessen's are the 5-cell edges. A 5-cell face has its three {{radic|5}} edges in three different Jessen's icosahedra. The Jessen's is not a cell, but one of the 120-cell's vertex figures, like the [[600-cell#Icosahedra|120 regular icosahedron vertex figures in the 600-cell]]. That is why we find 600 Jessen's, of course. The center point in this Jessen's illustration is another ''vertex'' of the 120-cell, not the empty center of a cell.{{Efn|The 13 vertices of the illustration which include its center point lie in the curved 3-space of the 3-sphere, on the 120-cell's surface. In 4-space, this object is an [[W:Icosahedral pyramid|icosahedral pyramid]] with a Jessen's icosahedron as its base, and the apical center vertex as its apex. The center point in the illustration is a vertex of the 120-cell, and the center of the curved Jessen's, and the apex of the icosahedral pyramid, but it is not the center point in 4-space of a flat 3-dimensional Jessen's icosahedron. The center point of the base Jessen's icosahedron is a point inside the 120-cell, not a 120-cell vertex on its surface. It lies in the same 3-dimensional flat-slice hyperplane as the 12 vertices of the base Jessen's icosahedron, directly below the 13th 120-cell vertex.}} Each Jessen's includes the central apex vertex, {{radic|2}} radii, {{radic|2}} edges and {{radic|5}} chords of a vertex figure around the 120-cell vertex at its center. The {{radic|2}} face edges are 24-cell edges (also tesseract edges), and the inscribed green cube is the 24-cell's cube vertex figure. The 8 {{radic|2}} face triangles occur in 8 distinct 24-cells that meet at the apex vertex.{{Efn|Eight 24-cells meet at each vertex of a [[24-cell#Radially equilateral honeycomb|honeycomb of 24-cells]]: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell.{{Efn|In the 600-cell, which contains [[600-cell#Twenty-five 24-cells|25 24-cells]], 5 24-cells meet at each vertex. Each pair of 24-cells at the vertex meets at one of 200 distinct great hexagon central planes. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 8 other 24-cells. In the 120-cell, which contains 10 600-cells (5 disjoint 600-cells two different ways) and 225 24-cells (25 disjoint 24-cells), 8 24-cells meet at each vertex. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 208 other 24-cells. But since in the 120-cell the great hexagons lie in pairs in one of 200 {12} central planes (containing 400 great hexagons), each 24-cell shares one of its {12} central ''planes'' with .. other 24-cells.}}}} This Jessen's vertex figure includes 5-cell edges and 24-cell edges (which are also tesseract edges), so it is descriptive of the relationship between those regular 4-polytopes, but it does not include any 120-cell edges or 600-cell edges, so it has nothing to say, by itself, about the <math>H_4</math> polytopes. It is only a tiny fraction of the 120-cell's full vertex figure, which is a staggeringly complex star: 600 chords of 30 distinct lengths meet at each of the 600 vertices. The {{radic|5}} chords are 5-cell edges, connecting vertices in different 24-cells. The 3 pairs of parallel 5-cell edges in each Jessen's lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th pair of parallel 5-cell edges orthogonal to all of them, in fact three more orthogonal pairs, since 6 orthogonal planes (not just 4) intersect at a point in 4-space. The Jessen's situation is that it lies completely orthogonal to another Jessen's, the vertex figure of the antipodal vertex, and its 3 orthogonal planes (xy, yz, zx) lie completely orthogonal to its antipodal Jessen's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} These 6 pairs of parallel 5-cell edges form a 24-point 4-polytope, composed of two completely orthogonal 12-point Jessen's, inscribed in two completely orthogonal rhombicosidodecahedra. This 24-point 4-polytope is not a 24-cell: the 24-cell is not a compound of two 12-point Jessen's. But it turns out that two completely orthogonal 12-point Jessen's indirectly define a 24-point 24-cell. We shall see that their 4-space intersection is a 24-cell. This finding, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, brings Fuller's study of [[w:Tesseract#Radial_equilateral_symmetry|radially equilateral]] vector equilibrium polytopes to its completion in the 24-cell. Fuller began with the hexagon, the 6-point vector equilibrium in 2 dimensions, the only polygon with its radius equal to its edge length. He studied the cuboctahedron, the 12-point vector equilibrium in 3 dimensions, the only polyhedron with its radius equal to its edge length, in all its flexible guises. He discovered its stable equilibrium as the the Jessen's shadfish, with its cube of 6 open mouths and 90° dihedral angles between all its faces, the geometric center of [[WikiJournal Preprints/Kinematics of the cuboctahedron|the cuboctahedron's kinematic transformation]] through the regular polyhedra: tetrahedron, octahedron, Jessen's, regular icosahedron, and cuboctahedron. Fuller's study of kinematic Euclidean geometry did not reach the 4-polytopes, and the ultimate 24-point vector equilibrium in 4 dimensions, the 24-cell, the unique <math>F_4</math> symmetry found only in 4 dimensions. But Fuller led us up to it, through the kinematics of infinitesimal mobility, and that route to it is our clue to the infinite calculus of dimensional expansion and contraction. We observe this geometry, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, only in the 120-cell. The 600-cell contains 12-point Jessen's, but no completely orthogonal pairs of them. The 24-cell individually, and the 25 24-cells in the 600-cell, are not occupied by a pair of 12-point Jessen's. The 24-point 24-cell is not, in fact, a compound of two 12-point Jessen's. While the 120-cell's ratio of disjoint 12-point Jessen's to disjoint 24-point 24-cells is <math>50/25 = 2/1</math>, the ratio of distinct 12-point Jessen's to distinct 24-point 24-cells is <math>600/225 = 8/3 </math>. We observe another geometry, of 24-cells in dual positions, only in the 120-cell. No two 24-cells in the 600-cell are in dual positions, but in the 120-cell with 225 distinct 24-cells (25 disjoint 24-cells), every 24-cell is in dual position to other 24-cells. The 24-cell is self-dual, and when two 24-cells of the same radius are in dual position, they are completely disjoint with respect to vertices, but they intersect at the midpoints of their 96 orthogonal edges. Since four orthogonal lines intersect at a point in 4-space, in addition to the midedge radius and the two intersecting edges there is a third intersecting edge through each point of contact: ''three'' 24-cells lie in dual positions to each other, with their orthogonal edges intersecting. Three ''pairs'' of 24-cells lie in orthogonal dual positions to each other, sharing no vertices, but the same 96 midedge points. We also observe this geometry, of 24-cells in dual positions, in the irregular {12} dodecagon central planes, which have two inscribed great {6} hexagons, offset from each other irregularly by a 15.5° arc on one side (a 120-cell edge chord) and a 44.5° arc on the other side. The 600-cell and the 24-cell contain only great {6} hexagon planes. The two inscribed great {6} hexagons in each {12} central plane belong to a pair of 24-cells in dual position. We observe inscribed 5-cells only in the 120-cell. The 600-cell has <math>5^2 = 25</math> distinct 24-cells inscribed in 120 vertices, and is a regular compound of <math>5</math> disjoint 24-cells in 10 different ways, but it has no inscribed 5-point 5-cells joining corresponding vertices of 5 of its 25 24-cells.{{Efn|The 600-cell does have inscribed 5-point great pentagons joining corresponding vertices of 5 of its 25 24-cells. The 600-cell has 2-dimensional pentads, but only the 120-cell has 4-dimensional pentads.}} The 120-cell has <math>5^2 \times 3^2 = 225</math> distinct 24-cells inscribed in 600 vertices, and is a regular compound of <math>5^2 = 25</math> disjoint 24-point 24-cells in 10 different ways, and it has 120 inscribed 5-cells joining corresponding vertices of 5 of its 225 24-cells. [[File:Great 5-cell √5 digons rectangle.png|thumb|400px|Three {{radic|5}} x {{radic|3}} rectangles (red) are found in 200 central planes of the radius {{radic|2}} 120-cell, and in its 600 Jessen's icosahedra, where 3 orthogonal rectangles comprise each 12-point Jessen's. Each central plane intersects {12} vertices in an irregular great dodecagon. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges (solid red), which form two opposing ''irregular'' great hexagons (truncated triangles) with the {{radic|5}} chords. The {12} central planes also contain four {{radic|6}} great triangles (green), inscribed in two {{radic|2}} ''regular'' great hexagons. 1200 smaller {{radic|5}} 5-cell ''face'' triangles (blue) occupy 600 other, non-central planes.]] The Jessen's eight {{radic|6}} triangle faces lie in eight great {6} hexagons in eight {12} central planes of the 120-cell. The Jessen's {{radic|5}} chords lie in great {4} rectangles ({{radic|5}} by {{radic|3}}) in orthogonal central planes of the Jessen's. These are ''also'' {12} central planes of the 120-cell. We can pick out the {{radic|5}} by {{radic|3}} rectangles in the {12} central plane chord diagrams (bounded by red dashed lines). The Jessen's vertex figure is bounded by eight {12} face planes, and divided by six orthogonal {12} central planes, and all 14 planes are {12} central planes of the 120-cell. The 5-cells' ''face'' planes are ''not'' central planes of the 120-cell. Recall that 10 distinct Jessen's are inscribed in each rhombicosidodecahedron, as two chiral sets of 5 completely disjoint Jessen's, such that two {{radic|5}} 5-cell edges meet at each vertex of the rhombicosidodecahedron. These are two of the four 5-cell edges that meet at each vertex of the 5-cell: edges of a 5-cell face, 20 of which are disjointly inscribed in each rhombicosidodecahedron. In each Jessen's the 6 {{radic|5}} reflex edges are disjoint, and in each rhombicosidodecahedron only two edges meet at each vertex, but in the 120-cell each {{radic|5}} chord meets three others, that lie in three other Jessen's. Each 5-cell face triangle has each edge in a distinct Jessen's, but the face triangle lies in just one rhombicosidodecahedron. The 1200 5-cell face triangles lie in opposing pairs, in one of 600 ''non-central'' hexagon ''face'' planes. Each of the 60 rhombicosidodecahedra is a compound of 10 Jessen's (5 disjoint Jessen's in two different ways), just the way the 120-cell is a compound of 10 600-cells (5 disjoint 600-cells in two different ways), and the 120-cell's dodecahedron cell is a compound of 10 600-cell tetrahedron cells (5 disjoint tetrahedra in two different ways). The 600 Jessen's in the 120-cell occur in bundles of 8 disjoint Jessen's, in 4 completely orthogonal pairs, each pair aligned with one of the four axes of the Cartesian coordinate system. Collectively they comprise 3 disjoint 24-cells in orthogonal dual position. They are [[24-cell#Clifford parallel polytopes|Clifford parallel 4-polytopes]], 3 completely disjoint 24-cells 90° apart, and two sets of 4 completely disjoint Jessen's 15.5° apart. Opposite triangle faces in a Jessen's occupy opposing positions in opposite great hexagons. In contrast, the two completely orthogonal Jessen's are completely disjoint, with completely orthogonal bounding planes that intersect only at one point, the center of the 120-cell. The corresponding {{radic|6}} triangle faces of two completely orthogonal Jessen's occupy completely orthogonal {12} central planes that share no vertices. If we look again at a single Jessen's, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) Jessen's lies in 4-space, it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (vertex figure) Jessen's is part of a 16-point (8-cell) tesseract containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already found it was part of a 24-point (24-cell) 4-polytope, which contains 3 16-point (8-cell) tesseracts. Each 12-point (6 {{radic|5}} reflex edge) Jessen's is one of 10 concentric Jessen's in a rhombicosidodecahedron, two sets of 5 disjoint Jessen's rotated with respect to each other isoclinically by 12° x 12° = 15.5°, with a total of 60 disjoint {{radic|5}} edges. Each 12-point (24 {{radic|6}} edge) Jessen's is one of 8 concentric Jessen's in two 24-cells in dual positions, rotated with respect to each other isoclinically by 41.4° x 41.4° = 90°, with a total of 192 {{radic|6}} edges.{{Efn|There are 96 {{radic|6}} chords in each 24-cell, linking every other vertex under its 96 {{radic|2}} edges.}} The 24-point 24-cell has 4 Hopf fibrations of 4 hexagonal great circle fibers, so it is a complex of 16 great hexagons, generally not orthogonal to each other, but containing 3 sets of 4 orthogonal great hexagons. Three Borromean link great rectangles are inscribed in each great hexagon, and three tesseracts are inscribed in each 24-cell. Four of the 6 orthogonal [[w:Borromean_rings|Borromean link]] great rectangles in each completely orthogonal pair of Jessen's are inscribed in each tesseract. == Conclusion == Thus we see what the 11-cell really is: an unexpected seventh regular convex 4-polytope falling between the 600-cell and 120-cell, a quasi-regular compound of 600-cell and 5-cell (an icosahedron-tetrahedron analogue), as the 24-cell is an unexpected sixth regular convex polytope falling between the 8-cell and 600-cell, a quasi-regular compound of 8-cell and 16-cell (a cube-octahedron analogue). Like the 5-cell, the 11-cell is a far-side 4-polytope with its long edges spanning the near and far halves of the 3-sphere. Unlike the 5-cell, the 11-cell's left and right rotational instances are not the same object: they have distinct cell polyhedra, which are duals. The 11-cell is a real regular convex 4-polytope, not just an [[W:abstract polytope|abstract 4-polytope]], but not just a singleton regular convex 4-polytope, and not just a single kind of cell honeycomb on the 3-sphere.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} Though it is all those things singly, it never occurs singly, but its multiple instances in the 120-cell compound to all those things, and significantly more. The 11-cell (singular) is the 11-vertex (17 cell) non-uniform Blind 4-polytope, with 11 non-uniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. The abstract regular 11-point (11-cell) has a realization in Euclidean 4-space as this convex 4-polytope, with regular facets and regular triangle faces. The 11-cell (plural) is subsumed in the 120-cell, as all the regular convex 4-polytopes are. The compound of eleven 11-cells (the ..-cell) and Schoute's compound of five 24-cells (the 600-cell) is the quasi-regular 137-point (..-cell) 4-polytope, an object of further study. The 11-cells' realization in the 120-cell as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (4-simplex and 4-orthoplex), which are both inscribed in the 11-point (17-cell), 137-point (..-cell) and 600-point (120-cell), but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationship between the regular ''n''-polytopes of different ''n''. The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies. == Build with the blocks == <blockquote>"The best of truths is of no use unless it has become one's most personal inner experience."{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: "The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery."}}}}</blockquote> <blockquote>"Even the very wise cannot see all ends."{{Sfn|Tolkien|1954|loc=Gandalf}}</blockquote> No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to show how I came by my understanding of these objects, since I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, in my imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am. {{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=2}} {{Regular convex 4-polytopes|columns=7|wiki=W:|radius=1}} == Acknowledgements == ... == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite web| url = https://www.youtube.com/watch?v=9sM44p385Ws| title = Vector Equilibrium | first = R. Buckminster | last = Fuller | author-link=W:Buckminster Fuller | year = 1975 | work= Everything I Know Sessions | place = Philadelphia}} * {{Citation|last=Christie|first=David Brooks|author-link=User:Dc.samizdat|year=2024|title=Bucky Fuller and the languages of geometry|title-link=User:Dc.samizdat#Bucky Fuller and the languages of geometry|journal=Wikiversity|ref={{SfnRef|Christie: On Fuller's use of language|2024}}}} * {{Citation|author-last=Moxness|author-first=J.G.|year=2022|author-link=W:User:Jgmoxness|title=120-Cell showing the individual 8 concentric hulls and in combination|title-link=Wikimedia:File:120-Cell showing the individual 8 concentric hulls and in combination.svg|journal=Wikimedia Commons|ref={{SfnRef|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)}}}} * {{Citation|author-last=Moxness|author-first=J.G.|year=2023|author-link=W:User:Jgmoxness|title=Archimedean and Catalan solid hulls with their Weyl orbit definitions|title-link=Wikimedia:File:Archimedean and Catalan solid hulls with their Weyl orbit definitions.svg|journal=Wikimedia Commons|ref={{SfnRef|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}}}} * {{Citation|author-last=Moxness|author-first=J.G.|year=2023|author-link=W:User:Jgmoxness|title=3D & 4D Solids using Quaternion Weyl Orbits from Coxeter-Dynkin Geometric Group Theory|journal=PowerPoint|url=https://theoryofeverything.org/TOE/JGM/Quaternion%20Coxeter-Dynkin%20Geometric%20Group%20Theory-2b.pdf|ref={{SfnRef|Moxness: Quaternion graphics software|2023}}}} === 11-cell === * {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1976 | title=Regularity of Graphs, Complexes and Designs | journal=Colloques Internationaux C.N.R.S. | publisher=Orsay | volume=260 | pages=191-197 | url=https://faculty.washington.edu/moishe/branko/BG111.Regularity%20of%20graphs,etc.pdf | ref=}} * {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1975 | title=Venn Diagrams and Independent Families of Sets | journal=Mathematics Magazine | volume=48 | issue=1 | url=https://maa.org/sites/default/files/pdf/upload_library/22/Ford/BrankoGrunbaum.pdf | doi=10.1080/0025570X.1975.11976431 }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147 }} *{{citation | last1 = Séquin | first1 = Carlo H. | author1-link = W:Carlo H. 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Séquin | title=A 10-Dimensional Jewel | journal=Gathering for Gardner G4GX | place=Atlanta GA | year=2012 | url=https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2012_G4GX_10D_jewel.pdf }} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} === Illustrations === * {{Citation|title=Tensegrity icosahedron structure|title-link=Wikimedia:File:Tensegrity Icosahedron.png|journal=Wikimedia Commons|last1=Burkhardt|first1=Bob|year=1994}} * {{Citation|author-last=Christie|author-first=David Brooks|year=2024|author-link=W:User:Dc.samizdat|title=Pentahemidemicube|title-link=Wikimedia:File:Pentahemidemicube.png|journal=Wikimedia Commons|ref={{SfnRef|Christie: Pentahemidemicube|2024}}}} * {{Citation|author-last=Christie|author-first=David Brooks|year=2024|author-link=W:User:Dc.samizdat|title=Pentahemicosahedron|title-link=Wikimedia:File:Pentahemicosahedron.png|journal=Wikimedia Commons|ref={{SfnRef|Christie: Pentahemicosahedron|2024}}}} * {{Citation|author=Cmglee|date=2019|author-link=W:User:Cmglee|title=Radially-symmetrical five-set Venn diagram devised by Branko Grünbaum|title-link=Wikimedia:File:Symmetrical 5-set Venn diagram.svg|journal=Wikimedia Commons|ref={{SfnRef|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled.}}}} * {{Citation|author-last=Cyp|year=2005|author-link=W:User:Cyp|title=Truncated tetrahedron, transparent, slowly turning, created with POV-ray|title-link=Wikimedia:File:Truncatedtetrahedron.gif|journal=Wikimedia Commons|ref={{SfnRef|Cyp: Truncated tetrahedron|2005}}}} * {{Cite book|last=Duveneck|first=Josephine Whitney|title=Life on Two Levels: An Autobiography|year=1978|publisher=William Kaufman|place=Los Altos, CA|ref={{SfnRef|Duveneck|1978}}}} * {{Citation|author-last=Hise|author-first=Jason|year=2011|author-link=W:User:JasonHise|title=A 3D projection of a 120-cell performing a simple rotation|title-link=Wikimedia:File:120-cell.gif|journal=Wikimedia Commons}} * {{Cite book|last=Huxley|first=Aldous|author-link=W:Aldous Huxley|title=Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization|date=1937|publisher=Harper and Brothers|ref={{SfnRef|Huxley|1937}}}} * {{Cite book|last=Jung|first=Carl Gustav|author-link=W:Carl Jung|title=Psychological Reflections: An Anthology of the Writings of C. G. Jung|date=1961|page=XVII|ref={{SfnRef|Jung|1961}}}} * {{Citation|author-last=Piesk|author-first=Tilman|date=2018|author-link=W:User:Watchduck|title=Nonuniform rhombicosidodecahedron as rectified rhombic triacontahedron max|title-link=Wikimedia:File:Nonuniform rhombicosidodecahedron as rectified rhombic triacontahedron max.png|journal=Wikimedia Commons|ref={{SfnRef|Piesk: Rhombicosidodecahedron|2018}}}} * {{Citation|author-last=Piesk|author-first=Tilman|date=2018|author-link=W:User:Watchduck|title=Polyhedron truncated 20 from yellow max|title-link=Wikimedia:File:Polyhedron truncated 20 from yellow max.png|journal=Wikimedia Commons|ref={{SfnRef|Piesk: Truncated icosahedron|2018}}}} * {{Citation|author-last=Ruen|author-first=Tom|year=2007|author-link=W:User:Tomruen|title=Hemi-icosahedron|title-link=Wikimedia:File:Hemi-icosahedron.png|journal=Wikimedia Commons|ref={{SfnRef|Ruen: Hemi-icosahedron|2007}}}} * {{Citation|title=Great grand stellated 120-cell|title-link=Wikimedia:File:Ortho solid 016-uniform polychoron p33-t0.png|journal=Wikimedia Commons|last1=Ruen|first1=Tom|year=2007|author-link=W:User:Tomruen|ref={{SfnRef|Ruen: Great grand stellated 120-cell|2007}}}} * {{Citation|author-last=Ruen|author-first=Tom|year=2019|author-link=W:User:Tomruen|title=Tetrahemihexahedron rotation|title-link=Wikimedia:File:Tetrahemihexahedron rotation.gif|journal=Wikimedia Commons|ref={{SfnRef|Ruen: Tetrahemihexahedron rotation|2019}}}} * {{Citation|title=Net of the bitruncated 5-cell|title-link=Wikimedia:File:Bitruncated 5-cell net.png|journal=Wikimedia Commons|last1=Ruen|first1=Tom|year=2007|author-link=W:User:Tomruen|ref={{SfnRef|Ruen: Net of the bitruncated 5-cell|2007}}}} * {{Citation|title=5-cell|title-link=5-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Christie|editor-first2=David Brooks|editor-link2=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen et al. eds. 5-cell|2024}}}} * {{Citation|title=16-cell|title-link=16-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Christie|editor-first2=David Brooks|editor-link2=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen et al. eds. 16-cell|2024}}}} * {{Citation|title=24-cell|title-link=24-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen & Goucher et al. eds. 24-cell|2024}}}} * {{Citation|title=600-cell|title-link=600-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|ref={{SfnRef|Ruen & Goucher et al. eds. 600-cell|2024}}}} * {{Citation|title=120-cell|title-link=120-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|ref={{SfnRef|Ruen & Goucher et al. eds. 120-cell|2024}}}} * {{Cite book|last=Sandperl|first=Ira|author-link=W:Ira Sandperl|title=A Little Kinder|year=1974|publisher=Science and Behavior Books|place=Palo Alto, CA|isbn=0-8314-0035-8|lccn=73-93870|url=https://www.allinoneboat.org/a-little-kinder-an-old-friend-moves-on/|ref={{SfnRef|Sandperl|1974}}}} * {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|orig-date=1954|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|date=1967|publisher=Houghton Mifflin|place=Boston|author-link=W:J.R.R.Tolkien|title-link=W:The Lord of the Rings|ref={{SfnRef|Tolkien|1954}}}} {{Refend}} opw24vuxuc39hzt8clds432hp75sg33 2816271 2816268 2026-06-19T01:40:17Z Dc.samizdat 2856930 /* Concentric 120-cells */ 2816271 wikitext text/x-wiki = A symmetrical arrangement of eleven 11-cells = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|March 2024 - June 2026}} <blockquote>[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]] ₅{3,5,3}₅, a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]] {3,5}₅, a hexad non-orientable polyhedron. The 11-cell is described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, we find that the 11-cell has a realization in Euclidean 4-space inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the convex regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra and 120 regular 5-cells. We find that the 120-cell also contains: a non-uniform icosahedral polyhedron that contains the realization of the abstract hemi-icosahedron; real 11-point 11-cells made from 11 of it; and a compound of eleven real 11-cells. We also find a quasi-regular compound of the compound of eleven 11-cells and [[w:Schoute|Schoute]]'s compound of five 24-cells (the 600-cell). We describe the real 11-point 11-cell 4-polytope; its compound of eleven 11-cells; the quasi-regular compound; and their relation to the regular polytopes.</blockquote> == Introduction == [[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hemi-icosahedral hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them. [[File:120-cell.gif|thumb|360px|The picture on the cover of the box of 4-dimensional building blocks.{{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]]."}} Only the 120-cell's own edges are shown. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and 11-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]] The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-hypercube, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the 600-point [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). The 5-cell has a fundamental relationship to all the other 4-polytopes, but not one as simple as compounding, so it is not immediately useful to children trying to learn to build with 4-dimensional building blocks. But the 16-cell is our very starting point, and the most frequently used tool in the box. Nevertheless, to build the 11-cell, we start with the 5-cell. The 5-cell and 11-cell are both self-reciprocal (their own duals). They are the only 4-polytopes where every cell shares a face with every other cell. The 5-cell is a tetrahedron surrounded by 4 other tetrahedra, in five different ways. The 11-cell is a hemi-icosahedron surrounded by 10 other hemi-icosahedra, in eleven different ways. The 5-cell has 5 vertices that form 5 tetrahedral cells, and a total of 10 triangular faces and 10 edges. The 11-cell has 11 vertices that form 11 hemi-icosahedral cells, each with 6 verticies 10 triangular faces and 15 edges, and a total of 55 triangular faces and 55 edges. == 5-cells and hemi-icosahedra in the 11-cell == [[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its <math>2^5</math> elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]] [[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two or more real elements found in different places in a concrete realization of the 11-cell.{{Sfn|Ruen: Hemi-icosahedron|2007}}]] The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we find a facet congruence between a 4-polytope and a 3-polytope we suspect a dimensional analogy. In the exceptional case of 5-cell and icosahedron, which share the same symmetry group <math>A_5</math>, we fully expect a dimensional analogy.{{Efn|There is an exceptional inter-dimensional duality between the regular icosahedron and the 5-cell because they share <math>A_5</math> symmetry. See this question asked on [https://math.stackexchange.com/questions/4235783/the-rotational-symmetry-groups-of-the-5-cell-and-the-icosahedron-are-isomorphi math.stackexchange.com 2021].}} Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization as the 6-point 5-simplex. Yet another real hexad is the 6-point 3-orthoplex; thus as a hexad the hemi-icosahedron is related by dimensional analogy to the 4-simplex (5-cell) from above, and to the 4-orthoplex (16-cell) from below, while those two simplest 4-polytope building blocks are only related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The cell of the 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, the 4-simplex (5-cell) and 4-orthoplex (16-cell), who are famously stand-offish with each other. Interesting! The cell of the 11-cell is an abstract hexad hemi-icosahedron with 5 central planes, most handsomely illustrated by Séquin.{{Sfn|Séquin|2012|loc=A 10-Dimensional Jewel}}{{Sfn|Séquin & Lanier|2007|p=3|loc=Figure 4: (b,c) two views of the hemi-icosahedron projected into 3D space|ps=; Séquin et. al. have a lovely colored illustration of the hemi-icosahedron, subdivided into 10 triangular faces by 5 central planes of its icosahedral symmetry, revealing rings of polytopes nestled in its interior. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}}{{Sfn|Séquin & Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point [[W:Hemi-isosahedron|hemi-isosahedron]] is the vertex figure of the 11-cell's dual 4-polytope the 57-point [[W:57-cell|57-cell]].}} The 11 hemi-icosahedral cells have 10 triangle faces each, and each cell is face-bonded to the other 10 cells. The 5-cell's 5 tetrahedral cells have 10 faces and 10 edges altogether, and each cell is face-bonded to the other 4 cells. If 11-cell faces correspond to 5-cell faces, then 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, that does not belong to the same hemi-icosahedral cell as its opposing face. He found that the 10 edges opposite each hemi-icosahedron's 10 faces are the 10 edges of a single 5-cell, which does not share any vertices, edges or faces with the hemi-icosahedron. For each cell of the 11-point 11-cell, there is exactly one 5-point 5-cell that is completely disjoint from the 6-point hemi-icosahedron cell.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; "We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary <math>a_4</math> [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron."}} There are 11 disjoint 5-cell 4-polytopes inscribed in each 11-cell, which also contains 11 hemi-icosahedral cells, 55 faces, 55 edges and 11 vertices. The real 11-cell is more complex than the abstract 11-cell representing it, because the real hemi-icosahedron is more complex and harder to find than the abstract hemi-icosahedron. Seeing the real 11-cell will be easier once we have identified the real hemi-icosahedron, and seen exactly where the 11-cell's real elements reside in the other 4-polytopes within the 120-cell with which the 11-cell intermingles. The 5-cell has 10 faces, and the 11-cell has 10 faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron is face-bonded to the other 10 hemi-icosahedra, and to 10 of the 11 5-cells, and there is exactly one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point 5-cell is the other 5 vertices of the 11-point 11-cell that are not vertices of this 6-point hemi-icosahedron: the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces which belong to 10 distinct hemi-icosahedra of the 11-cell, and there is just one hemi-icosahedron with which it does not share a face. In the abstract 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 completely disjoint 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction.{{Efn|We shall see that this small eliminated distance is in fact the length of a 120-cell edge, the shortest chordal distance found in the 120-cell.}} Seemingly adjacent hemi-icosahedron faces do not actually meet at an edge; there is a polygon separating them, which has been abstracted to an edge. The 10 hemi-icosahedron faces are 5-cell faces from 10 distinct 5-cells, and they do not actually touch each other: the 120 5-cells in the 120-cell are completely disjoint. In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of tetrahedral cells together, because each 11-cell face represents two actual 5-cell faces in different places. Each duplex 11-cell face bonds tetrahedra in two 5-cells in different places, without binding the 5-cells together (they are completely disjoint). One actual 5-cell face is one half of a duplex 11-cell face, so 110 5-cell faces are 55 duplex 11-cell faces. The 11-cell's 11 abstract vertices represent all 55 distinct vertices of the 11 disjoint 5-cells, so they must be abstract conflations of at least 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; 11-cells must be sharing vertices, not disjoint as the 5-cells are. == The real hemi-icosahedron == [[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right| Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; "Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, & 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 & 5 are each pairs of Truncated icosahedrons. Hulls 6 & 8 are Rhombicosidodecahedrons."}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes starting with a dodecahedron cell. Hull #8 with 60 vertices (lower right) is a central section of the 120-cell, the 8th and largest section starting with a cell.{{Efn|1=Although the 8 hulls are illustrated as the same size, in the 120-cell they have increasing size as numbered, and occur nested inside each other like Russian dolls. Only Hull #8 is a central section of the same radius as the 120-cell itself, analogous to the equator. Sections 1-7 occur in pairs on opposite sides of the central section, and are analogous to lines of latitude. Section 1 is simply a dodecahedral cell. The "Combined hulls" is for illustrative purposes only; no such compound polyhedron exists in the 120-cell.}}]] We shall see in subsequent sections that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the realization of the abstract hemi-icosahedron, as an actual polyhedron that occurs in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''² = ''j''² = ''k''² = ''ijk'' = −1}} as the [[W:Tesseract|16-point (8-cell) tesseract]] 4-polytope. He did not realize then that he had discovered the 4-hypercube polytope and [[W:Tesseractic honeycomb|its Euclidean honeycomb]], the (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because [[w:Quaternions_and_spatial_rotation|quaternions make rotations]] and projections in 3D or 4D space as simple as matrix multiplications.{{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 quaternions are 4-hypercube building blocks, analogous to the 3-hypercube wooden blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the [[24-cell]], but we digress). Moxness used his software to render illustrations of polyhedra inside the 120-cell, some of which he published. Notice his "Hull # = 8 with 60 vertices", lower right in his illustration of the 120-cell sections starting with a cell. It is a real icosahedron that occurs in the 120-cell, and we shall see that the abstract hemi-icosahedron represents it. Moxness's 60-point Hull #8 is a concrete realization of the 6-point hemi-icosahedron in spherical 3-space <math>S^3</math>, embedded in Euclidean 4-space <math>\mathbb{R}^4</math>. Its 12 little pentagon faces are 120-cell faces. It also has 20 triangle faces like any icosahedron, separated from each other by rectangles, but beware: those triangles are not the 5-cell faces. They are smaller equilateral triangles, of edge length <math>1</math> in a {{radic|2}}-radius 120-cell, where the 5-cell face triangles have edge length {{radic|5}}.{{Efn|The 41.4° chord of edge length 1 in a {{radic|2}}-radius 120-cell occurs only in the 120-cell; it is not the edge of any smaller regular 4-polytope inscribed in the 120-cell. The equilateral triangle faces of Moxness's Hull #8 rhombicosidodecahedron are not the 5-cell faces of edge length <small><math>\sqrt{5} \approx 2.236</math> </small>(104.5°), not the 16-cell faces of edge length <small><math>2</math></small> (90°), not the 24-cell faces of edge length <small><math>\sqrt{2} \approx 1.414</math></small> (60°), and not the 600-cell faces of edge length <small><math>\sqrt{2}/\phi \approx 0.874</math></small> (36°).|name=Moxness 60-point triangle faces}} [[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges, in 200 irregular {12} dodecagon central planes. The 5-cell ''faces'' do not lie in central planes.]] Edges of the larger 5-cell face triangles of length {{radic|5}} can also be found in Hull #8, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that six 120-cell edges (little pentagon edges) lie on a great circle, alternating with six rectangle diagonals. Also lying on this irregular {12} great circle are six 5-cell edges, invisible chords joining every other 120-cell edge and running under the 120-cell edge between them. The six long chords and six short edges form two opposing irregular {6} great hexagons (truncated triangles) of alternating 5-cell edges and 120-cell edges, as illustrated. The irregular great {12} lies on a great circle of Moxness's Hull #8, and also on a great circle of the 120-cell, because Hull #8 is the ''central'' cell-first section of the 120-cell.{{Efn|The cell-first central section of the 600-cell (and of the 24-cell) is a cuboctahedron with 24-cell edges. The 120-cell is the regular compound of 5 600-cells (and of 25 24-cells), so Moxness's Hull #8, as the cell-first central section of the 120-cell, is the regular compound of 5 cuboctahedra. Their 24-cell edges, like the 5-cell edges, are invisible chords of Hull #8 that lie below its surface, on the same irregular {12} great circles. Each 24-cell edge chord spans one 120-cell edge chord (one little pentagon edge) and one rectangle face diagonal chord. Six 24-cell edge chords form a regular great {6} hexagon, inscribed in the irregular great {12} dodecagon.|name=compound of 5 cuboctahedra}} There are 10 great dodecagon central planes and 60 5-cell edges in Moxness's Hull #8, and 200 great dodecagon central planes and 1200 5-cell edges in the 120-cell. [[File:Central cell-first section of the 120-cell with 5-cell face triangle.png|thumb|Orthogonal projection of the cell-first central section of the 120-cell, Hull #8 rendered by Moxness, with one of 20 inscribed 5-cell faces (black chords) drawn under portions of three of its ten great circle {12} dodecagons (green).{{Efn|The point of view in this rendering is not quite right to best illustrate that a rhombicosidodecahedron triangle face lies over the center of a 5-cell face parallel to it, such that it would be perfectly inscribed in the center of the larger black triangle in an orthogonal view.}}]] But the 5-cell ''faces'' do not lie in those central planes. We can locate them in the 60-point polyhedron where they lie parallel to and under each small face triangle of edge length <math>1</math>. Truncating at a triangle face of Moxness's Hull #8 exposes a deeper 5-cell triangle face.{{Efn|Each face triangle of edge length <math>1</math> is surrounded by 3 rectangles, and beyond each rectangle by another face triangle. The distant vertices of those 3 surrounding triangles form a {{radic|5}} triangle, a 5-cell face.}} There are 20 such 5-cell faces inscribed in the Hull #8 polyhedron, all completely disjoint. We find 60 vertices, 60 edges and 20 faces of various 5-cells in each Hull #8 polyhedron, but no whole tetrahedral cells of the 5-cells.{{Efn|The fourth vertex of each 5-cell tetrahedron lies opposite the small face triangle of edge length <math>1</math> that lies over the 5-cell face. Since Moxness's Hull #8 polyhedron has opposing triangle faces (like any icosahedron), the fourth vertex of the 5-cell tetrahedron lies over the center of the opposing face, outside the Hull #8 polyhedron. This is a vertex of some other Hull #8 polyhedron in the 120-cell. Each tetrahedral cell of a 5-cell spans four Hull #8 polyhedra, with one face inscribed in each, and one vertex outside of each.}} [[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point Hull #8 is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation of the icosahedron with smaller red pentagons and narrower rhombs. Rhombicosidodecahedra are also made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the unique 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 60 of Moxness's Hull #8 rhombicosidodecahedron. Each occupies a central hyperplane, and so is analogous to an equator dividing the sphere in half.]] Moxness's Hull #8 is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point [[W:icosidodecahedron|icosidodecahedron]], which has the same relationship to Moxness's Hull #8 that the 6-point hemi-icosahedron does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The regular 5-point 5-cell can be another abstraction of Moxness's 60-point Hull #8, 12-vertices-into-1. None of these contractions of Moxness's Hull #8 is an instance of her operation actually described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length, but she did explicitly describe contractions of the semi-regular Archimedean rhomibicosidodecahedron.}} Moxness was not the first person to find rhombicosidodecahedra in the 120-cell. Alicia Boole Stott identified the 6th section of the 120-cell beginning with a cell as the semi-regular rhombicosidodecahedron that is her ''e₂ expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e₂ expansion|p=7}} But that 6th section rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8, it is the semi-regular Archimedean solid (Moxness's Hull #6), with a single edge length and square faces. Moxness's Hull #8, with its two distinct edge lengths and rectangular faces, is Coxeter's 8₃, the 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; "Alicia Boole Stott (1860-1940) ... also constructed the sections i₃ of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1₃-7₃; but 8₃ is missing. Incidentally, Diagram XIII (our 6₃) is a rhombicosidodecahedron, the Archimedean solid."}} Coxeter was the first to describe the central section 8₃, and he gave its coordinates, but he did not identify it as an irregular rhombicosidodecahedron. His table entry for its description is empty (characteristically, since it is not a regular or semi-regular polyhedron), so he gives us no indication that he actually visualized it. Although Moxness was not the first to compute the 60-point 8₃ section, he may have been the first person to ''see'' it. The 30-point icosidodecahedron is the quasi-regular product of 5-point pentagon and 6-point hexagon, recalling Coxeter's original discovery of the 11-cell in pentads and hexads, and also the two child's building blocks: one so useless the 5-point (pentad) 5-cell, and the other so useful the 8-point 16-cell with its four orthogonal 6-point (hexad) octahedron central sections, which can be compounded into everything larger. Some children building with the 30-point icosidodecahedron notice that it occurs as the central section 4₀ of the 120-point 600-cell. It is less often noticed that Moxness's Hull #8 rhombicosidodecahedron is the central section 8₃ of the 600-point 120-cell. It occupies a flat 3-dimensional hyperplane that bisects the 120-cell, and since there are 120 dodecahedral cells, there are 60 such central hyperplanes, each perpendicular to an axis that connects the centers of two antipodal cells. The 60 central hyperplanes, each containing an instance of Moxness's Hull #8, are rotated with respect to each other. They intersect, with 6 rhombicosidodecahedra sharing each vertex and 3 sharing each edge, but each little pentagon face (120-cell face) belongs to just one rhombicosidodecahedron. The 60 central sections lie in isoclinic hyperplanes, that is, the rhombicosidodecahedra are rotated symmetrically with respect to each other, by two equal angles.{{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.)"}} Each pair of rhombicosidodecahedra intersect in a central plane containing an irregular {12} dodecagon, unless they are completely orthogonal and intersect only at the center of the 4-polytope. Each of the 120 dodecahedral cells lies in the closed, curved 3-dimensional space of the 3-sphere as the 1st and smallest section beginning with a cell (section 1₃), the innermost of a series of concentric polyhedral hulls of increasing size, which nest like Russian dolls around it. Moxness's Hull #8 rhombicosidodecahedron is the 8th and largest concentric hull beginning with a cell (section 8₃), a central section of the 120-cell that bisects the 3-sphere the way an equator bisects an ordinary sphere.{{Efn|The 120-cell's curved 3-space surface is a honeycomb of 120 dodecahedron cells. In this 3-space a dodecahedron cell lies inside at the center of each section 8₃ rhombicosidodecahedron, face-bonded to 12 other dodecahedron cells which surround it, also inside the rhombicosidodecahedron. We find the opposite pentagon faces of those 12 surrounding cells on the surface of the section 8₃ rhombicosidodecahedron. These twelve dodecahedra surrounding one dodecahedron partially fill the volume of the rhombicosidodecahedron, leaving 30 concavities in its surface at the rectangle faces, and 12 deeper concavities between them at the triangle faces. 30 more dodecahedra fit into the rectangle concavities, lying half inside and half outside the rhombicosidodecahedron. The diagonal of each rectangle face is a long diameter of a dodecahedron cell. 12 more dodecahedra fit into the triangle face concavities, lying ....|name=dodecahedral cells in the section 8 rhombicosidodecahedron}} Such a central polyhedron is the dimensional analog of an equatorial great circle polygon. Its 60 vertices lie in the same 3-dimensional hyperplane, a flat 3-dimensional section sliced through the center of the 120-cell. There are 60 distinct stacks of 15 parallel section ''n''₃ hyperplanes in the 120-cell, one stack spindled on each axis that connects a dodecahedron cell-center to its antipodal dodecahedron cell-center. Each central section 8₃ has ''two'' disjoint sets of smaller sections nested within it, that lie in opposite directions from the 120-cell's center along its 4th dimension axis. The largest-radius central slice lies in the center of the stack, and the smaller non-central section hyperplanes occur in parallel pairs on either side of the central slice. The 120-cell therefore contains 120 instances of each kind of non-central section 1₃ through 7₃, and 60 instances of the central section 8₃.{{Efn|A central section is concave on its inside and also on its outside: it has two insides. It may be helpful to imagine the central 60-point section as two mirror-image 60-point polyhedra whose points are coincident, but which are convex in opposite directions: the inside of one is the outside of the other. Each has seven smaller polyhedra nested within itself, but their two volumes are disjoint.}} [[File:Tensegrity Icosahedron.png|thumb|[[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|Tensegrity icosahedron]] structure.{{Sfn|Burkhardt|1994}} First built by [[W:Kenneth Snelson|Kenneth Snelson]] in 1949. Geometrically a [[w:Jessen's_icosahedron|Jessen's icosahedron]] with 6 reflex ''long'' edge struts, and 24 ''short'' edge tension cables around 8 equilateral triangle faces. 3 pairs of parallel struts lie in 3 orthogonal central planes.]] We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that the 60 Moxness's Hull #8 rhombicosidodecahedra are real polyhedra which the abstract hemi-icosahedra represent in some manner, but we have not yet identified 11 real face-bonded cells, at 11 distinct locations in the 120-cell, as an 11-cell. The abstract hemi-icosahedron's 10 faces correspond to actual 5-cell faces inscribed in real rhombicosidodecahedra, and its 15 edges correspond to 5-cell edges (of length {{radic|5}} in a {{radic|2}}-radius 120-cell) that occur as chords lurking under the surface of the rhombicosidodecahedra. [[File:Buckminster-Fuller-holding-a-geodesic-tensegrity-sphere.png|thumb|200px|Buckminster Fuller holding a 3-dimensional geodesic tensegrity 2-sphere, an infinitesimally mobile rigid polytope consisting of tension cable edges and disjoint compression strut chords.<ref>{{Cite journal|last=Álvarez Elipe|first=Dolores|title=Ensegrities and Tensioned Structures|journal=Journal of Architectural Environment & Structural Engineering Research|date=July 2020|volume=3|issue=3|url=https://www.researchgate.net/publication/343652287_Ensegrities_and_Tensioned_Structures}}</ref>]] A rhombicosidodecahedron is constructed from a regular icosahedron by truncating its vertices, making them into pentagon faces. The regular icosahedron frames all the regular and semi-regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter & Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Sfn|Coxeter, du Val, Flather & Petrie|1938|p=4|ps=; "Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "[[W:Golden section|golden section]]". Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''."}} Before we can move on to locating the 11 discrete hemi-icosahedral cells of the 11-cell in the 120-cell, it is important that we take notice of one more icosahedral symmetry of the hidden {{radic|5}} chords lurking below the surface of Moxness's Hull #8 rhombicosidodecahedron. The 12 little pentagon faces (120-cell faces) are connected to each other in parallel pairs, by 10 sets of six disjoint {{radic|5}} chords (5-cell edges). Each six-chord set is the six reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], which is to say that the six disjoint chords are the parallel-orthogonal strut chords of a [[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|tensegrity icosahedron]]. The six chords of each set are disjoint (they don't touch or form 5-cell faces), and they are symmetrically arranged as 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal {12} central planes.{{Efn|The Jessen's icosahedron has 8 equilateral triangle faces, which are not rhombicosidodecahedron triangle faces or 5-cell triangle faces, they are 24-cell triangle faces. Each 120-cell pentagon face lies at one end of 20 5-cell edges, from 20 distinct Jessen's icosahedra and five disjoint 5-cells: four at each pentagon vertex from each 5-cell.}} Five disjoint instances of the Jessen's icosahedron may be inscribed in each Moxness's Hull #8 rhombicosidodecahedron, their struts propping the rhombicosidodecahedron and the 120-cell itself open like a tensegrity structure.{{Efn|Moxness's Hull #8 rhombicosidodecahedron is a compound of five disjoint Jessen's icosahedra, because the 60 {{radic|5}} chords meet two-at-a-vertex and form 10 distinct Jessen's icosahedra: five disjoint Jessen's, in two different ways. The dimensionally analogous construction is the [[120-cell#Compound of five 600-cells|120-cell as a compound of five disjoint 600-cells]], in two different ways.}} But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, and work our way patiently outwards, before in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we approach that rare child Bucky Fuller's orthogonal 12-point tensegrity icosahedron, an [[WikiJournal Preprints/Kinematics of the cuboctahedron|in-folded cuboctahedron]], the unique pyritohedral fish swimming deep in the 3-sphere ocean. == Eleven == Each pair of rhombicosidodecahedra that are not completely orthogonal intersect in a central plane containing an irregular {12} dodecagon. Ten irregular great dodecagons occur in each 60-point (central section 8₃) rhombicosidodecahedron, with 2 dodecagons crossing orthogonally at each vertex. Each rhombicosidodecahedron shares a {12} central plane with ten other rhombicosidodecahedra. ''Groups of 11 rhombicosidodecahedra share central planes pairwise.'' Here, at last, we find eleven of something, a group which must comprise an 11-cell. There are eleven {12} central planes in the group, with one of the eleven absent from each rhombicosidodecahedron. {|class="wikitable floatright" width=450 !colspan=2|Perspective views{{Efn|1=These images are ''non-orthogonal'' orthographic projections of the chords described in the caption. Those chords do not lie in a plane parallel to the projection plane, so they appear foreshortened.{{Efn|name=orthogonal triacontagram projections}} Consecutive chords of the helical Petrie polygon slant toward and away from the viewer. Any three consecutive chords, but no four, are edges of the same cell, in the 4-polytope whose edges are the chord.{{Efn|name=Petrie polygon of a honeycomb}}}} of a compound of six disjoint 5-cells in dual position |- ![[W:Triacontagon#Triacontagram|{30/12}{{=}}6{5/2} compound]] ![[W:Triacontagon#Triacontagram|{30/8}{{=}}2{15/4} compound]]{{Efn|name=orthogonal triacontagram projections|1=The {30/''n''} triacontagrams can each be seen as an ''orthogonal projection'' of the 120-cell showing all instances of the {30/''n''} chord. Each chord lies orthogonal to the line of sight, in a plane parallel to the projection plane. The diameter of the image is the diameter of the 120-cell. For example, the {30/8}=2{15/4} triacontagram is an orthogonal projection showing the 120-cell's 1200 {30/8} chords, the edges of 120 5-cells. Each edge of the triacontagram covers 40 5-cell edges, and each vertex covers 20 120-cell vertices. This projection can also be viewed as a compound of six 5-cells and their 30 unique vertices. But viewed that way, only 30 of the 60 5-cell edges are visible. Two edges meet at each vertex, but the other two are invisible. They are visible in the orthogonal view, the {30/4}=2{15} projection.}} |- valign=top |[[File:Regular_star_figure_6(5,2).svg|240px]]<BR>The 6{5/2} compound of six 5-cells. The six disjoint pentagrams in this view are six disjoint 5-cells.{{Efn|name=5-cell edges do not intersect is S³}} The 120-cell, with 120 disjoint 5-cells, is a compound of 20 of these compounds. All edges are 5-cell edges, but only five of each 5-cell's ten edges are shown. The other five edges, connecting the points of the six 5-cell pentagrams, are shown in the 6{5} projection below, the orthogonal view:<BR>[[File:Regular_star_figure_6(5,1).svg|240px]]These two views look straight down the orthogonal axes of a [[w:Duocylinder|duocylinder]], from inside the curved 3-dimensional space of the 120-cell's surface. They are like looking down a column of 5-cells stacked on top of one another in curved 3-space, but the column is actually circular: it is bent into a torus in the fourth dimension. |[[File:Regular_star_figure_2(15,4).svg|240px]]<BR>The 2{15/4} rotation circuits of the 5-cell isoclinic rotation. In this view, all edges are 75.5° chords of length {{radic|3}}, the 180° complement chord of the 5-cell edges of length {{radic|5}}.{{Efn|These are not 15-gons of 5-cell edges. There are no skew {15} polygons of 5-cell edges in the 120-cell. The 120 5-cells are completely disjoint, so the largest circuit along 5-cell edges is a skew {5}. Each vertex in the 120-cell is {{radic|5}} away from four and only four other vertices. No {{radic|5}} chords connect disjoint 5-cells; they are connected by several other chords. The skew {15} polygons are the discrete continuous spiral paths of moving vertices during an isoclinic rotation, and their edges are {{radic|3}} chords connecting 5-cells, not 5-cell edges.}} Each skew {15} polygon is the spiral chord-path of half the 30 vertices during the isoclinic rotation. The twined vertex orbits lie skew in 4-space; they form a circular double helix of two 15-gon spiral isoclines, winding through all four dimensions. These two completely orthogonal views look straight down an axis of a double helix cylinder, from inside the curved 3-dimensional space of the 120-cell's surface. Since the duocylinder is bent into a [[w:Clifford_torus|Clifford torus]] in the fourth dimension, the sightline axis in curved 3-space is a geodesic great circle in 4-space.<BR>[[File:Regular_star_figure_2(15,2).svg|240px]] |- ![[W:Triacontagon#Triacontagram|{30/6}{{=}}6{5} compound]] ![[W:Triacontagon#Triacontagram|{30/4}{{=}}2{15/2} compound]] |- |colspan=2|Images by Tom Ruen in [[W:Triacontagon#Triacontagram|Triacontagram compounds and stars]].{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}} |} Each shared {12} central plane contains six disjoint 5-cell edges, from six completely disjoint 5-cells. Each rhombicosidodecahedron contains 60 5-cell edges, which form 20 disjoint 5-cell faces within the rhombicosidodecahedron, under and parallel to its own 20 smaller triangle faces. Four 5-cell edges meet at each vertex at the 5-cell's tetrahedral vertex figure. Two 5-cell edges of a face within the rhombicosidodecahedron meet two edges belonging to other faces of the 5-cell: edges and faces outside the rhombicosidodecahedron, in some neighboring rhombicosidodecahedron.{{Efn|name=orthogonal triacontagram projections}} Each 5-cell face is shared by two tetrahedral cells of one 5-cell. It has its three 104.5° {{radic|5}} edges in three distinct {12} central planes, and is parallel to a fourth {12} central plane. In each rhombicosidodecahedron there are ten sets of five parallel planes: a {12} central plane, a pair of 5-cell faces on either side of it (from disjoint 5-cells), and a pair of rhombicosidodecahedron triangle faces. Each rhombicosidodecahedron is sliced into five parallel planes, ten distinct ways. There is no face sharing between 5-cells: the 120 5-cells in the 120-cell are completely disjoint. 5-cells never share any elements, but they are related to each other positionally, in groups of six, in the '''characteristic rotation of the regular 5-cell'''. That rigid isoclinic rotation takes the six 5-cells within each group to each other's positions, and back to their original positions, in a circuit of 15 rotational displacements.{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439|ps=; in symmetry group 𝛢₄ the operation [15]𝑹_q3,q3 is the 15 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 5-cell; in symmetry group 𝛨₄ the operation [1200]𝑹_q3,q13 is the 1200 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 120-cell.}} Each displacement takes every 104.5° 5-cell edge of length {{radic|5}} to an edge 75.5° and {{radic|3}} away in another 5-cell in the group of six 5-cells. The 30 vertices of the six 5-cells rotate along 15-chord helical-circular isocline paths from 5-cell to 5-cell, before closing their circuits and returning the moving 5-cells to their original locations and orientations.{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. The orthogonal distance equals half the total Pythagorean distance. For example, when the {{radic|2}}-radius 5-cell rotates isoclinically 104.5° in the invariant central planes of its 104.5° edges of length {{radic|5}}, each vertex is displaced to another vertex 75.5° and {{radic|3}} away, moving {{radic|3/4}} in four orthogonal directions at once.|name=isoclinic 4-dimensional diagonal}} The six rotationally related 5-cells form a stellated compound, a non-convex 4-polytope with 30 star points.{{Efn|name=compound of six 5-cells}} The star compound, and the rotation of the 5-cell within it, are here illustrated by orthogonal projections from four different perspective viewpoints. To help us visualize the 4-polytopes within the 120-cell, we can examine 2-dimensional orthographic projections from various points of view. Such images filtered to include only chords of a single length are especially revealing, because they pick out the edges of a particular 4-polytope, or the isocline chords of its rotational orbits, the chords which link 4-polytopes together. No view of a single chord from a single point of view is sufficient by itself, but if we visualize various chords from various perspectives, we may imagine the 4-dimensional rotational geometry of interrelated objects within the 120-cell. The star compound as a whole has ten {12} central planes, like a rhombicosidodecahedron. Each {12} central plane contains one edge from each of the six 5-cells. Each {12} central plane is shared by two rhombicosidodecahedra in the group of eleven, and by six 5-cells in the group of six. == The eleventh chord == [[File:Major chord 11 of 135.5° in the 120-cell.png|thumb|The 120-cell contains 200 irregular {12} central planes containing 1200 135.5° {30/11} chords, six in each plane (shown in blue). They lie parallel to six 104.5° {30/8} chords (the 5-cell edges, shown in red), to which they are joined by 15.5° {30/1} 120-cell edges, and by 120° {30/10} great triangle edges (only one of the four great triangles is shown, in green).]] In addition to six 104.5° {30/8} 5-cell edge chords of length {{radic|5}}, the {12} central plane contains six 135.5° {30/11} chords of length <math>\phi^2</math>, parallel to the {{radic|5}} chords. The {30/11} chord spans an arc of five shorter chords: * 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} = 135.5° {30/11) * 15.5° {30/1} + 104.5° (30/8) + 15.5° {30/1} = 135.5° {30/11) * 15.5° {30/1} + 120° (30/10) = 135.5° {30/11) and its chord length is the linear sum of five shorter chords: * 1/𝜙^2 {30/1} + 1/𝜙^2 {30/1} + 1/𝜙 {30/2} + 1/𝜙 {30/2} + 1/𝜙 {30/2} = 𝜙^2 {30/11) Two distinct chords are always related to each other in two different ways: by their degrees-of-arc-difference, and by their linear difference chord. The 135.5° {30/11) chord is ''two'' 15.5° (30/1) 120-cell edge-arcs longer than the 104.5° (30/8) 5-cell edge chord. But the <math>\phi^2</math> {30/11} chord ''length'' is just ''one'' {30/1} 120-cell edge chord length longer than the {{radic|5}} {30/8} 5-cell edge chord.{{Efn|In a <small><math>\sqrt{2}</math></small>-radius 120-cell, the 15.5° {30/1} 120-cell edge chord has length <small><math>\phi^{-2}</math></small>. The 25.2° {30/2} pentagon face diagonal chord of length <small><math>\phi^{-1}</math></small> is <small><math>\phi</math></small> times the {30/1} edge length. The 41.1° 5-cell isocline chord of length <small><math>\sqrt{1}</math></small> is <small><math>\phi^2</math></small> times the {30/1} edge length. The 69.8° chord of length <small><math>\phi</math></small> is <small><math>\phi^3</math></small> times the {30/1} edge length. The 135.5° {30/11} 11-cell edge chord of length <small><math>\phi^2</math></small> is <small><math>\phi^4</math></small> times the {30/1} edge length.}} The {30/11} chord can be bisected into two shorter 120-cell chords in three different ways: * 15.5° {30/1} 120-cell edge + 104.5° {30/8} 5-cell edge = {30/11} chord * 25.2° {30/2} 120-cell pentagon face diagonal + 90° {30/15} 16-cell edge = {30/11} chord * 41.4° {30/1}+{30/2} chord + 69.8° {30/2}+{30/1}+{30/2} chord = {30/11} chord [[File:Regular_star_polygon_30-11.svg|thumb|The [[W:Triacontagon#Triacontagram|{30/11} regular triacontagram]] of the 11-cell rotation.{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}} In this 2-dimensional projection of a 30-edge 4-dimensional helix ring, the 30 chords pictured lie in 30 distinct central planes, and no two planes are orthogonal.]] The last of those bisections trisects the {30/11} chord into three distinct shorter chords: * 15.5° {30/1} + 25.2° {30/2} + 44.5° {30/4} chord = 135.5° {30/11} chord The {30/11} chords do not form triangle faces within the rhombicosidodecahedron the way the {30/8} chords do, but they do meet at a tetrahedral vertex figure. Groups of 11 rhombicosidodecahedra (an 11-cell) share central planes pairwise, including all the chords in the {12} central plane. When 11 things, all pairwise-adjacent to each other, are arranged in any circuit of 30 positions, there exists another pairwise circuit of 30 positions through every eleventh position, whether the things are 11 vertices, 11 rhombicosidodecahedra, or 11 [[w:Aardvark|aardvarks]] (although it might be unwieldy in practice to so arrange 11 live aardvarks, e.g. by tying them together pairwise with cords in both circuits). This intrinsic property of the [[w:Rational_number|rational number]] 30/11 is responsible for the existence of the {30/11} regular triacontagram (see illustration). The 11 rhombicosidodecahedra of the 11-cell are linked by a regular {30/11} triacontagram of 30 chords which runs through them. Each successive chord of the 30 in the triacontagram is shared by a distinct pair of rhombicosidodecahedra in the 11-cell group. An isoclinic rotation characteristic of the 11-cell takes the rhombicosidodecahedra in each 11-cell to each other's positions, pair by pair, in a circuit of 30 rotational displacements. It takes every {12} central plane to a Clifford parallel {12} central plane that is 44.5° away in two completely orthogonal angles. One 135.5° {30/11} chord separates each of the 12 vertex pairs. In this '''characteristic rotation of the 11-cell''' in its edge planes, the invariant planes are {12} central planes, the edges of the 11-cell are {30/11} chords, and the isocline chords of the vertex orbits are also {30/11} 11-cell edges, because the triacontagram is regular.{{Efn|In the 120-cell there are three ''regular isoclinic rotations'' in which the rotation edge and the isocline chord are the same chord. These rotations are each described by a [[W:Triacontagon#Triacontagram|regular triacontagram]]: the {30/7} rotation characteristic of the 16-cell in great square invariant planes, the {30/11} rotation characteristic of the 11-cell, and the {30/13} rotation.}} The 44.5° {30/4} chord of length <small><math>\sqrt{3}/\phi</math></small>, the 180° complement of the {30/11} chord, is the orthogonal distance between nearest parallel {30/11} chords.{{Efn|In its characteristic isoclinic rotation, a 4-polytope rotates an equal arc distance in each invariant {12} edge plane in each rotational displacement. In the 11-cell, every invariant plane rotates 44.5° (like a wheel), and tilts sideways 44.5° (like a coin flipping) in the completely orthogonal invariant plane, to occupy another invariant plane in the group of eleven. Each pair of original and destination {12} central planes are Clifford parallel and intersect only at one point (the center of the 4-polytope), but six other {12} central planes intersect them both. Two parallel {30/11} chords in each of the six spanning {12} central planes separate two vertex pairs in the original and destination planes, and these are the isocline chords over which the two vertices move in the rotation. None of the six spanning {12} central planes are contained in either the original or destination rhombicosidodecahedron. A total of ten {12} central planes span each original and destination rhombicosidodecahedron; they comprise a third rhombicosidodecahedron which does not belong to the group of eleven. The edges of an 11-cell and the isocline chords of an 11-cell are disjoint sets of {30/11} chords.}} The 60 vertices of each rhombicosidodecahedron rotate in parallel, on non-intersecting 30-chord spiral orbital paths, from rhombicosidodecahedron to rhombicosidodecahedron, before closing their circuits and returning the moving rhombicosidodecahedron to its original location and orientation. In this isoclinic rotation of a rigid 120-cell, the 60 rhombicosidodecahedra do this concurrently. Each of the 600 vertices moves on a 4-dimensionally-curved helical isocline, over a skew regular polygram of 30 {30/11} chords, in which a {30/11} chord connects every eleventh vertex of a {30} triacontagram. In the course of a complete revolution (the 30 rotational displacements of this isoclinic rotation), an 11-cell visits the positions of three 11-cells (including itself) 10 times each (in 10 different orientations), and returns to its original position and orientation.{{Sfn|Coxeter|1984|loc=§9. Eleven disjoint decagons}} At each step it occupies the same distinct group of 11 rhombicosidodecahedra sharing planes pairwise, and its 11 vertex positions are those of a distinct 11-cell in the group of eleven 11-cells. A group of 4-polytopes related by an isoclinic rotation is contained in a larger compound 4-polytope which subsumes them. This group of eleven 11-cells related by an isoclinic rotation is not a compound of eleven disjoint 11-cells (since they share vertices), but it is a compound of eleven non-disjoint 11-cells, in the same sense that a 24-cell is a compound of three non-disjoint 8-cell tesseracts. Consider the incidence of these 30-chord {30/11} triacontagram rotation paths, and their intersections. Each rhombicosidodecahedron has 60 vertices and 60 {30/11} chords, which rotate concurrently on Clifford parallel triacontagrams. The 120-cell has only 600 vertices and 1200 {30/11} chords, so at most 20 triacontagrams can be disjoint; some must intersect. But the 11 vertices of an individual 11-cell must be linked by disjoint 30-position {30/11} triacontagram helices, such that their rotation paths never intersect.{{Efn|The isoclines on which a 4-polytope's vertices rotate in parallel never intersect. Isoclinic rotation is a concurrent motion of Clifford parallel (disjoint) elements over Clifford parallel (non-intersecting) circles.}} Each 11-cell has two disjoint triacontagram helicies, its left and right isoclinic rotations, in each of its four discrete fibrations. The 120-cell has 60 distinct {30/11} triacontagram helices, which are 11 disjoint {30/11} triacontagram helices in 11 distinct ways. {{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} == Compounds in the 120-cell == [[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 for 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 other 5 regular convex 4-polytopes]]. {{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}} === How many building blocks, how many ways === The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy. The 24-point also compounds by <math>5^2</math> non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. [[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|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}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]] The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times <math>5^2</math> (75) disjoint instances of itself into the 600-point, which then magically contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point, and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the original 8-point. They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[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''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}} 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]], the final regular [[W:Stellation|stellation]] of the 120-cell, 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:Stellation core|stellation core]] deep inside. The compound of 120 regular 5-cells can be seen to be equivalent to the compound of 5 disjoint 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. === Building the building blocks themselves === We have built every regular 4-polytope except the 5-cell out of 16-cells, but we haven't made the 16-cell (or the 5-cell) out of anything. So far, we have just accepted them both a priori, like [[W:Euclid's postulates|Euclid's postulates]], and proceeded to build with them. But it turns out that while they are the two atomic regular 4-polytopes, they are not indivisible, and can be built up as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct [[w:Symmetry_group|symmetry group]]. Every regular convex ''n''-polytope can be subdivided into instances of its characteristic [[W:Orthoscheme|Schläfli orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-[[w:Orthoplex|orthoplex]]!) is an ''irregular'' ''n''-[[w:Simplex_(geometry)|simplex]] with faces that are various right triangles instead of congruent equilateral triangles. A characteristic ''n''-orthoscheme possesses the complete symmetry of its ''n''-polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements. It is the gene for the polytope, which can be replicated to construct the polytope.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope because 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 partitioned radially by its planes of symmetry (Coxeter's "reflecting circles") into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme and its chiral mirror image can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its <math>A_4</math> planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into <small><math>120\times 120 = 14400</math></small> of these 4-orthoschemes, so that is the symmetry order of the 120-cell. The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its <math>B_4</math> planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order <small><math>75\times 384 / 2 = 14400</math></small>. The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its <math>F_4</math> planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order <small><math>25\times 1152 / 2 = 14400</math></small>. The regular 120-point (600-cell) is subdivided into 14400 instances of its [[600-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its <math>H_4</math> planes of symmetry at once intersecting at its center, so its symmetry is of order 14400. The regular 600-point (120-cell) is the convex hull of the regular compound of 5 disjoint 600-cells (which have 5-fold reflective symmetry), so its symmetry is of order <small><math>5 \times 14400 / 5 = 14400</math></small>. === Building with sticks === [[File:15 major chords.png|thumb|300px|The 15 major chords {30/1} ... {30/15} join vertex pairs which are 1 to 15 edges apart on a skew {30} [[w:Petrie_polygon|Petrie polygon]] of the 120-cell.{{Efn|Drawing the fan of major chords with #1 and #11 at a different origin than all the others was an artistic choice, since all the chords are incident at every vertex. We could just as well have fanned all the chords from the same origin vertex, but this arrangement notices the important parallel relationship between #8 and #11, and calls attention to the 11-cell's maverick edge chord.|name=fan of 15 major chords}} The 15 minor chords (not shown) fall between two major chords, and their length is the sum of two other major chords; e.g. the 41.4° minor chord of length {30/1}+{30/2} falls between the 36° {30/3} and 44.5° {30/4} chords.]] We have seen how all the regular convex 4-polytopes except the 5-cell, including the largest one on the cover of the box, can be built from a box containing 675 16-cell building blocks, provided we can arrange the blocks on top of one another in 4-space, as interpenetrating objects. An alternate box, containing 120 regular 5-cell building blocks, builds the great grand stellated 120-cell (the picture on ''its'' cover), by the same method. In these boxes, the atomic building part is one of the two smallest regular 4-polytopes (5-cell or 16-cell), each generated by its characteristic isoclinic rotation as an expression of its symmetry group (<math>A_4</math> or <math>B_4</math>). All the regular convex 4-polytopes, including the largest one on the cover of the box, can also be built from a box containing a certain number of building sticks and rubber joints, provided we can connect the sticks together in 4-space with the rubber joints. In this box, the atomic building parts are 1-dimensional edges and chords of just 15 distinct arc-lengths. The regular 4-polytopes do not contain a vast variety of stick lengths, but only 30 of them: only 15 unique pairs of 180° complementary chords. The 15 ''major chords'' {30/1} ... {30/15} suffice to construct all the regular 4-polytopes. The 15 ''minor chords'' occur only in the 120-cell, not in any smaller regular 4-polytope; they emerge as a consequence of building the largest 4-polytope on the cover of the box from major chords. In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the unit-radius sphere of a distinct length <math>l</math>, and a distinct rational number <math>h</math>, a unique flavor. If the polytope is regular, it is a noteworthy distinctive flavor. The chord's length <math>l</math> is a square root, related to the rational number <math>h = k/d</math> and to the polygon <small><math>\{k/d\}</math></small> it represents, by a formula discovered by Steinbach.{{Sfn|Steinbach|1997|loc=''Golden Fields''; §1. The Diagonal Product Formula|pp=22-24|ps=; The product of two diagonals is a sum of a sequence of diagonals (in the fan, every other one) centered on the longer of the two, for all regular polygons. We may express products and quotients of diagonals <math>d_k</math> of an <math>n</math>-gon (with edge <math>d_0=1</math>) as linear combinations of diagonals.}} The chord length <math>l</math> is related to the number of sides of the regular polygon <small><math>\{k\}</math></small>, and to the winding number or density of the polygram (its denominator <math>d</math>).{{Sfn|Kappraff & Adamson|2004}} The largest <math>k</math> of any major chord in the 120-cell is 30, and the polygrams <small><math>\{30/d\}</math></small> represent all the skew Petrie polygons and characteristic isoclinic rotations of the regular 4-polytopes. == Concentric 120-cells == The 8-point 16-cell, not the 5-point 5-cell, is the smallest regular 4-polytope which compounds to every larger regular 4-polytope. The 5-point 5-cell is also an atomic building block, but one that compounds to nothing else regular except the leviathan 120-cell polytope: the picture on the cover of the box, that is built from everything in the box. In the [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|sequence of 4-polytope compounds]], we actually start with the 16-cell at the small end, and the 5-cell emerges only at the large end. To build with the 16-cell blocks, we simply put them on top of each other as interpenetrating compounds. We can build every other regular 4-polytope from them by that method, except the individual regular 5-cell. We can also try to build with the 5-cell that way, as when we tried to build a 4-polytope of 11 hemi-icosahedral cells from 11 5-cells, but that was rather hard going. We somehow found 5-cell edges and faces lurking inside hemi-icosahedral rhombicosidodecahedra, and 11 rhombicosidodecahedra sharing central planes pairwise, and even the edges and characteristic rotation of the 11-cell, but we didn't quite get all the way to a discrete 11-cell 4-polytope made from 11 5-cells. That's because ''compounding'' isn't the easiest method for building with the 5-cell. The 5-cell is the last building block hierarchically, not the first, and the most natural way to build with it is in reverse, by ''subdividing'' it, to find all the parts inscribed inside it. When we've taken the 5-cell apart, all the ways we possibly can, into certain ''irregular'' 4-polytopes found within it, we will have a new set of irregular 4-polytope building blocks, which compound to the 5-cells and everything else, including the 11-cells. Subdividing a polytope is done by a geometric operation called ''[[w:Truncation_(geometry)|truncation]]''. There are myriad ways to truncate a 5-cell, each corresponding to a distinct ''depth'' of truncation at a particular point on an edge, or a line on a face, or a face on a cell, where a piece of the 5-cell is cut off. The simplest truncations, such as [[w:Rectification_(geometry)|cutting off each vertex at the midedge of each incident edge]], have been very well-studied; but how should we proceed? Let us see what happens when we [[w:Truncated_5-cell|truncate the 5-cells]] found in the 120-cell, by the simplest kinds of truncation. These three semi-regular 10-cells are closely related truncations of the regular 5-cell: * The 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of six 5-cells. * The 20-point 10-cell [[w:Truncated_5-cell|truncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of four 5-cells. * The 10-point 10-cell [[w:Rectified_5-cell|rectified 5-cell]] is the convex hull, and the convex common core, of a stellated compound of two 5-cells. In the following sections, we explore the effect of performing these truncations on the 120-cell's 120 5-cells. We begin by identifying some promising truncation points on the 120-cell's 5-cell edge chords at which to cut. If we cut off the 120-cell's 600 vertices at some point on its 1200 5-cell edges, we create new vertices on the edges of the 120 5-cells, which lie on a smaller 3-sphere than the 120-cell. How many vertices does the smaller 4-polytope thus created have? That is, how many distinct 5-cell edge truncation points occur in the 120-cell? As many as 1200, the number of 5-cell edges, or perhaps 2400, if each edge is truncated at both ends. But also perhaps fewer; for example, if the 120-cell contains pairs of 5-cells with intersecting edges, and the edges intersect at the point on each edge where we make our cut. [[File:Great_(12)_chords_of_radius_√2.png|thumb|400px|Chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {{radic|2}} chords form two regular {6} hexagons (black).{{Efn|name=compound of 5 cuboctahedra}} The 120-cell edges form two irregular {6} hexagons (red truncated triangles) with the {{radic|5}} chords. The {6} intersection points (black) of the {{radic|5}} chords form a smaller red regular hexagon of radius {{radic|1}} (inscribed in the red circle).]]In the irregular {12} central plane chord diagram, we see six truncation points on the six 104.5° 5-cell edges of length {{Radic|5}}, where two co-planar 5-cell edges intersect, directly under the midpoint of a 44.5° chord (and under the intersection point of two 60° chords). The six truncation points lie on a red circle that is a circumference of the smaller 4-polytope created by this truncation. They form a red regular hexagon inscribed in the red circle. The edge length of this regular hexagon is {{radic|1}}. The two intersection points on the {{Radic|5}} chord divide it into its golden sections. The center section of the chord is <small><math>1</math></small>. The center section plus either of the smaller sections is <small><math>\phi = \tfrac{\sqrt{5} + 1}{2} \approx 1.618</math></small>, the larger golden section. Each of the two smaller sections is <small><math>\Phi = \phi - 1 = \tfrac{1}{\phi} \approx 0.618</math></small>, the smaller golden section.{{Efn|The bitruncated {30/8} chord of the 120-cell of radius <small><math>\sqrt{2}</math></small> provides a "proof by geometric picture" of the golden ratio formulas. Picture a 120-cell of radius <small><math>2\sqrt{2}</math></small> in which the {30/8} chord is <small><math>2\sqrt{5}</math></small> and the center section of the chord is <small><math>2</math></small>. Divide the lengths of its golden sections by <small><math>2</math></small> to get their radius <small><math>\sqrt{2}</math></small> lengths. The left section of the chord is: :<small><math>\tfrac{\sqrt{5} - 1}{2} \approx 0.618</math></small> The center section plus the right section is: :<small><math>\tfrac{1 + \sqrt{5}}{2} \approx 1.618</math></small> }} The smaller golden sections <small><math>\Phi \approx 0.618</math></small> of the 5-cell edge are the same length as the 120-cell's 25.2° pentagon face diagonal chords. No 25.2° chords appear in the {12} central plane diagram, because they do not lie in {12} central planes. Each 104.5° 5-cell edge chord of length {{Radic|5}} has ''two'' points of intersection with other 5-cell edges, exactly 60° apart, the ''arc'' of a 24-cell edge chord, but with ''length'' {{radic|1}}. The center segment of the 5-cell edge, between the two intersection points, is a 24-cell edge in the smaller 4-polytope, and the red hexagon is a [[24-cell#Great hexagons|24-cell's great hexagon]] in the smaller 4-polytope. Nine other of its great hexagons, in other planes, each intersect with an antipodal pair of these {6} vertices. The dihedral angles between hexagon planes in a 24-cell are 60°, and four great hexagons intersect at each vertex. The 1200 5-cell edges, with two intersection points each, are reduced to 600 distinct vertices, so the smaller 4-polytope is a smaller 120-cell. The larger 120-cell, of radius {{radic|2}}, is concentric to a smaller instance of itself, of radius {{radic|1}}. Each 120-cell contains 225 distinct (25 disjoint) inscribed 24-cells. The smaller 24-cells are the [[w:Inscribed_sphere|insphere]] duals of the larger 24-cells. The vertices of the smaller 120-cell are located at the octahedral cell centers of the 24-cells in the larger 120-cell. Four 5-cell edges meet in 600 tetrahedral vertex figures. Four orthogonally intersecting 5-cell edges of the larger 120-cell meet in cubic vertex figures of 24-cells in the smaller 120-cell. Two disjoint 5-cell tetrahedral vertex figures are inscribed in alternate positions in each 24-cell cubic vertex figure. The 24-cell edges of the smaller 120-cell are the 5-cell edges of the larger 120-cell, truncated at both ends. The distance between the two points of intersection on a {{radic|5}} chord is {{radic|1}}, the same length as the 41.4° chord. But the actual 41.4° chords of the 120-cell do not appear in this diagram at all, because they do not lie in the 200 irregular {12} dodecagon central planes. === Bitruncating the 5-cells === The smaller concentric 120-cell can be built from 5-cell building blocks, by applying a specific kind of truncation operation to the blocks of the larger 120-cell called [[w:Bitruncation|''bitruncation'']]. This reveals a smaller irregular 4-polytope inside each 5-cell called the [[w:Bitruncated_5-cell|bitruncated 5-cell]]. The smaller unit-radius 120-cell is the convex hull of a compound of 20 disjoint (and 60 distinct) bitruncated 5-cells, bitruncated from the 120 disjoint 5-cells of the larger {{Radic|2}}-radius 120-cell. Bitruncation of the 120 disjoint 5-cells is the same truncation of the 120-cell described in the previous section, at the two golden section truncation points on each 104.5° 5-cell edge where two co-planar 5-cell edges intersect. [[File:Truncatedtetrahedron.gif|thumb|A 12-point [[w:Truncated_tetrahedron|truncated tetrahedron]] cell of the 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]].{{Sfn|Cyp: Truncated tetrahedron|2005}} Its edges are 41.4° chords of length 1 in a {{radic|2}}-radius 120-cell (or length {{radic|1/2}} in a unit-radius 120-cell). The 120-cell contains 20 disjoint (60 distinct) bitruncated 5-cells, containing 600 distinct truncated tetrahedra.]] The bitruncated 5-cell is a 30-vertex convex 4-polytope with 10 [[W:Truncated tetrahedron|truncated tetrahedron]] cells that have faces of two kinds: 4 triangle faces opposite 4 hexagon faces. The bitruncated 5-cell has 60 edges of the same length, 20 triangle faces, and 20 hexagon faces. Its 20 hexagon face planes are not [[24-cell#Great hexagons|24-cell central plane hexagons]]; they intersect each other at their edges, not at their long diameters. Its edges are not 60° 24-cell edge chords (the {{radic|2}} or 1 radius chords), but shorter 41.4° chords (of length 1 or {{radic|1/2}}), which do not appear at all in the diagram above, because they do not lie in the {12} central planes. The long diameter of the hexagon faces is not a 180° 120-cell long diameter chord (of length 2{{radic|2}} or 2) but a 90° 16-cell edge chord (of length 2 or {{radic|2}}). Consequently, three 16-cell tetrahedron cells (from three disjoint 16-cells) are inscribed in each truncated tetrahedron, at the three vertices of each face triangle. The truncated tetrahedron cell is a truncation of a tetrahedron of the same size as the tetrahedral cells of the 120-cell's 5-cells. The four smaller tetrahedra truncated from the corners of the larger tetrahedron have edges which are 25.2° chords (of length 1/𝜙 or {{radic|0.19}}). The truncated tetrahedron edges (of length 1 or {{radic|1/2}}) are equal in length to the 41.4° center sections of the 104.5° 5-cell edge chords (of length {{radic|5}} or {{radic|5/2}}). The shorter diagonal of the hexagon faces is the 75.5° chord (of length {{radic|3}} or {{radic|1.5}}), which is the 180° complement of the 104.5° 5-cell edge chord. The dimensions of the truncated tetrahedron cell suggest that it was cut directly from a 5-cell tetrahedron cell, simply by cutting off the tetrahedron corners, but remarkably, that is not the case. The edges of the bitruncated 5-cell are not actually center sections of 5-cell edges, although they are exactly that length, because the edges of the bitruncated 5-cell do not lie in the same {12} central planes as the 5-cell edges. They are not colinear with 5-cell edges in any way, and only intersect 5-cell edges at vertices (the 5-cell edges' intersection points). Bitruncation of the 5-cells does ''not'' simply truncate each tetrahedron cell in place. By creating new edges which connect the intersection points of 5-cell edges, bitruncation does create 600 truncated tetrahedron cells perfectly sized to fit within the 600 original tetrahedron cells, but at new locations, not centered on an original 5-cell tetrahedron cell. These new locations lie on a smaller 3-sphere than the original locations. [[File:Bitruncated_5-cell_net.png|thumb|Net of the bitruncated 5-cell honeycomb. 10 truncated tetrahedron cells alternately colored red and yellow.{{Sfn|Ruen: Net of the bitruncated 5-cell|2007}}]] The 3-dimensional surface of each bitruncated 5-cell is a honeycomb of 10 truncated tetrahedron cells. The truncated tetrahedra are joined face-to-face in a 3-sphere-filling honeycomb (like the cells of any 4-polytope), at both their hexagon and triangle faces. Each hexagonal face of a cell is joined in complementary orientation to the neighboring cell. Three cells meet at each edge, which is shared by two hexagons and one triangle. Four cells meet at each vertex in a [[w:Tetragonal_disphenoid|tetragonal disphenoid]] vertex figure. The 30-point bitruncated 5-cell is the convex common core (spatial [[w:Intersection|intersection]]) of six 5-point 5-cells in dual position. These six 5-cells are completely disjoint: they share no vertices, but their edges intersect orthogonally, at two points on each edge. Four 5-cell edges, from four of the six 5-cells, cross orthogonally in 30 places, the two intersection points on 60 5-cell edges: the 30 vertices of a bitruncated 5-cell. The six 5-cells are three dual pairs (in two different ways) of the self-dual 5-cell: six pairs of duals reciprocated at their common midsphere. Each dual pair intersects at just one of the two intersection points on each edge.{{Sfn|Klitzing|2025|loc=''sted'' (Stellated Decachoron)|ps=; [https://bendwavy.org/klitzing/incmats/sted.htm ''sted''] is the compound of two [https://bendwavy.org/klitzing/incmats/pen.htm ''pen'' (Pentachoron)] in dual position. Their intersection core ("Admiral of the fleet") is [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)].}} We have seen these six 5-cells before, illustrated in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Eleven|§Eleven]]'' above; they are the compound of six completely disjoint 5-cells visited during each 5-cell's characteristic isoclinic rotation of period 15.{{Efn|1=The 5-cell edges of the six disjoint pentagrams in the {30/12}=6{5/2} triacontagram illustration do not appear to intersect, as the 5-cell edge chords of the bitruncated 5-cell compound are said to intersect. The {30/12}=6{5/2} projection is a perspective view from inside the curved 3-dimensional space of the 120-cell's surface, looking straight down a cylindrical column of six stacked 5-cells. None of the 5-cell edges intersect in that curved 3-space, except where they meet at the 30 120-cell vertices. The 60 5-cell edges do intersect orthogonally in 4-space, in groups of four, at 30 points which lie on a smaller 3-sphere than the 120-cell. None of those 4-space intersections are visible in these projections of points and lines on the 120-cell's 3-sphere surface.|name=5-cell edges do not intersect is S³}} The six 5-cell compound is a stellated 4-polytope with 30 star-points, inscribed in the 120-cell.{{Efn|The stellated compound of six 5-cells in dual position is three pairs of 5-cells reciprocated at their common midsphere. It is composed of dual pairs of the [[W:Compound of five tetrahedra|compound of five tetrahedra]], which form the [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]; its 30 tetrahedral cells are three such dual pairs. In the compound of five tetrahedra the edges of the tetrahedra do not intersect. In the compound of ten tetrahedra they intersect orthogonally, but not at their midpoints. Each edge has two points of intersection on it. The compound of ten tetrahedra is five pairs of dual tetrahedra reciprocated at their common midsphere. It is inscribed in a dodecahedron (its convex hull). Its ''stellation core'' is an icosahedron, but its ''common core'' where the tetrahedron edges intersect is a dodecahedron, the tetrahedrons' convex spatial intersection. The stellated compound of six 5-cells has the analogous property: it is inscribed in a bitruncated 5-cell (its convex hull), and its common core is a smaller bitruncated 5-cell. (Its stellation core is a [[W:Truncated 5-cell#Disphenoidal 30-cell|disphenoidal 30-cell]], its dual polytope.)|name=compound of six 5-cells}} It is 1/20th of the 600-point [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#How many building blocks, how many ways|great grand stellated 120-cell]], the compound of 120 5-cells. The convex hull of its 30 star-points is a bitruncated 5-cell. In this stellated compound of six 5-cells in dual position, the bitruncated 5-cell occurs in two places and two sizes: as both the convex hull, and the convex common core, of the six 5-cells. Inscribed in the larger 120-cell of radius {{radic|2}}, the convex hull of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length 1. The convex common core of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length {{radic|1/2}}, inscribed in the smaller 120-cell of radius 1. In the 120-cell, 120 disjoint 5-cell building blocks combine in dual position groups of six related by the 5-cell's isoclinic rotation, to make 60 bitruncated 5-cells inscribed in the self-dual 5-cells' midsphere (at their edge intersections), and also 60 larger bitruncated 5-cells inscribed in the 120-cell, with each of the 600 vertices shared by three bitruncated 5-cells. The 120-cell is the convex hull of a compound of 20 disjoint (60 distinct) 30-point bitruncated 5-cells, generated by the characteristic rotation of its 120 completely disjoint 5-cells.{{Sfn|Klitzing|2025|loc= ''teppix'' (tripesic hexacosachoron)|ps=; ''[https://bendwavy.org/klitzing/incmats/teppix.htm teppix]'' is a compound of 60 [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)] with 3 ''deca'' sharing each vertex.}}{{Efn|In the 120-cell, 600 tetrahedron cells of 120 completely disjoint 5-cells intersect at two truncation points on each edge. Those 2400 truncation points are the vertices of 200 disjoint (and 600 distinct) truncated tetrahedra, which are the cells of 20 disjoint (and 60 distinct) bitruncated 5-cells. The 60 bitruncated 5-cells share vertices, but not edges, faces or cells. Each bitruncated 5-cell finds its 30 vertices at the 30 intersection points of 4 orthogonal 5-cell edges, belonging to 6 disjoint 5-cells, in the original 120-cell. Each bitruncated 5-cell vertex lies on an edge of 4 disjoint original 5-cells. Each bitruncated 5-cell edge touches intersection points on all 6 disjoint original 5-cells, and is shared by 3 truncated tetrahedra of just one bitruncated 5-cell.}} In [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Concentric 120-cells|the previous section]] we saw that the six 5-cell edges in each central plane intersect at the {6} vertices of the red hexagon, a great hexagon of a 24-cell. Each 5-cell edge, truncated at both ends at those intersection points, is a 24-cell edge of one of the 24-cells inscribed in a smaller 120-cell: the 600 intersection points. In this section we have seen how that truncation of 5-cell edges at both ends is the bitruncation of the 5-cell, and those 5-cell edges, truncated at both ends, are the same length as edges of bitruncated 5-cells inscribed in the original 120-cell. Bitruncating the {{radic|2}}-radius 120-cell's 120 5-cells reveals a smaller unit-radius 120-cell. The 24-cell edges of the smaller 120-cell are 5-cell edges of a larger-radius-by-{{radic|2}} 120-cell, truncated at both ends. Both 120-cells have 24-point 24-cells and 30-point bitruncated 5-cells inscribed in them. The 60° edge length of the 24-cells equals the radius; it is {{radic|2}} times the 41.4° edge length of the bitruncated 5-cells. The 60° 24-cell edges lie in the {12} central planes with the 5-cell edges and the 120-cell edges; but the 41.4° bitruncated 5-cell edges do not. The 120-cell contains 25 disjoint (225 distinct) 24-cells, and 20 disjoint (60 distinct) bitruncated 5-cells. Although regular 5-cells do not combine to form any regular 4-polytope smaller than the 120-cell, the 5-cells do combine to form semi-regular bitruncated 5-cells which are subsumed in the 120-cell.{{Efn|Although only major chords occur in regular 4-polytopes smaller than the 120-cell, minor chords do occur in semi-regular 4-polytopes smaller than the 120-cell. Truncating the 5-cell creates minor chords, such as the 41.1° edges of the bitruncated 5-cell.}} The 41.4° edge of the 30-point bitruncated 5-cell is also the triangle face edge we found in the 60-point central [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The real hemi-icosahedron|section 8₃ (Moxness's Hull #8) rhombicosidodecahedron]]. There are 60 distinct section 8₃ rhombicosidodecahedra and 600 distinct truncated tetrahedron cells of 60 distinct (20 disjoint) bitruncated 5-cells, and they share triangle faces, but little else. The truncated tetrahedron cells cannot be inscribed in the rhombicosidodecahedra, and the only chords they share are the 41.4° triangle edge and the 75.5° chord (the 180° complement of the 104.5° 5-cell edge chord). The section 8₃ rhombicosidodecahedron's 20 triangle faces lie over the centers of 20 larger-by-√2 5-cell faces, parallel to them and to a {12} central plane. The 5-cell faces are inscribed in the rhombicosidodecahedron, but are not edge-bound to each other; the 20 faces belong to 10 completely disjoint 5-cells. The 5-cell edges (but not the 5-cell faces) lie in {12} central planes; the 5-cell faces, the bitruncated 5-cell edges and their triangle and hexagon faces do not. Each section 8₃ rhombicosidodecahedron is the intersection of ten {12} central planes, shared pairwise with ten other rhombicosidodecahedra; 11 rhombicosidodecahedra share ten {12} central planes pairwise, as cells of a 4-polytope share face planes pairwise. Each truncated tetrahedron cell of a bitruncated 5-cell shares none of the {12} central planes; it is the intersection of 6 great rectangles, with two parallel 41.1° edges lying in each, alternating with two parallel 138.6° chords (its hexagon face diameters). Each bitruncated 5-cell is the intersection of 30 great rectangle {4} central planes. A truncated tetrahedron is face-bonded to the outside of each triangle face of a rhombicosidodecahedron. Three of its hexagon faces stand on the long edge of a rectangle face, perpendicular to the rectangle. We find the 25.2° chord as the edge of the non-central section 6₃ (Moxness's Hull #6) rhombicosidodecahedron. Those 120 semi-regular rhombicosidodecahedra have only that single edge (of length 1/𝜙 in a {{radic|2}}-radius 120-cell, or 1/𝜙{{radic|2}} in a unit-radius 120-cell). This edge length is in the golden ratio to the 41.4° edge of the 30-point bitruncated 5-cells, which is also the triangle face edge of the central section 8₃ (Moxness's Hull #8) rhombicosidodecahedron. The 120 semi-regular section 6₃ rhombicosidodecahedra share their smaller edges with 720 pentagonal prisms, 1200 hexagonal prisms and 600 truncated tetrahedron cells, in a semi-regular honeycomb of the 120-cell discovered by Alicia Boole Stott and described in her 1910 paper.{{Sfn|Boole Stott|1910|loc=Table of Polytopes in S₄|ps=; <math>e_2e_3C_{120}\ RID\ P_5\ P_6\ tT</math>}} These truncated tetrahedra are 1/𝜙 smaller than the 600 cells of the bitruncated 5-cells. The 60 distinct section 8₃ rhombicosidodecahedra (Moxness's Hull #8) share pentagon faces. Each of the 120 dodecahedron cells lies just inside 12 distinct rhombicosidodecahedra which share its volume. Each rhombicosidodecahedron includes a ball of 13 dodecahedron cells, 12 around one at the center of the rhombicosidodecahedron, within its volume. The remainder of the rhombicosidodecahedron is filled by 30 dodecahedron cell fragments that fit into the concavities of the 13 cell ball of dodecahedra. These fragments have triangle and rectangle faces. === Rectifying the 16-cells === Bitruncation is not the only way to truncate a regular polytope, or even the simplest way. The simplest method of truncation is [[w:Rectification_(geometry)|''rectification'']], complete truncation at the midpoint of each edge. Moreover, the 5-cell is not the only 120-cell building block we can truncate. We saw how bitruncation of the {{radic|2}}-radius 120-cell's 5-cells reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 20 disjoint (60 distinct) bitruncated 5-cells. In the next paragraph we describe how rectification of the {{radic|2}}-radius 120-cell's 16-cells also reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 25 disjoint (225 distinct) 24-cells. Those two operations on the 120-cell are equivalent. They are the same truncation of the 120-cell, which bitruncates 5-cells into bitruncated 5-cells, and also rectifies 16-cells into 24-cells. This single truncation of the 120-cell captures the distant relationship of 5-cell building blocks to 16-cell building blocks. Rectifying a {{radic|2}}-radius 16-cell of edge 2 creates a unit-radius 24-cell of unit edge, which is the compound of three unit-radius 16-cells. Rectifying one of those inscribed unit-radius 16-cells of edge {{radic|2}} creates a smaller 24-cell of radius and edge {{radic|1/2}}, which is the [[24-cell#Relationships among interior polytopes|common core (intersection]]) of the unit 24-cell and its three inscribed 16-cells. Like the 120-cell itself, the 24-cell is concentric to a smaller instance of itself of {{radic|1/2}} its radius. The common core of each of the 24-cells inscribed in the 120-cell is the corresponding 24-cell in the smaller 120-cell. === Rectifying the 5-cells === In the previous section we bitruncated the 5-cells and rectified the 16-cells, as one combined truncation operation that yields a smaller 120-cell of {{radic|1/2}} the radius. We can also rectify the 5-cells; but that is another distinct truncation operation, that yields a smaller 4-polytope of {{radic|3/8}} the radius. [[File:Great (12) chords of rectified 5-cell.png|thumb|400px|5-cell edge chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {6} bitruncation points (two on each of the 104.5° {{radic|5}} 5-cell edges) lie on a smaller 120-cell of radius 1 (the red circle); they are bitruncated 5-cell vertices. The {6} rectification points (at the midpoints of the 5-cell edges) lie on a still smaller 1200-point 4-polytope of radius {{radic|0.75}} ≈ 0.866 (the magenta circle); they are rectified 5-cell vertices.]] Rectifying the 5-cell creates the 10-point 10-cell semi-regular [[W:Rectified 5-cell|rectified 5-cell]], with 5 tetrahedral cells and 5 octahedral cells. It has 30 edges and 30 equilateral triangle faces. The 3-dimensional surface of the rectified 5-cell is an alternating [[W:Tetrahedral-octahedral honeycomb|tetrahedral-octahedral honeycomb]] of just 5 tetrahedra and 5 octahedra, tessellating the 3-sphere. Its vertex figure is the cuboctahedron. The rectified 5-cell is a [[w:Blind_polytope|Blind polytope]], because it is convex with only regular facets. It is a bistratic lace tower which has exactly three vertex layers with the same Coxeter symmetry, aligned on top of each other.{{Sfn|Klitzing|2025|loc=''[https://bendwavy.org/klitzing/incmats/rap.htm rap (rectified pentachoron)]''}} If the 120 5-cells in a radius {{radic|2}} 120-cell are rectified, the rectified 5-cells lie on a smaller 4-polytope of radius {{radic|3/4}} (the magenta circle in the diagram), inscribed at the 1200 midedges of the 5-cells.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the ''unit-radius'' 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the unit-radius 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of the [[24-cell#Characteristic orthoscheme|24-cell's 4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[24-cell#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the ''area'' of the equilateral triangle face of the unit-edge, unit-radius 24-cell.|name=root 3/4}} This smaller 4-polytope is not a smaller 120-cell; it is the convex hull of a 1200-point compound of two 120-cells. The rectified 5-cell does not occur inscribed in the 120-cell; it only occurs in this compound of two 120-cells, 240 regular 5-cells, and 120 rectified 5-cells. The rectified 5-cell with its 80.4° edge chord does not occur anywhere in a single 120-cell, so the rectified 5-cell's edges are not the edges of any polytope found in the 120-cell. The rectified 5-cell's significance to the 120-cell is well-hidden, but we shall see that it has an indirect role as a building block of the 11-cells in the 120-cell. Each 10-point rectified 5-cell is the convex hull of a stellated compound of two completely orthogonal 5-point 5-cells: five pairs of antipodal vertices. Their edges intersect at the midedge, and they are ''not'' in dual position (not reciprocated at their common 3-sphere). In this stellated compound of two completely orthogonal 5-cells (which does not occur in the 120-cell), the rectified 5-cell occurs in two places and two sizes: as both the convex hull of the vertices, and the convex common core of the midedge intersections. The edge length of the rectified 5-cells in the smaller 1200-point 4-polytope of radius {{radic|3/4}} is {{radic|5/4}}. The edge length of a unit-radius rectified 5-cell is {{radic|5/3}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|3}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}. In the 120-cell of radius {{radic|2}}, the 104.5° {{radic|5}} chord is the 5-cell edge, and the 75.5° {{radic|3}} chord is the distance between two parallel 5-cell edges (belonging to two disjoint 5-cells). The 104.5° and 75.5° chords are 180° complements, so they form great rectangles in the {12} central planes of the 120-cell (the red rectangles in the diagram). In the 1200-point compound of two 120-cells of radius {{radic|3}} where 120 rectified 5-cells occur, the {{radic|3}} chord is the ''radius'' (not the 75.5° chord), and the {{radic|5}} chord is the ''rectified'' 5-cell edge of arc 80.4° (not the 104.5° regular 5-cell edge). === Truncating the 5-cells === [[File:Great (12) chords of unit thirds radius.png|thumb|400px|Truncating the 120-cell's 5-cells at ''one-third'' of their edge length produces a smaller 120-cell of ''one-half'' the radius, with vertices at {6} one-third intersection points of the 120° {{Radic|6}} chords (''not'' of the 104.5° {{Radic|5}} 5-cell edge chords). The green {6} hexagon is a 24-cell great hexagon in the resulting smaller-by-one-half 1200-point 4-polytopes. Because there are {12} such intersection points in each {12} central plane, there are two chiral ways to perform this truncation, which produce disjoint 1200-point 4-polytopes.]] A third simple way to truncate the 5-cell is at one-third of its edge length. This truncation of the 5-cell creates a 20-point, 10-cell semi-regular 4-polytope, known somewhat ambiguously as ''the'' [[w:Truncated_5-cell|truncated 5-cell]], with 5 truncated tetrahedron cells (like the bitruncated 5-cell's), and 5 regular tetrahedron cells (like the rectified 5-cell's). The 3-dimensional surface of the truncated 5-cell is an alternating honeycomb of 5 truncated tetrahedra and 5 regular tetrahedra. It resembles the smaller rectified 5-cell with truncated tetrahedra instead of octahedra, or the larger bitruncated 5-cell with half its truncated tetrahedra replaced by regular tetrahedra. When the regular 5-cell is truncated at ''one-third'' of its edge length, the radius and edge length of the the resulting truncated 5-cell are ''one-half'' the regular 5-cell's radius and edge length. When the 120 5-cells in a 120-cell of radius 2 are truncated at one-third of their edge length, the truncated 5-cells lie on a smaller 120-cell of radius 1. The edge length of the unit-radius truncated 5-cell is {{radic|5/8}}, one-half the unit-radius 5-cell's edge length of {{radic|5/2}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|8}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}, and the rectified 5-cell is characterized by the ratio {{radic|5}} to {{radic|3}}. The 20-point truncated 5-cell is the convex common core of a stellated compound of four 5-cells (the four 5-cells' spatial intersection). The convex common core has half the radius of the convex hull of the compound. The four 5-cells are orthogonal (aligned on the four orthogonal axes), but none of their 20 vertices are antipodal. The 5-cells are ''not'' in dual position (not reciprocated at their common 3-sphere). The 5-cell edges do ''not'' intersect, but truncating the 120-cell's 5-cell edge chords at their one-third points truncates the 120-cell's other chords similarly. It is the 120-cell's 120° chords (of length {{Radic|6}} in a {{Radic|2}}-radius 120-cell, or {{Radic|3}} in a unit-radius 120-cell) which intersect each other at their one-third points. Four edges (one from each 5-cell) intersect orthogonally at just ''one'' of the two one-third intersection points on each of the 2400 120° chords that join vertices of two disjoint 5-cells. There are two chiral ways to perform this truncation of the 120-cell; they use the alternate intersection points on each edge, and produce disjoint 600-point 120-cells. The 52.25° edge chord of the truncated 5-cell (one-half the 5-cell's 104.5° edge chord) is not among the [[120-cell#Chords|chords of the 120-cell]], so the truncated 5-cell does not occur inscribed in the 120-cell; it occurs only in a compound of four 120-cells, and 480 regular 5-cells, and 120 truncated 5-cells. In the stellated compound of four orthogonal 5-cells (which does not occur in the 120-cell), the truncated 5-cell occurs in two places and two sizes: as both the convex hull of the 20 vertices, and the convex common core (of half the radius of the convex hull) of the 20 intersection points of four orthogonal 120° chords. == The perfection of Fuller's cyclic design == [[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 orthogonal axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]] This section is not an historical digression, but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]''}} After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session<ref>{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}</ref> where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.<ref>{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}</ref>|name=Snelson and Fuller}} A tensegrity icosahedron is an icosahedral geodesic sphere whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables. The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 10 of in Moxness's Hull #8 rhombicosidodecahedron, the real cell of the 11-cell. The Jessen's was named by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a [[W:Cubist|cubist]] shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point ''vector equilibrium'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point vector equilibrium, and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposite equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (a hexad), then backing away again while still rotating in the same directions. All this was overlaid with Fuller's own deep commentary, in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing "vector edge cube" (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}} Earlier, we noticed 10 Jessen's inscribed in each 60-point rhombicosidodecahedron central section of the 120-cell (each real hemi-icosahedron). Each rhombicosidodecahedron is a compound of 5 disjoint Jessen's, in two different ways, just the way the 120-cell is a compound of 5 disjoint 600-cells, in two different ways. In the rhombicosidodecahedron each regular icosahedron vertex has been replaced by the five vertices of a little pentagon face (a 120-cell face), and the regular icosahedron has been replaced by 5 disjoint (10 distinct) Jessen's icosahedra.{{Efn|name=compound of 5 cuboctahedra}} The 3 pairs of parallel 5-cell edges in each Jessen's lie a bit uncertainly, infinitesimally mobile and [[Kinematics of the cuboctahedron#Elastic-edge transformation|behaving like the struts of a tensegrity icosahedron]], so we can push any parallel pair of them apart or together infinitesimally, making each Jessen's icosahedron expand or contract infinitesimally. All 600 Jessen's, all 60 rhombicosidodecahedra, and the 120-cell itself expand or contract infinitesimally, together.{{Efn|name=tensegrity 120-cell}} Expansion and contraction are Boole Stott's operators of dimensional analogy, and that infinitesimal mobility is the infinite calculus of an inter-dimensional symmetry. The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three rectangular chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables. [[File:Jessen's √2 radius dimensions.png|thumb|400px|Moxness's 60-point section 8₃ rhombicosidodecahedron is a compound of 5 of this 12-point Jessen's icosahedron, shown here in a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed {{radic|1.5}} green cube, and its 8 equilateral triangle faces are 24-cell faces. This is a ''vertex figure'' of the 120-cell. The center point is also a vertex of the 120-cell.]] As a matter of convenience in this paper, we have used {{radic|2}}-radius metrics for 3-sphere polytopes, so e.g. the 5-cell edge is {{radic|5}}, where in unit-radius coordinates it would be {{Radic|5/2}}. Here we give two illustrations of the Jessen's using two different metrics: the 2-sphere Jessen's has a {{radic|5}} diameter, and the 3-sphere Jessen's has a {{radic|2}} radius. This reveals a curiously cyclic way in which our 2-sphere and 3-sphere metrics correspond. In the embedding into 4-space the characteristic root factors of the Jessen's seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord. We might have expected to find the 6-point hemi-icosahedron's 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the Jessen's are the 5-cell edges. A 5-cell face has its three {{radic|5}} edges in three different Jessen's icosahedra. The Jessen's is not a cell, but one of the 120-cell's vertex figures, like the [[600-cell#Icosahedra|120 regular icosahedron vertex figures in the 600-cell]]. That is why we find 600 Jessen's, of course. The center point in this Jessen's illustration is another ''vertex'' of the 120-cell, not the empty center of a cell.{{Efn|The 13 vertices of the illustration which include its center point lie in the curved 3-space of the 3-sphere, on the 120-cell's surface. In 4-space, this object is an [[W:Icosahedral pyramid|icosahedral pyramid]] with a Jessen's icosahedron as its base, and the apical center vertex as its apex. The center point in the illustration is a vertex of the 120-cell, and the center of the curved Jessen's, and the apex of the icosahedral pyramid, but it is not the center point in 4-space of a flat 3-dimensional Jessen's icosahedron. The center point of the base Jessen's icosahedron is a point inside the 120-cell, not a 120-cell vertex on its surface. It lies in the same 3-dimensional flat-slice hyperplane as the 12 vertices of the base Jessen's icosahedron, directly below the 13th 120-cell vertex.}} Each Jessen's includes the central apex vertex, {{radic|2}} radii, {{radic|2}} edges and {{radic|5}} chords of a vertex figure around the 120-cell vertex at its center. The {{radic|2}} face edges are 24-cell edges (also tesseract edges), and the inscribed green cube is the 24-cell's cube vertex figure. The 8 {{radic|2}} face triangles occur in 8 distinct 24-cells that meet at the apex vertex.{{Efn|Eight 24-cells meet at each vertex of a [[24-cell#Radially equilateral honeycomb|honeycomb of 24-cells]]: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell.{{Efn|In the 600-cell, which contains [[600-cell#Twenty-five 24-cells|25 24-cells]], 5 24-cells meet at each vertex. Each pair of 24-cells at the vertex meets at one of 200 distinct great hexagon central planes. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 8 other 24-cells. In the 120-cell, which contains 10 600-cells (5 disjoint 600-cells two different ways) and 225 24-cells (25 disjoint 24-cells), 8 24-cells meet at each vertex. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 208 other 24-cells. But since in the 120-cell the great hexagons lie in pairs in one of 200 {12} central planes (containing 400 great hexagons), each 24-cell shares one of its {12} central ''planes'' with .. other 24-cells.}}}} This Jessen's vertex figure includes 5-cell edges and 24-cell edges (which are also tesseract edges), so it is descriptive of the relationship between those regular 4-polytopes, but it does not include any 120-cell edges or 600-cell edges, so it has nothing to say, by itself, about the <math>H_4</math> polytopes. It is only a tiny fraction of the 120-cell's full vertex figure, which is a staggeringly complex star: 600 chords of 30 distinct lengths meet at each of the 600 vertices. The {{radic|5}} chords are 5-cell edges, connecting vertices in different 24-cells. The 3 pairs of parallel 5-cell edges in each Jessen's lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th pair of parallel 5-cell edges orthogonal to all of them, in fact three more orthogonal pairs, since 6 orthogonal planes (not just 4) intersect at a point in 4-space. The Jessen's situation is that it lies completely orthogonal to another Jessen's, the vertex figure of the antipodal vertex, and its 3 orthogonal planes (xy, yz, zx) lie completely orthogonal to its antipodal Jessen's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} These 6 pairs of parallel 5-cell edges form a 24-point 4-polytope, composed of two completely orthogonal 12-point Jessen's, inscribed in two completely orthogonal rhombicosidodecahedra. This 24-point 4-polytope is not a 24-cell: the 24-cell is not a compound of two 12-point Jessen's. But it turns out that two completely orthogonal 12-point Jessen's indirectly define a 24-point 24-cell. We shall see that their 4-space intersection is a 24-cell. This finding, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, brings Fuller's study of [[w:Tesseract#Radial_equilateral_symmetry|radially equilateral]] vector equilibrium polytopes to its completion in the 24-cell. Fuller began with the hexagon, the 6-point vector equilibrium in 2 dimensions, the only polygon with its radius equal to its edge length. He studied the cuboctahedron, the 12-point vector equilibrium in 3 dimensions, the only polyhedron with its radius equal to its edge length, in all its flexible guises. He discovered its stable equilibrium as the the Jessen's shadfish, with its cube of 6 open mouths and 90° dihedral angles between all its faces, the geometric center of [[WikiJournal Preprints/Kinematics of the cuboctahedron|the cuboctahedron's kinematic transformation]] through the regular polyhedra: tetrahedron, octahedron, Jessen's, regular icosahedron, and cuboctahedron. Fuller's study of kinematic Euclidean geometry did not reach the 4-polytopes, and the ultimate 24-point vector equilibrium in 4 dimensions, the 24-cell, the unique <math>F_4</math> symmetry found only in 4 dimensions. But Fuller led us up to it, through the kinematics of infinitesimal mobility, and that route to it is our clue to the infinite calculus of dimensional expansion and contraction. We observe this geometry, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, only in the 120-cell. The 600-cell contains 12-point Jessen's, but no completely orthogonal pairs of them. The 24-cell individually, and the 25 24-cells in the 600-cell, are not occupied by a pair of 12-point Jessen's. The 24-point 24-cell is not, in fact, a compound of two 12-point Jessen's. While the 120-cell's ratio of disjoint 12-point Jessen's to disjoint 24-point 24-cells is <math>50/25 = 2/1</math>, the ratio of distinct 12-point Jessen's to distinct 24-point 24-cells is <math>600/225 = 8/3 </math>. We observe another geometry, of 24-cells in dual positions, only in the 120-cell. No two 24-cells in the 600-cell are in dual positions, but in the 120-cell with 225 distinct 24-cells (25 disjoint 24-cells), every 24-cell is in dual position to other 24-cells. The 24-cell is self-dual, and when two 24-cells of the same radius are in dual position, they are completely disjoint with respect to vertices, but they intersect at the midpoints of their 96 orthogonal edges. Since four orthogonal lines intersect at a point in 4-space, in addition to the midedge radius and the two intersecting edges there is a third intersecting edge through each point of contact: ''three'' 24-cells lie in dual positions to each other, with their orthogonal edges intersecting. Three ''pairs'' of 24-cells lie in orthogonal dual positions to each other, sharing no vertices, but the same 96 midedge points. We also observe this geometry, of 24-cells in dual positions, in the irregular {12} dodecagon central planes, which have two inscribed great {6} hexagons, offset from each other irregularly by a 15.5° arc on one side (a 120-cell edge chord) and a 44.5° arc on the other side. The 600-cell and the 24-cell contain only great {6} hexagon planes. The two inscribed great {6} hexagons in each {12} central plane belong to a pair of 24-cells in dual position. We observe inscribed 5-cells only in the 120-cell. The 600-cell has <math>5^2 = 25</math> distinct 24-cells inscribed in 120 vertices, and is a regular compound of <math>5</math> disjoint 24-cells in 10 different ways, but it has no inscribed 5-point 5-cells joining corresponding vertices of 5 of its 25 24-cells.{{Efn|The 600-cell does have inscribed 5-point great pentagons joining corresponding vertices of 5 of its 25 24-cells. The 600-cell has 2-dimensional pentads, but only the 120-cell has 4-dimensional pentads.}} The 120-cell has <math>5^2 \times 3^2 = 225</math> distinct 24-cells inscribed in 600 vertices, and is a regular compound of <math>5^2 = 25</math> disjoint 24-point 24-cells in 10 different ways, and it has 120 inscribed 5-cells joining corresponding vertices of 5 of its 225 24-cells. [[File:Great 5-cell √5 digons rectangle.png|thumb|400px|Three {{radic|5}} x {{radic|3}} rectangles (red) are found in 200 central planes of the radius {{radic|2}} 120-cell, and in its 600 Jessen's icosahedra, where 3 orthogonal rectangles comprise each 12-point Jessen's. Each central plane intersects {12} vertices in an irregular great dodecagon. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges (solid red), which form two opposing ''irregular'' great hexagons (truncated triangles) with the {{radic|5}} chords. The {12} central planes also contain four {{radic|6}} great triangles (green), inscribed in two {{radic|2}} ''regular'' great hexagons. 1200 smaller {{radic|5}} 5-cell ''face'' triangles (blue) occupy 600 other, non-central planes.]] The Jessen's eight {{radic|6}} triangle faces lie in eight great {6} hexagons in eight {12} central planes of the 120-cell. The Jessen's {{radic|5}} chords lie in great {4} rectangles ({{radic|5}} by {{radic|3}}) in orthogonal central planes of the Jessen's. These are ''also'' {12} central planes of the 120-cell. We can pick out the {{radic|5}} by {{radic|3}} rectangles in the {12} central plane chord diagrams (bounded by red dashed lines). The Jessen's vertex figure is bounded by eight {12} face planes, and divided by six orthogonal {12} central planes, and all 14 planes are {12} central planes of the 120-cell. The 5-cells' ''face'' planes are ''not'' central planes of the 120-cell. Recall that 10 distinct Jessen's are inscribed in each rhombicosidodecahedron, as two chiral sets of 5 completely disjoint Jessen's, such that two {{radic|5}} 5-cell edges meet at each vertex of the rhombicosidodecahedron. These are two of the four 5-cell edges that meet at each vertex of the 5-cell: edges of a 5-cell face, 20 of which are disjointly inscribed in each rhombicosidodecahedron. In each Jessen's the 6 {{radic|5}} reflex edges are disjoint, and in each rhombicosidodecahedron only two edges meet at each vertex, but in the 120-cell each {{radic|5}} chord meets three others, that lie in three other Jessen's. Each 5-cell face triangle has each edge in a distinct Jessen's, but the face triangle lies in just one rhombicosidodecahedron. The 1200 5-cell face triangles lie in opposing pairs, in one of 600 ''non-central'' hexagon ''face'' planes. Each of the 60 rhombicosidodecahedra is a compound of 10 Jessen's (5 disjoint Jessen's in two different ways), just the way the 120-cell is a compound of 10 600-cells (5 disjoint 600-cells in two different ways), and the 120-cell's dodecahedron cell is a compound of 10 600-cell tetrahedron cells (5 disjoint tetrahedra in two different ways). The 600 Jessen's in the 120-cell occur in bundles of 8 disjoint Jessen's, in 4 completely orthogonal pairs, each pair aligned with one of the four axes of the Cartesian coordinate system. Collectively they comprise 3 disjoint 24-cells in orthogonal dual position. They are [[24-cell#Clifford parallel polytopes|Clifford parallel 4-polytopes]], 3 completely disjoint 24-cells 90° apart, and two sets of 4 completely disjoint Jessen's 15.5° apart. Opposite triangle faces in a Jessen's occupy opposing positions in opposite great hexagons. In contrast, the two completely orthogonal Jessen's are completely disjoint, with completely orthogonal bounding planes that intersect only at one point, the center of the 120-cell. The corresponding {{radic|6}} triangle faces of two completely orthogonal Jessen's occupy completely orthogonal {12} central planes that share no vertices. If we look again at a single Jessen's, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) Jessen's lies in 4-space, it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (vertex figure) Jessen's is part of a 16-point (8-cell) tesseract containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already found it was part of a 24-point (24-cell) 4-polytope, which contains 3 16-point (8-cell) tesseracts. Each 12-point (6 {{radic|5}} reflex edge) Jessen's is one of 10 concentric Jessen's in a rhombicosidodecahedron, two sets of 5 disjoint Jessen's rotated with respect to each other isoclinically by 12° x 12° = 15.5°, with a total of 60 disjoint {{radic|5}} edges. Each 12-point (24 {{radic|6}} edge) Jessen's is one of 8 concentric Jessen's in two 24-cells in dual positions, rotated with respect to each other isoclinically by 41.4° x 41.4° = 90°, with a total of 192 {{radic|6}} edges.{{Efn|There are 96 {{radic|6}} chords in each 24-cell, linking every other vertex under its 96 {{radic|2}} edges.}} The 24-point 24-cell has 4 Hopf fibrations of 4 hexagonal great circle fibers, so it is a complex of 16 great hexagons, generally not orthogonal to each other, but containing 3 sets of 4 orthogonal great hexagons. Three Borromean link great rectangles are inscribed in each great hexagon, and three tesseracts are inscribed in each 24-cell. Four of the 6 orthogonal [[w:Borromean_rings|Borromean link]] great rectangles in each completely orthogonal pair of Jessen's are inscribed in each tesseract. == Conclusion == Thus we see what the 11-cell really is: an unexpected seventh regular convex 4-polytope falling between the 600-cell and 120-cell, a quasi-regular compound of 600-cell and 5-cell (an icosahedron-tetrahedron analogue), as the 24-cell is an unexpected sixth regular convex polytope falling between the 8-cell and 600-cell, a quasi-regular compound of 8-cell and 16-cell (a cube-octahedron analogue). Like the 5-cell, the 11-cell is a far-side 4-polytope with its long edges spanning the near and far halves of the 3-sphere. Unlike the 5-cell, the 11-cell's left and right rotational instances are not the same object: they have distinct cell polyhedra, which are duals. The 11-cell is a real regular convex 4-polytope, not just an [[W:abstract polytope|abstract 4-polytope]], but not just a singleton regular convex 4-polytope, and not just a single kind of cell honeycomb on the 3-sphere.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} Though it is all those things singly, it never occurs singly, but its multiple instances in the 120-cell compound to all those things, and significantly more. The 11-cell (singular) is the 11-vertex (17 cell) non-uniform Blind 4-polytope, with 11 non-uniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. The abstract regular 11-point (11-cell) has a realization in Euclidean 4-space as this convex 4-polytope, with regular facets and regular triangle faces. The 11-cell (plural) is subsumed in the 120-cell, as all the regular convex 4-polytopes are. The compound of eleven 11-cells (the ..-cell) and Schoute's compound of five 24-cells (the 600-cell) is the quasi-regular 137-point (..-cell) 4-polytope, an object of further study. The 11-cells' realization in the 120-cell as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (4-simplex and 4-orthoplex), which are both inscribed in the 11-point (17-cell), 137-point (..-cell) and 600-point (120-cell), but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationship between the regular ''n''-polytopes of different ''n''. The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies. == Build with the blocks == <blockquote>"The best of truths is of no use unless it has become one's most personal inner experience."{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: "The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery."}}}}</blockquote> <blockquote>"Even the very wise cannot see all ends."{{Sfn|Tolkien|1954|loc=Gandalf}}</blockquote> No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to show how I came by my understanding of these objects, since I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, in my imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am. {{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=2}} {{Regular convex 4-polytopes|columns=7|wiki=W:|radius=1}} == Acknowledgements == ... == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite web| url = https://www.youtube.com/watch?v=9sM44p385Ws| title = Vector Equilibrium | first = R. Buckminster | last = Fuller | author-link=W:Buckminster Fuller | year = 1975 | work= Everything I Know Sessions | place = Philadelphia}} * {{Citation|last=Christie|first=David Brooks|author-link=User:Dc.samizdat|year=2024|title=Bucky Fuller and the languages of geometry|title-link=User:Dc.samizdat#Bucky Fuller and the languages of geometry|journal=Wikiversity|ref={{SfnRef|Christie: On Fuller's use of language|2024}}}} * {{Citation|author-last=Moxness|author-first=J.G.|year=2022|author-link=W:User:Jgmoxness|title=120-Cell showing the individual 8 concentric hulls and in combination|title-link=Wikimedia:File:120-Cell showing the individual 8 concentric hulls and in combination.svg|journal=Wikimedia Commons|ref={{SfnRef|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)}}}} * {{Citation|author-last=Moxness|author-first=J.G.|year=2023|author-link=W:User:Jgmoxness|title=Archimedean and Catalan solid hulls with their Weyl orbit definitions|title-link=Wikimedia:File:Archimedean and Catalan solid hulls with their Weyl orbit definitions.svg|journal=Wikimedia Commons|ref={{SfnRef|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}}}} * {{Citation|author-last=Moxness|author-first=J.G.|year=2023|author-link=W:User:Jgmoxness|title=3D & 4D Solids using Quaternion Weyl Orbits from Coxeter-Dynkin Geometric Group Theory|journal=PowerPoint|url=https://theoryofeverything.org/TOE/JGM/Quaternion%20Coxeter-Dynkin%20Geometric%20Group%20Theory-2b.pdf|ref={{SfnRef|Moxness: Quaternion graphics software|2023}}}} === 11-cell === * {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1976 | title=Regularity of Graphs, Complexes and Designs | journal=Colloques Internationaux C.N.R.S. | publisher=Orsay | volume=260 | pages=191-197 | url=https://faculty.washington.edu/moishe/branko/BG111.Regularity%20of%20graphs,etc.pdf | ref=}} * {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1975 | title=Venn Diagrams and Independent Families of Sets | journal=Mathematics Magazine | volume=48 | issue=1 | url=https://maa.org/sites/default/files/pdf/upload_library/22/Ford/BrankoGrunbaum.pdf | doi=10.1080/0025570X.1975.11976431 }} * {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147 }} *{{citation | last1 = Séquin | first1 = Carlo H. | author1-link = W:Carlo H. Séquin | last2 = Lanier | first2 = Jaron | author2-link = W:Jaron Lanier | title = Hyperseeing the Regular Hendacachoron | year = 2007 | journal = ISAMA | publisher=Texas A & M | pp=159-166 | issue=May 2007 | url=https://people.eecs.berkeley.edu/~sequin/PAPERS/2007_ISAMA_11Cell.pdf | ref={{SfnRef|Séquin & Lanier|2007}}}} *{{citation | last1 = Séquin | first1 = Carlo H. | author1-link = W:Carlo H. Séquin | last2 = Hamlin | first2 = James F. | title = The Regular 4-dimensional 57-cell | doi = 10.1145/1278780.1278784 | location = New York, NY, USA | publisher = ACM | series = SIGGRAPH '07 | journal = ACM SIGGRAPH 2007 Sketches | year = 2007| s2cid = 37594016 | url = https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2007_SIGGRAPH_57Cell.pdf | ref={{SfnRef|Séquin & Hamlin|2007}}}} *{{citation | last=Séquin | first=Carlo H. | author-link = W:Carlo H. Séquin | title=A 10-Dimensional Jewel | journal=Gathering for Gardner G4GX | place=Atlanta GA | year=2012 | url=https://people.eecs.berkeley.edu/%7Esequin/PAPERS/2012_G4GX_10D_jewel.pdf }} === [[Polyscheme|Polyschemes]] === {{Regular convex 4-polytopes Refs|wiki=W:}} === Illustrations === * {{Citation|title=Tensegrity icosahedron structure|title-link=Wikimedia:File:Tensegrity Icosahedron.png|journal=Wikimedia Commons|last1=Burkhardt|first1=Bob|year=1994}} * {{Citation|author-last=Christie|author-first=David Brooks|year=2024|author-link=W:User:Dc.samizdat|title=Pentahemidemicube|title-link=Wikimedia:File:Pentahemidemicube.png|journal=Wikimedia Commons|ref={{SfnRef|Christie: Pentahemidemicube|2024}}}} * {{Citation|author-last=Christie|author-first=David Brooks|year=2024|author-link=W:User:Dc.samizdat|title=Pentahemicosahedron|title-link=Wikimedia:File:Pentahemicosahedron.png|journal=Wikimedia Commons|ref={{SfnRef|Christie: Pentahemicosahedron|2024}}}} * {{Citation|author=Cmglee|date=2019|author-link=W:User:Cmglee|title=Radially-symmetrical five-set Venn diagram devised by Branko Grünbaum|title-link=Wikimedia:File:Symmetrical 5-set Venn diagram.svg|journal=Wikimedia Commons|ref={{SfnRef|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled.}}}} * {{Citation|author-last=Cyp|year=2005|author-link=W:User:Cyp|title=Truncated tetrahedron, transparent, slowly turning, created with POV-ray|title-link=Wikimedia:File:Truncatedtetrahedron.gif|journal=Wikimedia Commons|ref={{SfnRef|Cyp: Truncated tetrahedron|2005}}}} * {{Cite book|last=Duveneck|first=Josephine Whitney|title=Life on Two Levels: An Autobiography|year=1978|publisher=William Kaufman|place=Los Altos, CA|ref={{SfnRef|Duveneck|1978}}}} * {{Citation|author-last=Hise|author-first=Jason|year=2011|author-link=W:User:JasonHise|title=A 3D projection of a 120-cell performing a simple rotation|title-link=Wikimedia:File:120-cell.gif|journal=Wikimedia Commons}} * {{Cite book|last=Huxley|first=Aldous|author-link=W:Aldous Huxley|title=Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization|date=1937|publisher=Harper and Brothers|ref={{SfnRef|Huxley|1937}}}} * {{Cite book|last=Jung|first=Carl Gustav|author-link=W:Carl Jung|title=Psychological Reflections: An Anthology of the Writings of C. G. Jung|date=1961|page=XVII|ref={{SfnRef|Jung|1961}}}} * {{Citation|author-last=Piesk|author-first=Tilman|date=2018|author-link=W:User:Watchduck|title=Nonuniform rhombicosidodecahedron as rectified rhombic triacontahedron max|title-link=Wikimedia:File:Nonuniform rhombicosidodecahedron as rectified rhombic triacontahedron max.png|journal=Wikimedia Commons|ref={{SfnRef|Piesk: Rhombicosidodecahedron|2018}}}} * {{Citation|author-last=Piesk|author-first=Tilman|date=2018|author-link=W:User:Watchduck|title=Polyhedron truncated 20 from yellow max|title-link=Wikimedia:File:Polyhedron truncated 20 from yellow max.png|journal=Wikimedia Commons|ref={{SfnRef|Piesk: Truncated icosahedron|2018}}}} * {{Citation|author-last=Ruen|author-first=Tom|year=2007|author-link=W:User:Tomruen|title=Hemi-icosahedron|title-link=Wikimedia:File:Hemi-icosahedron.png|journal=Wikimedia Commons|ref={{SfnRef|Ruen: Hemi-icosahedron|2007}}}} * {{Citation|title=Great grand stellated 120-cell|title-link=Wikimedia:File:Ortho solid 016-uniform polychoron p33-t0.png|journal=Wikimedia Commons|last1=Ruen|first1=Tom|year=2007|author-link=W:User:Tomruen|ref={{SfnRef|Ruen: Great grand stellated 120-cell|2007}}}} * {{Citation|author-last=Ruen|author-first=Tom|year=2019|author-link=W:User:Tomruen|title=Tetrahemihexahedron rotation|title-link=Wikimedia:File:Tetrahemihexahedron rotation.gif|journal=Wikimedia Commons|ref={{SfnRef|Ruen: Tetrahemihexahedron rotation|2019}}}} * {{Citation|title=Net of the bitruncated 5-cell|title-link=Wikimedia:File:Bitruncated 5-cell net.png|journal=Wikimedia Commons|last1=Ruen|first1=Tom|year=2007|author-link=W:User:Tomruen|ref={{SfnRef|Ruen: Net of the bitruncated 5-cell|2007}}}} * {{Citation|title=5-cell|title-link=5-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Christie|editor-first2=David Brooks|editor-link2=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen et al. eds. 5-cell|2024}}}} * {{Citation|title=16-cell|title-link=16-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Christie|editor-first2=David Brooks|editor-link2=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen et al. eds. 16-cell|2024}}}} * {{Citation|title=24-cell|title-link=24-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|year=2024|ref={{SfnRef|Ruen & Goucher et al. eds. 24-cell|2024}}}} * {{Citation|title=600-cell|title-link=600-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|ref={{SfnRef|Ruen & Goucher et al. eds. 600-cell|2024}}}} * {{Citation|title=120-cell|title-link=120-cell|journal=Polyscheme|publisher=Wikiversity|editor-last1=Ruen|editor-first1=Tom|editor-link1=W:User:Tomruen|editor-last2=Goucher|editor-first2=A.P.|editor-link2=W:User:Cloudswrest|editor-last3=Christie|editor-first3=David Brooks|editor-link3=W:User:Dc.samizdat|editor-last4=Moxness|editor-first4=J. Gregory|editor-link4=W:User:Jgmoxness|year=2024|ref={{SfnRef|Ruen & Goucher et al. eds. 120-cell|2024}}}} * {{Cite book|last=Sandperl|first=Ira|author-link=W:Ira Sandperl|title=A Little Kinder|year=1974|publisher=Science and Behavior Books|place=Palo Alto, CA|isbn=0-8314-0035-8|lccn=73-93870|url=https://www.allinoneboat.org/a-little-kinder-an-old-friend-moves-on/|ref={{SfnRef|Sandperl|1974}}}} * {{Cite book|last=Tolkien|first=J.R.R.|title=The Lord of the Rings|orig-date=1954|volume=The Fellowship of the Ring|chapter=The Shadow of the Past|page=69|edition=2nd|date=1967|publisher=Houghton Mifflin|place=Boston|author-link=W:J.R.R.Tolkien|title-link=W:The Lord of the Rings|ref={{SfnRef|Tolkien|1954}}}} {{Refend}} qp48uj64m8rtmffhtce7558393ch15r 2816273 2816271 2026-06-19T01:42:44Z Dc.samizdat 2856930 /* Concentric 120-cells */ 2816273 wikitext text/x-wiki = A symmetrical arrangement of eleven 11-cells = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|March 2024 - June 2026}} <blockquote>[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]] ₅{3,5,3}₅, a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]] {3,5}₅, a hexad non-orientable polyhedron. The 11-cell is described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, we find that the 11-cell has a realization in Euclidean 4-space inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the convex regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra and 120 regular 5-cells. We find that the 120-cell also contains: a non-uniform icosahedral polyhedron that contains the realization of the abstract hemi-icosahedron; real 11-point 11-cells made from 11 of it; and a compound of eleven real 11-cells. We also find a quasi-regular compound of the compound of eleven 11-cells and [[w:Schoute|Schoute]]'s compound of five 24-cells (the 600-cell). We describe the real 11-point 11-cell 4-polytope; its compound of eleven 11-cells; the quasi-regular compound; and their relation to the regular polytopes.</blockquote> == Introduction == [[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hemi-icosahedral hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them. [[File:120-cell.gif|thumb|360px|The picture on the cover of the box of 4-dimensional building blocks.{{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]]."}} Only the 120-cell's own edges are shown. The complex interior parts of the 120-cell, all its inscribed 600-cells, 24-cells, 8-cells, 16-cells, 5-cells and 11-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]] The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-hypercube, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the 600-point [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). The 5-cell has a fundamental relationship to all the other 4-polytopes, but not one as simple as compounding, so it is not immediately useful to children trying to learn to build with 4-dimensional building blocks. But the 16-cell is our very starting point, and the most frequently used tool in the box. Nevertheless, to build the 11-cell, we start with the 5-cell. The 5-cell and 11-cell are both self-reciprocal (their own duals). They are the only 4-polytopes where every cell shares a face with every other cell. The 5-cell is a tetrahedron surrounded by 4 other tetrahedra, in five different ways. The 11-cell is a hemi-icosahedron surrounded by 10 other hemi-icosahedra, in eleven different ways. The 5-cell has 5 vertices that form 5 tetrahedral cells, and a total of 10 triangular faces and 10 edges. The 11-cell has 11 vertices that form 11 hemi-icosahedral cells, each with 6 verticies 10 triangular faces and 15 edges, and a total of 55 triangular faces and 55 edges. == 5-cells and hemi-icosahedra in the 11-cell == [[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its <math>2^5</math> elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]] [[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two or more real elements found in different places in a concrete realization of the 11-cell.{{Sfn|Ruen: Hemi-icosahedron|2007}}]] The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we find a facet congruence between a 4-polytope and a 3-polytope we suspect a dimensional analogy. In the exceptional case of 5-cell and icosahedron, which share the same symmetry group <math>A_5</math>, we fully expect a dimensional analogy.{{Efn|There is an exceptional inter-dimensional duality between the regular icosahedron and the 5-cell because they share <math>A_5</math> symmetry. See this question asked on [https://math.stackexchange.com/questions/4235783/the-rotational-symmetry-groups-of-the-5-cell-and-the-icosahedron-are-isomorphi math.stackexchange.com 2021].}} Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization as the 6-point 5-simplex. Yet another real hexad is the 6-point 3-orthoplex; thus as a hexad the hemi-icosahedron is related by dimensional analogy to the 4-simplex (5-cell) from above, and to the 4-orthoplex (16-cell) from below, while those two simplest 4-polytope building blocks are only related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The cell of the 11-cell has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, the 4-simplex (5-cell) and 4-orthoplex (16-cell), who are famously stand-offish with each other. Interesting! The cell of the 11-cell is an abstract hexad hemi-icosahedron with 5 central planes, most handsomely illustrated by Séquin.{{Sfn|Séquin|2012|loc=A 10-Dimensional Jewel}}{{Sfn|Séquin & Lanier|2007|p=3|loc=Figure 4: (b,c) two views of the hemi-icosahedron projected into 3D space|ps=; Séquin et. al. have a lovely colored illustration of the hemi-icosahedron, subdivided into 10 triangular faces by 5 central planes of its icosahedral symmetry, revealing rings of polytopes nestled in its interior. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}}{{Sfn|Séquin & Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point [[W:Hemi-isosahedron|hemi-isosahedron]] is the vertex figure of the 11-cell's dual 4-polytope the 57-point [[W:57-cell|57-cell]].}} The 11 hemi-icosahedral cells have 10 triangle faces each, and each cell is face-bonded to the other 10 cells. The 5-cell's 5 tetrahedral cells have 10 faces and 10 edges altogether, and each cell is face-bonded to the other 4 cells. If 11-cell faces correspond to 5-cell faces, then 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, that does not belong to the same hemi-icosahedral cell as its opposing face. He found that the 10 edges opposite each hemi-icosahedron's 10 faces are the 10 edges of a single 5-cell, which does not share any vertices, edges or faces with the hemi-icosahedron. For each cell of the 11-point 11-cell, there is exactly one 5-point 5-cell that is completely disjoint from the 6-point hemi-icosahedron cell.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; "We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary <math>a_4</math> [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron."}} There are 11 disjoint 5-cell 4-polytopes inscribed in each 11-cell, which also contains 11 hemi-icosahedral cells, 55 faces, 55 edges and 11 vertices. The real 11-cell is more complex than the abstract 11-cell representing it, because the real hemi-icosahedron is more complex and harder to find than the abstract hemi-icosahedron. Seeing the real 11-cell will be easier once we have identified the real hemi-icosahedron, and seen exactly where the 11-cell's real elements reside in the other 4-polytopes within the 120-cell with which the 11-cell intermingles. The 5-cell has 10 faces, and the 11-cell has 10 faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron is face-bonded to the other 10 hemi-icosahedra, and to 10 of the 11 5-cells, and there is exactly one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point 5-cell is the other 5 vertices of the 11-point 11-cell that are not vertices of this 6-point hemi-icosahedron: the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces which belong to 10 distinct hemi-icosahedra of the 11-cell, and there is just one hemi-icosahedron with which it does not share a face. In the abstract 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 completely disjoint 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction.{{Efn|We shall see that this small eliminated distance is in fact the length of a 120-cell edge, the shortest chordal distance found in the 120-cell.}} Seemingly adjacent hemi-icosahedron faces do not actually meet at an edge; there is a polygon separating them, which has been abstracted to an edge. The 10 hemi-icosahedron faces are 5-cell faces from 10 distinct 5-cells, and they do not actually touch each other: the 120 5-cells in the 120-cell are completely disjoint. In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of tetrahedral cells together, because each 11-cell face represents two actual 5-cell faces in different places. Each duplex 11-cell face bonds tetrahedra in two 5-cells in different places, without binding the 5-cells together (they are completely disjoint). One actual 5-cell face is one half of a duplex 11-cell face, so 110 5-cell faces are 55 duplex 11-cell faces. The 11-cell's 11 abstract vertices represent all 55 distinct vertices of the 11 disjoint 5-cells, so they must be abstract conflations of at least 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; 11-cells must be sharing vertices, not disjoint as the 5-cells are. == The real hemi-icosahedron == [[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right| Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; "Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, & 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 & 5 are each pairs of Truncated icosahedrons. Hulls 6 & 8 are Rhombicosidodecahedrons."}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes starting with a dodecahedron cell. Hull #8 with 60 vertices (lower right) is a central section of the 120-cell, the 8th and largest section starting with a cell.{{Efn|1=Although the 8 hulls are illustrated as the same size, in the 120-cell they have increasing size as numbered, and occur nested inside each other like Russian dolls. Only Hull #8 is a central section of the same radius as the 120-cell itself, analogous to the equator. Sections 1-7 occur in pairs on opposite sides of the central section, and are analogous to lines of latitude. Section 1 is simply a dodecahedral cell. The "Combined hulls" is for illustrative purposes only; no such compound polyhedron exists in the 120-cell.}}]] We shall see in subsequent sections that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the realization of the abstract hemi-icosahedron, as an actual polyhedron that occurs in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''² = ''j''² = ''k''² = ''ijk'' = −1}} as the [[W:Tesseract|16-point (8-cell) tesseract]] 4-polytope. He did not realize then that he had discovered the 4-hypercube polytope and [[W:Tesseractic honeycomb|its Euclidean honeycomb]], the (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because [[w:Quaternions_and_spatial_rotation|quaternions make rotations]] and projections in 3D or 4D space as simple as matrix multiplications.{{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 quaternions are 4-hypercube building blocks, analogous to the 3-hypercube wooden blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the [[24-cell]], but we digress). Moxness used his software to render illustrations of polyhedra inside the 120-cell, some of which he published. Notice his "Hull # = 8 with 60 vertices", lower right in his illustration of the 120-cell sections starting with a cell. It is a real icosahedron that occurs in the 120-cell, and we shall see that the abstract hemi-icosahedron represents it. Moxness's 60-point Hull #8 is a concrete realization of the 6-point hemi-icosahedron in spherical 3-space <math>S^3</math>, embedded in Euclidean 4-space <math>\mathbb{R}^4</math>. Its 12 little pentagon faces are 120-cell faces. It also has 20 triangle faces like any icosahedron, separated from each other by rectangles, but beware: those triangles are not the 5-cell faces. They are smaller equilateral triangles, of edge length <math>1</math> in a {{radic|2}}-radius 120-cell, where the 5-cell face triangles have edge length {{radic|5}}.{{Efn|The 41.4° chord of edge length 1 in a {{radic|2}}-radius 120-cell occurs only in the 120-cell; it is not the edge of any smaller regular 4-polytope inscribed in the 120-cell. The equilateral triangle faces of Moxness's Hull #8 rhombicosidodecahedron are not the 5-cell faces of edge length <small><math>\sqrt{5} \approx 2.236</math> </small>(104.5°), not the 16-cell faces of edge length <small><math>2</math></small> (90°), not the 24-cell faces of edge length <small><math>\sqrt{2} \approx 1.414</math></small> (60°), and not the 600-cell faces of edge length <small><math>\sqrt{2}/\phi \approx 0.874</math></small> (36°).|name=Moxness 60-point triangle faces}} [[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges, in 200 irregular {12} dodecagon central planes. The 5-cell ''faces'' do not lie in central planes.]] Edges of the larger 5-cell face triangles of length {{radic|5}} can also be found in Hull #8, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that six 120-cell edges (little pentagon edges) lie on a great circle, alternating with six rectangle diagonals. Also lying on this irregular {12} great circle are six 5-cell edges, invisible chords joining every other 120-cell edge and running under the 120-cell edge between them. The six long chords and six short edges form two opposing irregular {6} great hexagons (truncated triangles) of alternating 5-cell edges and 120-cell edges, as illustrated. The irregular great {12} lies on a great circle of Moxness's Hull #8, and also on a great circle of the 120-cell, because Hull #8 is the ''central'' cell-first section of the 120-cell.{{Efn|The cell-first central section of the 600-cell (and of the 24-cell) is a cuboctahedron with 24-cell edges. The 120-cell is the regular compound of 5 600-cells (and of 25 24-cells), so Moxness's Hull #8, as the cell-first central section of the 120-cell, is the regular compound of 5 cuboctahedra. Their 24-cell edges, like the 5-cell edges, are invisible chords of Hull #8 that lie below its surface, on the same irregular {12} great circles. Each 24-cell edge chord spans one 120-cell edge chord (one little pentagon edge) and one rectangle face diagonal chord. Six 24-cell edge chords form a regular great {6} hexagon, inscribed in the irregular great {12} dodecagon.|name=compound of 5 cuboctahedra}} There are 10 great dodecagon central planes and 60 5-cell edges in Moxness's Hull #8, and 200 great dodecagon central planes and 1200 5-cell edges in the 120-cell. [[File:Central cell-first section of the 120-cell with 5-cell face triangle.png|thumb|Orthogonal projection of the cell-first central section of the 120-cell, Hull #8 rendered by Moxness, with one of 20 inscribed 5-cell faces (black chords) drawn under portions of three of its ten great circle {12} dodecagons (green).{{Efn|The point of view in this rendering is not quite right to best illustrate that a rhombicosidodecahedron triangle face lies over the center of a 5-cell face parallel to it, such that it would be perfectly inscribed in the center of the larger black triangle in an orthogonal view.}}]] But the 5-cell ''faces'' do not lie in those central planes. We can locate them in the 60-point polyhedron where they lie parallel to and under each small face triangle of edge length <math>1</math>. Truncating at a triangle face of Moxness's Hull #8 exposes a deeper 5-cell triangle face.{{Efn|Each face triangle of edge length <math>1</math> is surrounded by 3 rectangles, and beyond each rectangle by another face triangle. The distant vertices of those 3 surrounding triangles form a {{radic|5}} triangle, a 5-cell face.}} There are 20 such 5-cell faces inscribed in the Hull #8 polyhedron, all completely disjoint. We find 60 vertices, 60 edges and 20 faces of various 5-cells in each Hull #8 polyhedron, but no whole tetrahedral cells of the 5-cells.{{Efn|The fourth vertex of each 5-cell tetrahedron lies opposite the small face triangle of edge length <math>1</math> that lies over the 5-cell face. Since Moxness's Hull #8 polyhedron has opposing triangle faces (like any icosahedron), the fourth vertex of the 5-cell tetrahedron lies over the center of the opposing face, outside the Hull #8 polyhedron. This is a vertex of some other Hull #8 polyhedron in the 120-cell. Each tetrahedral cell of a 5-cell spans four Hull #8 polyhedra, with one face inscribed in each, and one vertex outside of each.}} [[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point Hull #8 is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation of the icosahedron with smaller red pentagons and narrower rhombs. Rhombicosidodecahedra are also made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the unique 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 60 of Moxness's Hull #8 rhombicosidodecahedron. Each occupies a central hyperplane, and so is analogous to an equator dividing the sphere in half.]] Moxness's Hull #8 is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point [[W:icosidodecahedron|icosidodecahedron]], which has the same relationship to Moxness's Hull #8 that the 6-point hemi-icosahedron does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The regular 5-point 5-cell can be another abstraction of Moxness's 60-point Hull #8, 12-vertices-into-1. None of these contractions of Moxness's Hull #8 is an instance of her operation actually described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length, but she did explicitly describe contractions of the semi-regular Archimedean rhomibicosidodecahedron.}} Moxness was not the first person to find rhombicosidodecahedra in the 120-cell. Alicia Boole Stott identified the 6th section of the 120-cell beginning with a cell as the semi-regular rhombicosidodecahedron that is her ''e₂ expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e₂ expansion|p=7}} But that 6th section rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8, it is the semi-regular Archimedean solid (Moxness's Hull #6), with a single edge length and square faces. Moxness's Hull #8, with its two distinct edge lengths and rectangular faces, is Coxeter's 8₃, the 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; "Alicia Boole Stott (1860-1940) ... also constructed the sections i₃ of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1₃-7₃; but 8₃ is missing. Incidentally, Diagram XIII (our 6₃) is a rhombicosidodecahedron, the Archimedean solid."}} Coxeter was the first to describe the central section 8₃, and he gave its coordinates, but he did not identify it as an irregular rhombicosidodecahedron. His table entry for its description is empty (characteristically, since it is not a regular or semi-regular polyhedron), so he gives us no indication that he actually visualized it. Although Moxness was not the first to compute the 60-point 8₃ section, he may have been the first person to ''see'' it. The 30-point icosidodecahedron is the quasi-regular product of 5-point pentagon and 6-point hexagon, recalling Coxeter's original discovery of the 11-cell in pentads and hexads, and also the two child's building blocks: one so useless the 5-point (pentad) 5-cell, and the other so useful the 8-point 16-cell with its four orthogonal 6-point (hexad) octahedron central sections, which can be compounded into everything larger. Some children building with the 30-point icosidodecahedron notice that it occurs as the central section 4₀ of the 120-point 600-cell. It is less often noticed that Moxness's Hull #8 rhombicosidodecahedron is the central section 8₃ of the 600-point 120-cell. It occupies a flat 3-dimensional hyperplane that bisects the 120-cell, and since there are 120 dodecahedral cells, there are 60 such central hyperplanes, each perpendicular to an axis that connects the centers of two antipodal cells. The 60 central hyperplanes, each containing an instance of Moxness's Hull #8, are rotated with respect to each other. They intersect, with 6 rhombicosidodecahedra sharing each vertex and 3 sharing each edge, but each little pentagon face (120-cell face) belongs to just one rhombicosidodecahedron. The 60 central sections lie in isoclinic hyperplanes, that is, the rhombicosidodecahedra are rotated symmetrically with respect to each other, by two equal angles.{{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.)"}} Each pair of rhombicosidodecahedra intersect in a central plane containing an irregular {12} dodecagon, unless they are completely orthogonal and intersect only at the center of the 4-polytope. Each of the 120 dodecahedral cells lies in the closed, curved 3-dimensional space of the 3-sphere as the 1st and smallest section beginning with a cell (section 1₃), the innermost of a series of concentric polyhedral hulls of increasing size, which nest like Russian dolls around it. Moxness's Hull #8 rhombicosidodecahedron is the 8th and largest concentric hull beginning with a cell (section 8₃), a central section of the 120-cell that bisects the 3-sphere the way an equator bisects an ordinary sphere.{{Efn|The 120-cell's curved 3-space surface is a honeycomb of 120 dodecahedron cells. In this 3-space a dodecahedron cell lies inside at the center of each section 8₃ rhombicosidodecahedron, face-bonded to 12 other dodecahedron cells which surround it, also inside the rhombicosidodecahedron. We find the opposite pentagon faces of those 12 surrounding cells on the surface of the section 8₃ rhombicosidodecahedron. These twelve dodecahedra surrounding one dodecahedron partially fill the volume of the rhombicosidodecahedron, leaving 30 concavities in its surface at the rectangle faces, and 12 deeper concavities between them at the triangle faces. 30 more dodecahedra fit into the rectangle concavities, lying half inside and half outside the rhombicosidodecahedron. The diagonal of each rectangle face is a long diameter of a dodecahedron cell. 12 more dodecahedra fit into the triangle face concavities, lying ....|name=dodecahedral cells in the section 8 rhombicosidodecahedron}} Such a central polyhedron is the dimensional analog of an equatorial great circle polygon. Its 60 vertices lie in the same 3-dimensional hyperplane, a flat 3-dimensional section sliced through the center of the 120-cell. There are 60 distinct stacks of 15 parallel section ''n''₃ hyperplanes in the 120-cell, one stack spindled on each axis that connects a dodecahedron cell-center to its antipodal dodecahedron cell-center. Each central section 8₃ has ''two'' disjoint sets of smaller sections nested within it, that lie in opposite directions from the 120-cell's center along its 4th dimension axis. The largest-radius central slice lies in the center of the stack, and the smaller non-central section hyperplanes occur in parallel pairs on either side of the central slice. The 120-cell therefore contains 120 instances of each kind of non-central section 1₃ through 7₃, and 60 instances of the central section 8₃.{{Efn|A central section is concave on its inside and also on its outside: it has two insides. It may be helpful to imagine the central 60-point section as two mirror-image 60-point polyhedra whose points are coincident, but which are convex in opposite directions: the inside of one is the outside of the other. Each has seven smaller polyhedra nested within itself, but their two volumes are disjoint.}} [[File:Tensegrity Icosahedron.png|thumb|[[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|Tensegrity icosahedron]] structure.{{Sfn|Burkhardt|1994}} First built by [[W:Kenneth Snelson|Kenneth Snelson]] in 1949. Geometrically a [[w:Jessen's_icosahedron|Jessen's icosahedron]] with 6 reflex ''long'' edge struts, and 24 ''short'' edge tension cables around 8 equilateral triangle faces. 3 pairs of parallel struts lie in 3 orthogonal central planes.]] We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that the 60 Moxness's Hull #8 rhombicosidodecahedra are real polyhedra which the abstract hemi-icosahedra represent in some manner, but we have not yet identified 11 real face-bonded cells, at 11 distinct locations in the 120-cell, as an 11-cell. The abstract hemi-icosahedron's 10 faces correspond to actual 5-cell faces inscribed in real rhombicosidodecahedra, and its 15 edges correspond to 5-cell edges (of length {{radic|5}} in a {{radic|2}}-radius 120-cell) that occur as chords lurking under the surface of the rhombicosidodecahedra. [[File:Buckminster-Fuller-holding-a-geodesic-tensegrity-sphere.png|thumb|200px|Buckminster Fuller holding a 3-dimensional geodesic tensegrity 2-sphere, an infinitesimally mobile rigid polytope consisting of tension cable edges and disjoint compression strut chords.<ref>{{Cite journal|last=Álvarez Elipe|first=Dolores|title=Ensegrities and Tensioned Structures|journal=Journal of Architectural Environment & Structural Engineering Research|date=July 2020|volume=3|issue=3|url=https://www.researchgate.net/publication/343652287_Ensegrities_and_Tensioned_Structures}}</ref>]] A rhombicosidodecahedron is constructed from a regular icosahedron by truncating its vertices, making them into pentagon faces. The regular icosahedron frames all the regular and semi-regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter & Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Sfn|Coxeter, du Val, Flather & Petrie|1938|p=4|ps=; "Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the "[[W:Golden section|golden section]]". Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the icosahedral group), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''."}} Before we can move on to locating the 11 discrete hemi-icosahedral cells of the 11-cell in the 120-cell, it is important that we take notice of one more icosahedral symmetry of the hidden {{radic|5}} chords lurking below the surface of Moxness's Hull #8 rhombicosidodecahedron. The 12 little pentagon faces (120-cell faces) are connected to each other in parallel pairs, by 10 sets of six disjoint {{radic|5}} chords (5-cell edges). Each six-chord set is the six reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], which is to say that the six disjoint chords are the parallel-orthogonal strut chords of a [[WikiJournal Preprints/Kinematics of the cuboctahedron#Elastic-edge transformation|tensegrity icosahedron]]. The six chords of each set are disjoint (they don't touch or form 5-cell faces), and they are symmetrically arranged as 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal {12} central planes.{{Efn|The Jessen's icosahedron has 8 equilateral triangle faces, which are not rhombicosidodecahedron triangle faces or 5-cell triangle faces, they are 24-cell triangle faces. Each 120-cell pentagon face lies at one end of 20 5-cell edges, from 20 distinct Jessen's icosahedra and five disjoint 5-cells: four at each pentagon vertex from each 5-cell.}} Five disjoint instances of the Jessen's icosahedron may be inscribed in each Moxness's Hull #8 rhombicosidodecahedron, their struts propping the rhombicosidodecahedron and the 120-cell itself open like a tensegrity structure.{{Efn|Moxness's Hull #8 rhombicosidodecahedron is a compound of five disjoint Jessen's icosahedra, because the 60 {{radic|5}} chords meet two-at-a-vertex and form 10 distinct Jessen's icosahedra: five disjoint Jessen's, in two different ways. The dimensionally analogous construction is the [[120-cell#Compound of five 600-cells|120-cell as a compound of five disjoint 600-cells]], in two different ways.}} But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again, and work our way patiently outwards, before in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we approach that rare child Bucky Fuller's orthogonal 12-point tensegrity icosahedron, an [[WikiJournal Preprints/Kinematics of the cuboctahedron|in-folded cuboctahedron]], the unique pyritohedral fish swimming deep in the 3-sphere ocean. == Eleven == Each pair of rhombicosidodecahedra that are not completely orthogonal intersect in a central plane containing an irregular {12} dodecagon. Ten irregular great dodecagons occur in each 60-point (central section 8₃) rhombicosidodecahedron, with 2 dodecagons crossing orthogonally at each vertex. Each rhombicosidodecahedron shares a {12} central plane with ten other rhombicosidodecahedra. ''Groups of 11 rhombicosidodecahedra share central planes pairwise.'' Here, at last, we find eleven of something, a group which must comprise an 11-cell. There are eleven {12} central planes in the group, with one of the eleven absent from each rhombicosidodecahedron. {|class="wikitable floatright" width=450 !colspan=2|Perspective views{{Efn|1=These images are ''non-orthogonal'' orthographic projections of the chords described in the caption. Those chords do not lie in a plane parallel to the projection plane, so they appear foreshortened.{{Efn|name=orthogonal triacontagram projections}} Consecutive chords of the helical Petrie polygon slant toward and away from the viewer. Any three consecutive chords, but no four, are edges of the same cell, in the 4-polytope whose edges are the chord.{{Efn|name=Petrie polygon of a honeycomb}}}} of a compound of six disjoint 5-cells in dual position |- ![[W:Triacontagon#Triacontagram|{30/12}{{=}}6{5/2} compound]] ![[W:Triacontagon#Triacontagram|{30/8}{{=}}2{15/4} compound]]{{Efn|name=orthogonal triacontagram projections|1=The {30/''n''} triacontagrams can each be seen as an ''orthogonal projection'' of the 120-cell showing all instances of the {30/''n''} chord. Each chord lies orthogonal to the line of sight, in a plane parallel to the projection plane. The diameter of the image is the diameter of the 120-cell. For example, the {30/8}=2{15/4} triacontagram is an orthogonal projection showing the 120-cell's 1200 {30/8} chords, the edges of 120 5-cells. Each edge of the triacontagram covers 40 5-cell edges, and each vertex covers 20 120-cell vertices. This projection can also be viewed as a compound of six 5-cells and their 30 unique vertices. But viewed that way, only 30 of the 60 5-cell edges are visible. Two edges meet at each vertex, but the other two are invisible. They are visible in the orthogonal view, the {30/4}=2{15} projection.}} |- valign=top |[[File:Regular_star_figure_6(5,2).svg|240px]]<BR>The 6{5/2} compound of six 5-cells. The six disjoint pentagrams in this view are six disjoint 5-cells.{{Efn|name=5-cell edges do not intersect is S³}} The 120-cell, with 120 disjoint 5-cells, is a compound of 20 of these compounds. All edges are 5-cell edges, but only five of each 5-cell's ten edges are shown. The other five edges, connecting the points of the six 5-cell pentagrams, are shown in the 6{5} projection below, the orthogonal view:<BR>[[File:Regular_star_figure_6(5,1).svg|240px]]These two views look straight down the orthogonal axes of a [[w:Duocylinder|duocylinder]], from inside the curved 3-dimensional space of the 120-cell's surface. They are like looking down a column of 5-cells stacked on top of one another in curved 3-space, but the column is actually circular: it is bent into a torus in the fourth dimension. |[[File:Regular_star_figure_2(15,4).svg|240px]]<BR>The 2{15/4} rotation circuits of the 5-cell isoclinic rotation. In this view, all edges are 75.5° chords of length {{radic|3}}, the 180° complement chord of the 5-cell edges of length {{radic|5}}.{{Efn|These are not 15-gons of 5-cell edges. There are no skew {15} polygons of 5-cell edges in the 120-cell. The 120 5-cells are completely disjoint, so the largest circuit along 5-cell edges is a skew {5}. Each vertex in the 120-cell is {{radic|5}} away from four and only four other vertices. No {{radic|5}} chords connect disjoint 5-cells; they are connected by several other chords. The skew {15} polygons are the discrete continuous spiral paths of moving vertices during an isoclinic rotation, and their edges are {{radic|3}} chords connecting 5-cells, not 5-cell edges.}} Each skew {15} polygon is the spiral chord-path of half the 30 vertices during the isoclinic rotation. The twined vertex orbits lie skew in 4-space; they form a circular double helix of two 15-gon spiral isoclines, winding through all four dimensions. These two completely orthogonal views look straight down an axis of a double helix cylinder, from inside the curved 3-dimensional space of the 120-cell's surface. Since the duocylinder is bent into a [[w:Clifford_torus|Clifford torus]] in the fourth dimension, the sightline axis in curved 3-space is a geodesic great circle in 4-space.<BR>[[File:Regular_star_figure_2(15,2).svg|240px]] |- ![[W:Triacontagon#Triacontagram|{30/6}{{=}}6{5} compound]] ![[W:Triacontagon#Triacontagram|{30/4}{{=}}2{15/2} compound]] |- |colspan=2|Images by Tom Ruen in [[W:Triacontagon#Triacontagram|Triacontagram compounds and stars]].{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}} |} Each shared {12} central plane contains six disjoint 5-cell edges, from six completely disjoint 5-cells. Each rhombicosidodecahedron contains 60 5-cell edges, which form 20 disjoint 5-cell faces within the rhombicosidodecahedron, under and parallel to its own 20 smaller triangle faces. Four 5-cell edges meet at each vertex at the 5-cell's tetrahedral vertex figure. Two 5-cell edges of a face within the rhombicosidodecahedron meet two edges belonging to other faces of the 5-cell: edges and faces outside the rhombicosidodecahedron, in some neighboring rhombicosidodecahedron.{{Efn|name=orthogonal triacontagram projections}} Each 5-cell face is shared by two tetrahedral cells of one 5-cell. It has its three 104.5° {{radic|5}} edges in three distinct {12} central planes, and is parallel to a fourth {12} central plane. In each rhombicosidodecahedron there are ten sets of five parallel planes: a {12} central plane, a pair of 5-cell faces on either side of it (from disjoint 5-cells), and a pair of rhombicosidodecahedron triangle faces. Each rhombicosidodecahedron is sliced into five parallel planes, ten distinct ways. There is no face sharing between 5-cells: the 120 5-cells in the 120-cell are completely disjoint. 5-cells never share any elements, but they are related to each other positionally, in groups of six, in the '''characteristic rotation of the regular 5-cell'''. That rigid isoclinic rotation takes the six 5-cells within each group to each other's positions, and back to their original positions, in a circuit of 15 rotational displacements.{{Sfn|Mamone, Pileio & Levitt|2010|loc=§4.5 Regular Convex 4-Polytopes, Table 2, Symmetry operations|pp=1438-1439|ps=; in symmetry group 𝛢₄ the operation [15]𝑹_q3,q3 is the 15 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 5-cell; in symmetry group 𝛨₄ the operation [1200]𝑹_q3,q13 is the 1200 distinct rotational displacements which comprise the class of pentadecagram isoclinic rotations of the 120-cell.}} Each displacement takes every 104.5° 5-cell edge of length {{radic|5}} to an edge 75.5° and {{radic|3}} away in another 5-cell in the group of six 5-cells. The 30 vertices of the six 5-cells rotate along 15-chord helical-circular isocline paths from 5-cell to 5-cell, before closing their circuits and returning the moving 5-cells to their original locations and orientations.{{Efn|In an [[24-cell#Isoclinic rotations|isoclinic rotation]], each point anywhere in the 4-polytope moves an equal distance in four orthogonal directions at once, on a [[W:8-cell#Radial equilateral symmetry|4-dimensional diagonal]]. The point is displaced a total [[W:Pythagorean distance|Pythagorean distance]] equal to the square root of four times the square of that distance. The orthogonal distance equals half the total Pythagorean distance. For example, when the {{radic|2}}-radius 5-cell rotates isoclinically 104.5° in the invariant central planes of its 104.5° edges of length {{radic|5}}, each vertex is displaced to another vertex 75.5° and {{radic|3}} away, moving {{radic|3/4}} in four orthogonal directions at once.|name=isoclinic 4-dimensional diagonal}} The six rotationally related 5-cells form a stellated compound, a non-convex 4-polytope with 30 star points.{{Efn|name=compound of six 5-cells}} The star compound, and the rotation of the 5-cell within it, are here illustrated by orthogonal projections from four different perspective viewpoints. To help us visualize the 4-polytopes within the 120-cell, we can examine 2-dimensional orthographic projections from various points of view. Such images filtered to include only chords of a single length are especially revealing, because they pick out the edges of a particular 4-polytope, or the isocline chords of its rotational orbits, the chords which link 4-polytopes together. No view of a single chord from a single point of view is sufficient by itself, but if we visualize various chords from various perspectives, we may imagine the 4-dimensional rotational geometry of interrelated objects within the 120-cell. The star compound as a whole has ten {12} central planes, like a rhombicosidodecahedron. Each {12} central plane contains one edge from each of the six 5-cells. Each {12} central plane is shared by two rhombicosidodecahedra in the group of eleven, and by six 5-cells in the group of six. == The eleventh chord == [[File:Major chord 11 of 135.5° in the 120-cell.png|thumb|The 120-cell contains 200 irregular {12} central planes containing 1200 135.5° {30/11} chords, six in each plane (shown in blue). They lie parallel to six 104.5° {30/8} chords (the 5-cell edges, shown in red), to which they are joined by 15.5° {30/1} 120-cell edges, and by 120° {30/10} great triangle edges (only one of the four great triangles is shown, in green).]] In addition to six 104.5° {30/8} 5-cell edge chords of length {{radic|5}}, the {12} central plane contains six 135.5° {30/11} chords of length <math>\phi^2</math>, parallel to the {{radic|5}} chords. The {30/11} chord spans an arc of five shorter chords: * 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} + 44.5° {30/4} + 15.5° {30/1} = 135.5° {30/11) * 15.5° {30/1} + 104.5° (30/8) + 15.5° {30/1} = 135.5° {30/11) * 15.5° {30/1} + 120° (30/10) = 135.5° {30/11) and its chord length is the linear sum of five shorter chords: * 1/𝜙^2 {30/1} + 1/𝜙^2 {30/1} + 1/𝜙 {30/2} + 1/𝜙 {30/2} + 1/𝜙 {30/2} = 𝜙^2 {30/11) Two distinct chords are always related to each other in two different ways: by their degrees-of-arc-difference, and by their linear difference chord. The 135.5° {30/11) chord is ''two'' 15.5° (30/1) 120-cell edge-arcs longer than the 104.5° (30/8) 5-cell edge chord. But the <math>\phi^2</math> {30/11} chord ''length'' is just ''one'' {30/1} 120-cell edge chord length longer than the {{radic|5}} {30/8} 5-cell edge chord.{{Efn|In a <small><math>\sqrt{2}</math></small>-radius 120-cell, the 15.5° {30/1} 120-cell edge chord has length <small><math>\phi^{-2}</math></small>. The 25.2° {30/2} pentagon face diagonal chord of length <small><math>\phi^{-1}</math></small> is <small><math>\phi</math></small> times the {30/1} edge length. The 41.1° 5-cell isocline chord of length <small><math>\sqrt{1}</math></small> is <small><math>\phi^2</math></small> times the {30/1} edge length. The 69.8° chord of length <small><math>\phi</math></small> is <small><math>\phi^3</math></small> times the {30/1} edge length. The 135.5° {30/11} 11-cell edge chord of length <small><math>\phi^2</math></small> is <small><math>\phi^4</math></small> times the {30/1} edge length.}} The {30/11} chord can be bisected into two shorter 120-cell chords in three different ways: * 15.5° {30/1} 120-cell edge + 104.5° {30/8} 5-cell edge = {30/11} chord * 25.2° {30/2} 120-cell pentagon face diagonal + 90° {30/15} 16-cell edge = {30/11} chord * 41.4° {30/1}+{30/2} chord + 69.8° {30/2}+{30/1}+{30/2} chord = {30/11} chord [[File:Regular_star_polygon_30-11.svg|thumb|The [[W:Triacontagon#Triacontagram|{30/11} regular triacontagram]] of the 11-cell rotation.{{Sfn|Ruen: Triacontagon|2011|loc=§Triacontagram compounds and stars}} In this 2-dimensional projection of a 30-edge 4-dimensional helix ring, the 30 chords pictured lie in 30 distinct central planes, and no two planes are orthogonal.]] The last of those bisections trisects the {30/11} chord into three distinct shorter chords: * 15.5° {30/1} + 25.2° {30/2} + 44.5° {30/4} chord = 135.5° {30/11} chord The {30/11} chords do not form triangle faces within the rhombicosidodecahedron the way the {30/8} chords do, but they do meet at a tetrahedral vertex figure. Groups of 11 rhombicosidodecahedra (an 11-cell) share central planes pairwise, including all the chords in the {12} central plane. When 11 things, all pairwise-adjacent to each other, are arranged in any circuit of 30 positions, there exists another pairwise circuit of 30 positions through every eleventh position, whether the things are 11 vertices, 11 rhombicosidodecahedra, or 11 [[w:Aardvark|aardvarks]] (although it might be unwieldy in practice to so arrange 11 live aardvarks, e.g. by tying them together pairwise with cords in both circuits). This intrinsic property of the [[w:Rational_number|rational number]] 30/11 is responsible for the existence of the {30/11} regular triacontagram (see illustration). The 11 rhombicosidodecahedra of the 11-cell are linked by a regular {30/11} triacontagram of 30 chords which runs through them. Each successive chord of the 30 in the triacontagram is shared by a distinct pair of rhombicosidodecahedra in the 11-cell group. An isoclinic rotation characteristic of the 11-cell takes the rhombicosidodecahedra in each 11-cell to each other's positions, pair by pair, in a circuit of 30 rotational displacements. It takes every {12} central plane to a Clifford parallel {12} central plane that is 44.5° away in two completely orthogonal angles. One 135.5° {30/11} chord separates each of the 12 vertex pairs. In this '''characteristic rotation of the 11-cell''' in its edge planes, the invariant planes are {12} central planes, the edges of the 11-cell are {30/11} chords, and the isocline chords of the vertex orbits are also {30/11} 11-cell edges, because the triacontagram is regular.{{Efn|In the 120-cell there are three ''regular isoclinic rotations'' in which the rotation edge and the isocline chord are the same chord. These rotations are each described by a [[W:Triacontagon#Triacontagram|regular triacontagram]]: the {30/7} rotation characteristic of the 16-cell in great square invariant planes, the {30/11} rotation characteristic of the 11-cell, and the {30/13} rotation.}} The 44.5° {30/4} chord of length <small><math>\sqrt{3}/\phi</math></small>, the 180° complement of the {30/11} chord, is the orthogonal distance between nearest parallel {30/11} chords.{{Efn|In its characteristic isoclinic rotation, a 4-polytope rotates an equal arc distance in each invariant {12} edge plane in each rotational displacement. In the 11-cell, every invariant plane rotates 44.5° (like a wheel), and tilts sideways 44.5° (like a coin flipping) in the completely orthogonal invariant plane, to occupy another invariant plane in the group of eleven. Each pair of original and destination {12} central planes are Clifford parallel and intersect only at one point (the center of the 4-polytope), but six other {12} central planes intersect them both. Two parallel {30/11} chords in each of the six spanning {12} central planes separate two vertex pairs in the original and destination planes, and these are the isocline chords over which the two vertices move in the rotation. None of the six spanning {12} central planes are contained in either the original or destination rhombicosidodecahedron. A total of ten {12} central planes span each original and destination rhombicosidodecahedron; they comprise a third rhombicosidodecahedron which does not belong to the group of eleven. The edges of an 11-cell and the isocline chords of an 11-cell are disjoint sets of {30/11} chords.}} The 60 vertices of each rhombicosidodecahedron rotate in parallel, on non-intersecting 30-chord spiral orbital paths, from rhombicosidodecahedron to rhombicosidodecahedron, before closing their circuits and returning the moving rhombicosidodecahedron to its original location and orientation. In this isoclinic rotation of a rigid 120-cell, the 60 rhombicosidodecahedra do this concurrently. Each of the 600 vertices moves on a 4-dimensionally-curved helical isocline, over a skew regular polygram of 30 {30/11} chords, in which a {30/11} chord connects every eleventh vertex of a {30} triacontagram. In the course of a complete revolution (the 30 rotational displacements of this isoclinic rotation), an 11-cell visits the positions of three 11-cells (including itself) 10 times each (in 10 different orientations), and returns to its original position and orientation.{{Sfn|Coxeter|1984|loc=§9. Eleven disjoint decagons}} At each step it occupies the same distinct group of 11 rhombicosidodecahedra sharing planes pairwise, and its 11 vertex positions are those of a distinct 11-cell in the group of eleven 11-cells. A group of 4-polytopes related by an isoclinic rotation is contained in a larger compound 4-polytope which subsumes them. This group of eleven 11-cells related by an isoclinic rotation is not a compound of eleven disjoint 11-cells (since they share vertices), but it is a compound of eleven non-disjoint 11-cells, in the same sense that a 24-cell is a compound of three non-disjoint 8-cell tesseracts. Consider the incidence of these 30-chord {30/11} triacontagram rotation paths, and their intersections. Each rhombicosidodecahedron has 60 vertices and 60 {30/11} chords, which rotate concurrently on Clifford parallel triacontagrams. The 120-cell has only 600 vertices and 1200 {30/11} chords, so at most 20 triacontagrams can be disjoint; some must intersect. But the 11 vertices of an individual 11-cell must be linked by disjoint 30-position {30/11} triacontagram helices, such that their rotation paths never intersect.{{Efn|The isoclines on which a 4-polytope's vertices rotate in parallel never intersect. Isoclinic rotation is a concurrent motion of Clifford parallel (disjoint) elements over Clifford parallel (non-intersecting) circles.}} Each 11-cell has two disjoint triacontagram helicies, its left and right isoclinic rotations, in each of its four discrete fibrations. The 120-cell has 60 distinct {30/11} triacontagram helices, which are 11 disjoint {30/11} triacontagram helices in 11 distinct ways. {{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} == Compounds in the 120-cell == [[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 for 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 other 5 regular convex 4-polytopes]]. {{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=1}} === How many building blocks, how many ways === The 120-cell is the convex hull of a compound of 75 disjoint 16-cells, of 25 disjoint 24-cells, of 5 disjoint 600-cells, and of 120 disjoint regular 5-cells. Children building the 120-cell up from their 16-cell building blocks will soon learn to protect their sanity by thinking of these nesting 4-polytopes by their alternate names, as ''n''-points symmetrically distributed on the 3-sphere, as synonyms for their conventional names, as ''n''-cells tiling the 3-sphere. They are the 8-point (16-cell), the 16-point (8-cell) tesseract, the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell). The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block, which compounds to everything else. The 8-point compounds by 2 into the 16-point, and by 3 into the 24-point; what could be simpler? The 16-point compounds into the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our children are happily building, and their castle makes sense to them. Then things get hairy. The 24-point also compounds by <math>5^2</math> non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. Whichever way the child builds it, the resulting 120-point, magically, contains 25 distinct 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point, which then magically contains 75 distinct 8-points. [[File:Ortho solid 016-uniform polychoron p33-t0.png|thumb|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}} discovered by [[W:Ludwig Schläfli|Ludwig Schläfli]]. Named by [[W:John Horton Conway|John Horton Conway]], extending the naming system by [[W:Arthur Cayley|Arthur Cayley]] for the [[W:Kepler-Poinsot polyhedron#Characteristics|Kepler-Poinsot solids]], and the only one containing all three modifiers in the name.]] The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times <math>5^2</math> (75) disjoint instances of itself into the 600-point, which then magically contains <math>3^2</math> times <math>5^2</math> (225) distinct instances of the 24-point, and <math>3^3</math> times <math>5^2</math> (675) distinct instances of the original 8-point. They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[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''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 120-point 600-cell as perspicaciously as Coxeter did from the permutations of the 11-point 11-cell.}} 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]], the final regular [[W:Stellation|stellation]] of the 120-cell, 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:Stellation core|stellation core]] deep inside. The compound of 120 regular 5-cells can be seen to be equivalent to the compound of 5 disjoint 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. === Building the building blocks themselves === We have built every regular 4-polytope except the 5-cell out of 16-cells, but we haven't made the 16-cell (or the 5-cell) out of anything. So far, we have just accepted them both a priori, like [[W:Euclid's postulates|Euclid's postulates]], and proceeded to build with them. But it turns out that while they are the two atomic regular 4-polytopes, they are not indivisible, and can be built up as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct [[w:Symmetry_group|symmetry group]]. Every regular convex ''n''-polytope can be subdivided into instances of its characteristic [[W:Orthoscheme|Schläfli orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-[[w:Orthoplex|orthoplex]]!) is an ''irregular'' ''n''-[[w:Simplex_(geometry)|simplex]] with faces that are various right triangles instead of congruent equilateral triangles. A characteristic ''n''-orthoscheme possesses the complete symmetry of its ''n''-polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements. It is the gene for the polytope, which can be replicated to construct the polytope.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope because 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 partitioned radially by its planes of symmetry (Coxeter's "reflecting circles") into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme and its chiral mirror image can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its <math>A_4</math> planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into <small><math>120\times 120 = 14400</math></small> of these 4-orthoschemes, so that is the symmetry order of the 120-cell. The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16-cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its <math>B_4</math> planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order <small><math>75\times 384 / 2 = 14400</math></small>. The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its <math>F_4</math> planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order <small><math>25\times 1152 / 2 = 14400</math></small>. The regular 120-point (600-cell) is subdivided into 14400 instances of its [[600-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its <math>H_4</math> planes of symmetry at once intersecting at its center, so its symmetry is of order 14400. The regular 600-point (120-cell) is the convex hull of the regular compound of 5 disjoint 600-cells (which have 5-fold reflective symmetry), so its symmetry is of order <small><math>5 \times 14400 / 5 = 14400</math></small>. === Building with sticks === [[File:15 major chords.png|thumb|300px|The 15 major chords {30/1} ... {30/15} join vertex pairs which are 1 to 15 edges apart on a skew {30} [[w:Petrie_polygon|Petrie polygon]] of the 120-cell.{{Efn|Drawing the fan of major chords with #1 and #11 at a different origin than all the others was an artistic choice, since all the chords are incident at every vertex. We could just as well have fanned all the chords from the same origin vertex, but this arrangement notices the important parallel relationship between #8 and #11, and calls attention to the 11-cell's maverick edge chord.|name=fan of 15 major chords}} The 15 minor chords (not shown) fall between two major chords, and their length is the sum of two other major chords; e.g. the 41.4° minor chord of length {30/1}+{30/2} falls between the 36° {30/3} and 44.5° {30/4} chords.]] We have seen how all the regular convex 4-polytopes except the 5-cell, including the largest one on the cover of the box, can be built from a box containing 675 16-cell building blocks, provided we can arrange the blocks on top of one another in 4-space, as interpenetrating objects. An alternate box, containing 120 regular 5-cell building blocks, builds the great grand stellated 120-cell (the picture on ''its'' cover), by the same method. In these boxes, the atomic building part is one of the two smallest regular 4-polytopes (5-cell or 16-cell), each generated by its characteristic isoclinic rotation as an expression of its symmetry group (<math>A_4</math> or <math>B_4</math>). All the regular convex 4-polytopes, including the largest one on the cover of the box, can also be built from a box containing a certain number of building sticks and rubber joints, provided we can connect the sticks together in 4-space with the rubber joints. In this box, the atomic building parts are 1-dimensional edges and chords of just 15 distinct arc-lengths. The regular 4-polytopes do not contain a vast variety of stick lengths, but only 30 of them: only 15 unique pairs of 180° complementary chords. The 15 ''major chords'' {30/1} ... {30/15} suffice to construct all the regular 4-polytopes. The 15 ''minor chords'' occur only in the 120-cell, not in any smaller regular 4-polytope; they emerge as a consequence of building the largest 4-polytope on the cover of the box from major chords. In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the unit-radius sphere of a distinct length <math>l</math>, and a distinct rational number <math>h</math>, a unique flavor. If the polytope is regular, it is a noteworthy distinctive flavor. The chord's length <math>l</math> is a square root, related to the rational number <math>h = k/d</math> and to the polygon <small><math>\{k/d\}</math></small> it represents, by a formula discovered by Steinbach.{{Sfn|Steinbach|1997|loc=''Golden Fields''; §1. The Diagonal Product Formula|pp=22-24|ps=; The product of two diagonals is a sum of a sequence of diagonals (in the fan, every other one) centered on the longer of the two, for all regular polygons. We may express products and quotients of diagonals <math>d_k</math> of an <math>n</math>-gon (with edge <math>d_0=1</math>) as linear combinations of diagonals.}} The chord length <math>l</math> is related to the number of sides of the regular polygon <small><math>\{k\}</math></small>, and to the winding number or density of the polygram (its denominator <math>d</math>).{{Sfn|Kappraff & Adamson|2004}} The largest <math>k</math> of any major chord in the 120-cell is 30, and the polygrams <small><math>\{30/d\}</math></small> represent all the skew Petrie polygons and characteristic isoclinic rotations of the regular 4-polytopes. == Concentric 120-cells == The 8-point 16-cell, not the 5-point 5-cell, is the smallest regular 4-polytope which compounds to every larger regular 4-polytope. The 5-point 5-cell is also an atomic building block, but one that compounds to nothing else regular except the leviathan 120-cell polytope: the picture on the cover of the box, that is built from everything in the box. In the [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|sequence of 4-polytope compounds]], we actually start with the 16-cell at the small end, and the 5-cell emerges only at the large end. To build with the 16-cell blocks, we simply put them on top of each other as interpenetrating compounds. We can build every other regular 4-polytope from them by that method, except the individual regular 5-cell. We can also try to build with the 5-cell that way, as when we tried to build a 4-polytope of 11 hemi-icosahedral cells from 11 5-cells, but that was rather hard going. We somehow found 5-cell edges and faces lurking inside hemi-icosahedral rhombicosidodecahedra, and 11 rhombicosidodecahedra sharing central planes pairwise, and even the edges and characteristic rotation of the 11-cell, but we didn't quite get all the way to a discrete 11-cell 4-polytope made from 11 5-cells. That's because ''compounding'' isn't the easiest method for building with the 5-cell. The 5-cell is the last building block hierarchically, not the first, and the most natural way to build with it is in reverse, by ''subdividing'' it, to find all the parts inscribed inside it. When we've taken the 5-cell apart, all the ways we possibly can, into certain ''irregular'' 4-polytopes found within it, we will have a new set of irregular 4-polytope building blocks, which compound to the 5-cells and everything else, including the 11-cells. Subdividing a polytope is done by a geometric operation called ''[[w:Truncation_(geometry)|truncation]]''. There are myriad ways to truncate a 5-cell, each corresponding to a distinct ''depth'' of truncation at a particular point on an edge, or a line on a face, or a face on a cell, where a piece of the 5-cell is cut off. The simplest truncations, such as [[w:Rectification_(geometry)|cutting off each vertex at the midedge of each incident edge]], have been very well-studied; but how should we proceed? Let us see what happens when we [[w:Truncated_5-cell|truncate the 5-cells]] found in the 120-cell, by the simplest kinds of truncation. These three semi-regular 10-cells are closely related truncations of the regular 5-cell: * The 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of six 5-cells. * The 20-point 10-cell [[w:Truncated_5-cell|truncated 5-cell]] is the convex hull, and the convex common core, of a stellated compound of four 5-cells. * The 10-point 10-cell [[w:Rectified_5-cell|rectified 5-cell]] is the convex hull, and the convex common core, of a stellated compound of two 5-cells. In the following sections, we explore the effect of performing these truncations on the 120-cell's 120 5-cells. We begin by identifying some promising truncation points on the 120-cell's 5-cell edge chords at which to cut. If we cut off the 120-cell's 600 vertices at some point on its 1200 5-cell edges, we create new vertices on the edges of the 120 5-cells, which lie on a smaller 3-sphere than the 120-cell. How many vertices does the smaller 4-polytope thus created have? That is, how many distinct 5-cell edge truncation points occur in the 120-cell? As many as 1200, the number of 5-cell edges, or perhaps 2400, if each edge is truncated at both ends. But also perhaps fewer; for example, if the 120-cell contains pairs of 5-cells with intersecting edges, and the edges intersect at the point on each edge where we make our cut. [[File:Great_(12)_chords_of_radius_√2.png|thumb|400px|Chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {{radic|2}} chords form two regular {6} hexagons (black).{{Efn|name=compound of 5 cuboctahedra}} The 120-cell edges form two irregular {6} hexagons (red truncated triangles) with the {{radic|5}} chords. The {6} intersection points (black) of the {{radic|5}} chords form a smaller red regular hexagon of radius {{radic|1}} (inscribed in the red circle).]]In the irregular {12} central plane chord diagram, we see six truncation points on the six 104.5° 5-cell edges of length {{Radic|5}}, where two co-planar 5-cell edges intersect, directly under the midpoint of a 44.5° chord (and under the intersection point of two 60° chords). The six truncation points lie on a red circle that is a circumference of the smaller 4-polytope created by this truncation. They form a red regular hexagon inscribed in the red circle. The edge length of this regular hexagon is {{radic|1}}. The two intersection points on the {{Radic|5}} chord divide it into its golden sections. The center section of the chord is <small><math>1</math></small>. The center section plus either of the smaller sections is <small><math>\phi = \tfrac{\sqrt{5} + 1}{2} \approx 1.618</math></small>, the larger golden section. Each of the two smaller sections is <small><math>\Phi = \phi - 1 = \tfrac{1}{\phi} \approx 0.618</math></small>, the smaller golden section.{{Efn|The bitruncated {30/8} chord of the 120-cell of radius <small><math>\sqrt{2}</math></small> provides a "proof by geometric picture" of the golden ratio formulas. First, consider a 120-cell of radius <small><math>2\sqrt{2}</math></small> in which the {30/8} chord is <small><math>2\sqrt{5}</math></small> and the center section of the chord is <small><math>2</math></small>. Divide the lengths of its golden sections by <small><math>2</math></small> to get their radius <small><math>\sqrt{2}</math></small> lengths. The left section of the chord is: :<small><math>\tfrac{\sqrt{5} - 1}{2} \approx 0.618</math></small> The center section plus the right section is: :<small><math>\tfrac{1 + \sqrt{5}}{2} \approx 1.618</math></small> }} The smaller golden sections <small><math>\Phi \approx 0.618</math></small> of the 5-cell edge are the same length as the 120-cell's 25.2° pentagon face diagonal chords. No 25.2° chords appear in the {12} central plane diagram, because they do not lie in {12} central planes. Each 104.5° 5-cell edge chord of length {{Radic|5}} has ''two'' points of intersection with other 5-cell edges, exactly 60° apart, the ''arc'' of a 24-cell edge chord, but with ''length'' {{radic|1}}. The center segment of the 5-cell edge, between the two intersection points, is a 24-cell edge in the smaller 4-polytope, and the red hexagon is a [[24-cell#Great hexagons|24-cell's great hexagon]] in the smaller 4-polytope. Nine other of its great hexagons, in other planes, each intersect with an antipodal pair of these {6} vertices. The dihedral angles between hexagon planes in a 24-cell are 60°, and four great hexagons intersect at each vertex. The 1200 5-cell edges, with two intersection points each, are reduced to 600 distinct vertices, so the smaller 4-polytope is a smaller 120-cell. The larger 120-cell, of radius {{radic|2}}, is concentric to a smaller instance of itself, of radius {{radic|1}}. Each 120-cell contains 225 distinct (25 disjoint) inscribed 24-cells. The smaller 24-cells are the [[w:Inscribed_sphere|insphere]] duals of the larger 24-cells. The vertices of the smaller 120-cell are located at the octahedral cell centers of the 24-cells in the larger 120-cell. Four 5-cell edges meet in 600 tetrahedral vertex figures. Four orthogonally intersecting 5-cell edges of the larger 120-cell meet in cubic vertex figures of 24-cells in the smaller 120-cell. Two disjoint 5-cell tetrahedral vertex figures are inscribed in alternate positions in each 24-cell cubic vertex figure. The 24-cell edges of the smaller 120-cell are the 5-cell edges of the larger 120-cell, truncated at both ends. The distance between the two points of intersection on a {{radic|5}} chord is {{radic|1}}, the same length as the 41.4° chord. But the actual 41.4° chords of the 120-cell do not appear in this diagram at all, because they do not lie in the 200 irregular {12} dodecagon central planes. === Bitruncating the 5-cells === The smaller concentric 120-cell can be built from 5-cell building blocks, by applying a specific kind of truncation operation to the blocks of the larger 120-cell called [[w:Bitruncation|''bitruncation'']]. This reveals a smaller irregular 4-polytope inside each 5-cell called the [[w:Bitruncated_5-cell|bitruncated 5-cell]]. The smaller unit-radius 120-cell is the convex hull of a compound of 20 disjoint (and 60 distinct) bitruncated 5-cells, bitruncated from the 120 disjoint 5-cells of the larger {{Radic|2}}-radius 120-cell. Bitruncation of the 120 disjoint 5-cells is the same truncation of the 120-cell described in the previous section, at the two golden section truncation points on each 104.5° 5-cell edge where two co-planar 5-cell edges intersect. [[File:Truncatedtetrahedron.gif|thumb|A 12-point [[w:Truncated_tetrahedron|truncated tetrahedron]] cell of the 30-point 10-cell [[w:Bitruncated_5-cell|bitruncated 5-cell]].{{Sfn|Cyp: Truncated tetrahedron|2005}} Its edges are 41.4° chords of length 1 in a {{radic|2}}-radius 120-cell (or length {{radic|1/2}} in a unit-radius 120-cell). The 120-cell contains 20 disjoint (60 distinct) bitruncated 5-cells, containing 600 distinct truncated tetrahedra.]] The bitruncated 5-cell is a 30-vertex convex 4-polytope with 10 [[W:Truncated tetrahedron|truncated tetrahedron]] cells that have faces of two kinds: 4 triangle faces opposite 4 hexagon faces. The bitruncated 5-cell has 60 edges of the same length, 20 triangle faces, and 20 hexagon faces. Its 20 hexagon face planes are not [[24-cell#Great hexagons|24-cell central plane hexagons]]; they intersect each other at their edges, not at their long diameters. Its edges are not 60° 24-cell edge chords (the {{radic|2}} or 1 radius chords), but shorter 41.4° chords (of length 1 or {{radic|1/2}}), which do not appear at all in the diagram above, because they do not lie in the {12} central planes. The long diameter of the hexagon faces is not a 180° 120-cell long diameter chord (of length 2{{radic|2}} or 2) but a 90° 16-cell edge chord (of length 2 or {{radic|2}}). Consequently, three 16-cell tetrahedron cells (from three disjoint 16-cells) are inscribed in each truncated tetrahedron, at the three vertices of each face triangle. The truncated tetrahedron cell is a truncation of a tetrahedron of the same size as the tetrahedral cells of the 120-cell's 5-cells. The four smaller tetrahedra truncated from the corners of the larger tetrahedron have edges which are 25.2° chords (of length 1/𝜙 or {{radic|0.19}}). The truncated tetrahedron edges (of length 1 or {{radic|1/2}}) are equal in length to the 41.4° center sections of the 104.5° 5-cell edge chords (of length {{radic|5}} or {{radic|5/2}}). The shorter diagonal of the hexagon faces is the 75.5° chord (of length {{radic|3}} or {{radic|1.5}}), which is the 180° complement of the 104.5° 5-cell edge chord. The dimensions of the truncated tetrahedron cell suggest that it was cut directly from a 5-cell tetrahedron cell, simply by cutting off the tetrahedron corners, but remarkably, that is not the case. The edges of the bitruncated 5-cell are not actually center sections of 5-cell edges, although they are exactly that length, because the edges of the bitruncated 5-cell do not lie in the same {12} central planes as the 5-cell edges. They are not colinear with 5-cell edges in any way, and only intersect 5-cell edges at vertices (the 5-cell edges' intersection points). Bitruncation of the 5-cells does ''not'' simply truncate each tetrahedron cell in place. By creating new edges which connect the intersection points of 5-cell edges, bitruncation does create 600 truncated tetrahedron cells perfectly sized to fit within the 600 original tetrahedron cells, but at new locations, not centered on an original 5-cell tetrahedron cell. These new locations lie on a smaller 3-sphere than the original locations. [[File:Bitruncated_5-cell_net.png|thumb|Net of the bitruncated 5-cell honeycomb. 10 truncated tetrahedron cells alternately colored red and yellow.{{Sfn|Ruen: Net of the bitruncated 5-cell|2007}}]] The 3-dimensional surface of each bitruncated 5-cell is a honeycomb of 10 truncated tetrahedron cells. The truncated tetrahedra are joined face-to-face in a 3-sphere-filling honeycomb (like the cells of any 4-polytope), at both their hexagon and triangle faces. Each hexagonal face of a cell is joined in complementary orientation to the neighboring cell. Three cells meet at each edge, which is shared by two hexagons and one triangle. Four cells meet at each vertex in a [[w:Tetragonal_disphenoid|tetragonal disphenoid]] vertex figure. The 30-point bitruncated 5-cell is the convex common core (spatial [[w:Intersection|intersection]]) of six 5-point 5-cells in dual position. These six 5-cells are completely disjoint: they share no vertices, but their edges intersect orthogonally, at two points on each edge. Four 5-cell edges, from four of the six 5-cells, cross orthogonally in 30 places, the two intersection points on 60 5-cell edges: the 30 vertices of a bitruncated 5-cell. The six 5-cells are three dual pairs (in two different ways) of the self-dual 5-cell: six pairs of duals reciprocated at their common midsphere. Each dual pair intersects at just one of the two intersection points on each edge.{{Sfn|Klitzing|2025|loc=''sted'' (Stellated Decachoron)|ps=; [https://bendwavy.org/klitzing/incmats/sted.htm ''sted''] is the compound of two [https://bendwavy.org/klitzing/incmats/pen.htm ''pen'' (Pentachoron)] in dual position. Their intersection core ("Admiral of the fleet") is [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)].}} We have seen these six 5-cells before, illustrated in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Eleven|§Eleven]]'' above; they are the compound of six completely disjoint 5-cells visited during each 5-cell's characteristic isoclinic rotation of period 15.{{Efn|1=The 5-cell edges of the six disjoint pentagrams in the {30/12}=6{5/2} triacontagram illustration do not appear to intersect, as the 5-cell edge chords of the bitruncated 5-cell compound are said to intersect. The {30/12}=6{5/2} projection is a perspective view from inside the curved 3-dimensional space of the 120-cell's surface, looking straight down a cylindrical column of six stacked 5-cells. None of the 5-cell edges intersect in that curved 3-space, except where they meet at the 30 120-cell vertices. The 60 5-cell edges do intersect orthogonally in 4-space, in groups of four, at 30 points which lie on a smaller 3-sphere than the 120-cell. None of those 4-space intersections are visible in these projections of points and lines on the 120-cell's 3-sphere surface.|name=5-cell edges do not intersect is S³}} The six 5-cell compound is a stellated 4-polytope with 30 star-points, inscribed in the 120-cell.{{Efn|The stellated compound of six 5-cells in dual position is three pairs of 5-cells reciprocated at their common midsphere. It is composed of dual pairs of the [[W:Compound of five tetrahedra|compound of five tetrahedra]], which form the [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]; its 30 tetrahedral cells are three such dual pairs. In the compound of five tetrahedra the edges of the tetrahedra do not intersect. In the compound of ten tetrahedra they intersect orthogonally, but not at their midpoints. Each edge has two points of intersection on it. The compound of ten tetrahedra is five pairs of dual tetrahedra reciprocated at their common midsphere. It is inscribed in a dodecahedron (its convex hull). Its ''stellation core'' is an icosahedron, but its ''common core'' where the tetrahedron edges intersect is a dodecahedron, the tetrahedrons' convex spatial intersection. The stellated compound of six 5-cells has the analogous property: it is inscribed in a bitruncated 5-cell (its convex hull), and its common core is a smaller bitruncated 5-cell. (Its stellation core is a [[W:Truncated 5-cell#Disphenoidal 30-cell|disphenoidal 30-cell]], its dual polytope.)|name=compound of six 5-cells}} It is 1/20th of the 600-point [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#How many building blocks, how many ways|great grand stellated 120-cell]], the compound of 120 5-cells. The convex hull of its 30 star-points is a bitruncated 5-cell. In this stellated compound of six 5-cells in dual position, the bitruncated 5-cell occurs in two places and two sizes: as both the convex hull, and the convex common core, of the six 5-cells. Inscribed in the larger 120-cell of radius {{radic|2}}, the convex hull of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length 1. The convex common core of every six 5-cell compound is a bitruncated 5-cell with 60 edges of length {{radic|1/2}}, inscribed in the smaller 120-cell of radius 1. In the 120-cell, 120 disjoint 5-cell building blocks combine in dual position groups of six related by the 5-cell's isoclinic rotation, to make 60 bitruncated 5-cells inscribed in the self-dual 5-cells' midsphere (at their edge intersections), and also 60 larger bitruncated 5-cells inscribed in the 120-cell, with each of the 600 vertices shared by three bitruncated 5-cells. The 120-cell is the convex hull of a compound of 20 disjoint (60 distinct) 30-point bitruncated 5-cells, generated by the characteristic rotation of its 120 completely disjoint 5-cells.{{Sfn|Klitzing|2025|loc= ''teppix'' (tripesic hexacosachoron)|ps=; ''[https://bendwavy.org/klitzing/incmats/teppix.htm teppix]'' is a compound of 60 [https://bendwavy.org/klitzing/incmats/deca.htm ''deca'' (decachoron aka bitruncated pentachoron)] with 3 ''deca'' sharing each vertex.}}{{Efn|In the 120-cell, 600 tetrahedron cells of 120 completely disjoint 5-cells intersect at two truncation points on each edge. Those 2400 truncation points are the vertices of 200 disjoint (and 600 distinct) truncated tetrahedra, which are the cells of 20 disjoint (and 60 distinct) bitruncated 5-cells. The 60 bitruncated 5-cells share vertices, but not edges, faces or cells. Each bitruncated 5-cell finds its 30 vertices at the 30 intersection points of 4 orthogonal 5-cell edges, belonging to 6 disjoint 5-cells, in the original 120-cell. Each bitruncated 5-cell vertex lies on an edge of 4 disjoint original 5-cells. Each bitruncated 5-cell edge touches intersection points on all 6 disjoint original 5-cells, and is shared by 3 truncated tetrahedra of just one bitruncated 5-cell.}} In [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Concentric 120-cells|the previous section]] we saw that the six 5-cell edges in each central plane intersect at the {6} vertices of the red hexagon, a great hexagon of a 24-cell. Each 5-cell edge, truncated at both ends at those intersection points, is a 24-cell edge of one of the 24-cells inscribed in a smaller 120-cell: the 600 intersection points. In this section we have seen how that truncation of 5-cell edges at both ends is the bitruncation of the 5-cell, and those 5-cell edges, truncated at both ends, are the same length as edges of bitruncated 5-cells inscribed in the original 120-cell. Bitruncating the {{radic|2}}-radius 120-cell's 120 5-cells reveals a smaller unit-radius 120-cell. The 24-cell edges of the smaller 120-cell are 5-cell edges of a larger-radius-by-{{radic|2}} 120-cell, truncated at both ends. Both 120-cells have 24-point 24-cells and 30-point bitruncated 5-cells inscribed in them. The 60° edge length of the 24-cells equals the radius; it is {{radic|2}} times the 41.4° edge length of the bitruncated 5-cells. The 60° 24-cell edges lie in the {12} central planes with the 5-cell edges and the 120-cell edges; but the 41.4° bitruncated 5-cell edges do not. The 120-cell contains 25 disjoint (225 distinct) 24-cells, and 20 disjoint (60 distinct) bitruncated 5-cells. Although regular 5-cells do not combine to form any regular 4-polytope smaller than the 120-cell, the 5-cells do combine to form semi-regular bitruncated 5-cells which are subsumed in the 120-cell.{{Efn|Although only major chords occur in regular 4-polytopes smaller than the 120-cell, minor chords do occur in semi-regular 4-polytopes smaller than the 120-cell. Truncating the 5-cell creates minor chords, such as the 41.1° edges of the bitruncated 5-cell.}} The 41.4° edge of the 30-point bitruncated 5-cell is also the triangle face edge we found in the 60-point central [[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The real hemi-icosahedron|section 8₃ (Moxness's Hull #8) rhombicosidodecahedron]]. There are 60 distinct section 8₃ rhombicosidodecahedra and 600 distinct truncated tetrahedron cells of 60 distinct (20 disjoint) bitruncated 5-cells, and they share triangle faces, but little else. The truncated tetrahedron cells cannot be inscribed in the rhombicosidodecahedra, and the only chords they share are the 41.4° triangle edge and the 75.5° chord (the 180° complement of the 104.5° 5-cell edge chord). The section 8₃ rhombicosidodecahedron's 20 triangle faces lie over the centers of 20 larger-by-√2 5-cell faces, parallel to them and to a {12} central plane. The 5-cell faces are inscribed in the rhombicosidodecahedron, but are not edge-bound to each other; the 20 faces belong to 10 completely disjoint 5-cells. The 5-cell edges (but not the 5-cell faces) lie in {12} central planes; the 5-cell faces, the bitruncated 5-cell edges and their triangle and hexagon faces do not. Each section 8₃ rhombicosidodecahedron is the intersection of ten {12} central planes, shared pairwise with ten other rhombicosidodecahedra; 11 rhombicosidodecahedra share ten {12} central planes pairwise, as cells of a 4-polytope share face planes pairwise. Each truncated tetrahedron cell of a bitruncated 5-cell shares none of the {12} central planes; it is the intersection of 6 great rectangles, with two parallel 41.1° edges lying in each, alternating with two parallel 138.6° chords (its hexagon face diameters). Each bitruncated 5-cell is the intersection of 30 great rectangle {4} central planes. A truncated tetrahedron is face-bonded to the outside of each triangle face of a rhombicosidodecahedron. Three of its hexagon faces stand on the long edge of a rectangle face, perpendicular to the rectangle. We find the 25.2° chord as the edge of the non-central section 6₃ (Moxness's Hull #6) rhombicosidodecahedron. Those 120 semi-regular rhombicosidodecahedra have only that single edge (of length 1/𝜙 in a {{radic|2}}-radius 120-cell, or 1/𝜙{{radic|2}} in a unit-radius 120-cell). This edge length is in the golden ratio to the 41.4° edge of the 30-point bitruncated 5-cells, which is also the triangle face edge of the central section 8₃ (Moxness's Hull #8) rhombicosidodecahedron. The 120 semi-regular section 6₃ rhombicosidodecahedra share their smaller edges with 720 pentagonal prisms, 1200 hexagonal prisms and 600 truncated tetrahedron cells, in a semi-regular honeycomb of the 120-cell discovered by Alicia Boole Stott and described in her 1910 paper.{{Sfn|Boole Stott|1910|loc=Table of Polytopes in S₄|ps=; <math>e_2e_3C_{120}\ RID\ P_5\ P_6\ tT</math>}} These truncated tetrahedra are 1/𝜙 smaller than the 600 cells of the bitruncated 5-cells. The 60 distinct section 8₃ rhombicosidodecahedra (Moxness's Hull #8) share pentagon faces. Each of the 120 dodecahedron cells lies just inside 12 distinct rhombicosidodecahedra which share its volume. Each rhombicosidodecahedron includes a ball of 13 dodecahedron cells, 12 around one at the center of the rhombicosidodecahedron, within its volume. The remainder of the rhombicosidodecahedron is filled by 30 dodecahedron cell fragments that fit into the concavities of the 13 cell ball of dodecahedra. These fragments have triangle and rectangle faces. === Rectifying the 16-cells === Bitruncation is not the only way to truncate a regular polytope, or even the simplest way. The simplest method of truncation is [[w:Rectification_(geometry)|''rectification'']], complete truncation at the midpoint of each edge. Moreover, the 5-cell is not the only 120-cell building block we can truncate. We saw how bitruncation of the {{radic|2}}-radius 120-cell's 5-cells reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 20 disjoint (60 distinct) bitruncated 5-cells. In the next paragraph we describe how rectification of the {{radic|2}}-radius 120-cell's 16-cells also reveals the smaller unit-radius 120-cell, as the convex hull of a compound of 25 disjoint (225 distinct) 24-cells. Those two operations on the 120-cell are equivalent. They are the same truncation of the 120-cell, which bitruncates 5-cells into bitruncated 5-cells, and also rectifies 16-cells into 24-cells. This single truncation of the 120-cell captures the distant relationship of 5-cell building blocks to 16-cell building blocks. Rectifying a {{radic|2}}-radius 16-cell of edge 2 creates a unit-radius 24-cell of unit edge, which is the compound of three unit-radius 16-cells. Rectifying one of those inscribed unit-radius 16-cells of edge {{radic|2}} creates a smaller 24-cell of radius and edge {{radic|1/2}}, which is the [[24-cell#Relationships among interior polytopes|common core (intersection]]) of the unit 24-cell and its three inscribed 16-cells. Like the 120-cell itself, the 24-cell is concentric to a smaller instance of itself of {{radic|1/2}} its radius. The common core of each of the 24-cells inscribed in the 120-cell is the corresponding 24-cell in the smaller 120-cell. === Rectifying the 5-cells === In the previous section we bitruncated the 5-cells and rectified the 16-cells, as one combined truncation operation that yields a smaller 120-cell of {{radic|1/2}} the radius. We can also rectify the 5-cells; but that is another distinct truncation operation, that yields a smaller 4-polytope of {{radic|3/8}} the radius. [[File:Great (12) chords of rectified 5-cell.png|thumb|400px|5-cell edge chords of the radius {{radic|2}} 120-cell in one of its 200 irregular {12} dodecagon central planes. The {6} bitruncation points (two on each of the 104.5° {{radic|5}} 5-cell edges) lie on a smaller 120-cell of radius 1 (the red circle); they are bitruncated 5-cell vertices. The {6} rectification points (at the midpoints of the 5-cell edges) lie on a still smaller 1200-point 4-polytope of radius {{radic|0.75}} ≈ 0.866 (the magenta circle); they are rectified 5-cell vertices.]] Rectifying the 5-cell creates the 10-point 10-cell semi-regular [[W:Rectified 5-cell|rectified 5-cell]], with 5 tetrahedral cells and 5 octahedral cells. It has 30 edges and 30 equilateral triangle faces. The 3-dimensional surface of the rectified 5-cell is an alternating [[W:Tetrahedral-octahedral honeycomb|tetrahedral-octahedral honeycomb]] of just 5 tetrahedra and 5 octahedra, tessellating the 3-sphere. Its vertex figure is the cuboctahedron. The rectified 5-cell is a [[w:Blind_polytope|Blind polytope]], because it is convex with only regular facets. It is a bistratic lace tower which has exactly three vertex layers with the same Coxeter symmetry, aligned on top of each other.{{Sfn|Klitzing|2025|loc=''[https://bendwavy.org/klitzing/incmats/rap.htm rap (rectified pentachoron)]''}} If the 120 5-cells in a radius {{radic|2}} 120-cell are rectified, the rectified 5-cells lie on a smaller 4-polytope of radius {{radic|3/4}} (the magenta circle in the diagram), inscribed at the 1200 midedges of the 5-cells.{{Efn|{{radic|3/4}} ≈ 0.866 is the long radius of the {{radic|2}}-edge regular tetrahedron (the ''unit-radius'' 16-cell's cell). Those four tetrahedron radii are not orthogonal, and they radiate symmetrically compressed into 3 dimensions (not 4). The four orthogonal {{radic|3/4}} ≈ 0.866 displacements summing to a 120° degree displacement in the unit-radius 24-cell's characteristic isoclinic rotation{{Efn|name=isoclinic 4-dimensional diagonal}} are not as easy to visualize as radii, but they can be imagined as successive orthogonal steps in a path extending in all 4 dimensions, along the orthogonal edges of the [[24-cell#Characteristic orthoscheme|24-cell's 4-orthoscheme]]. In an actual left (or right) isoclinic rotation the four orthogonal {{radic|3/4}} ≈ 0.866 steps of each 120° displacement are concurrent, not successive, so they ''are'' actually symmetrical radii in 4 dimensions. In fact they are four orthogonal [[24-cell#Characteristic orthoscheme|mid-edge radii of a unit-radius 24-cell]] centered at the rotating vertex. Finally, in 2 dimensional units, {{radic|3/4}} ≈ 0.866 is the ''area'' of the equilateral triangle face of the unit-edge, unit-radius 24-cell.|name=root 3/4}} This smaller 4-polytope is not a smaller 120-cell; it is the convex hull of a 1200-point compound of two 120-cells. The rectified 5-cell does not occur inscribed in the 120-cell; it only occurs in this compound of two 120-cells, 240 regular 5-cells, and 120 rectified 5-cells. The rectified 5-cell with its 80.4° edge chord does not occur anywhere in a single 120-cell, so the rectified 5-cell's edges are not the edges of any polytope found in the 120-cell. The rectified 5-cell's significance to the 120-cell is well-hidden, but we shall see that it has an indirect role as a building block of the 11-cells in the 120-cell. Each 10-point rectified 5-cell is the convex hull of a stellated compound of two completely orthogonal 5-point 5-cells: five pairs of antipodal vertices. Their edges intersect at the midedge, and they are ''not'' in dual position (not reciprocated at their common 3-sphere). In this stellated compound of two completely orthogonal 5-cells (which does not occur in the 120-cell), the rectified 5-cell occurs in two places and two sizes: as both the convex hull of the vertices, and the convex common core of the midedge intersections. The edge length of the rectified 5-cells in the smaller 1200-point 4-polytope of radius {{radic|3/4}} is {{radic|5/4}}. The edge length of a unit-radius rectified 5-cell is {{radic|5/3}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|3}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}. In the 120-cell of radius {{radic|2}}, the 104.5° {{radic|5}} chord is the 5-cell edge, and the 75.5° {{radic|3}} chord is the distance between two parallel 5-cell edges (belonging to two disjoint 5-cells). The 104.5° and 75.5° chords are 180° complements, so they form great rectangles in the {12} central planes of the 120-cell (the red rectangles in the diagram). In the 1200-point compound of two 120-cells of radius {{radic|3}} where 120 rectified 5-cells occur, the {{radic|3}} chord is the ''radius'' (not the 75.5° chord), and the {{radic|5}} chord is the ''rectified'' 5-cell edge of arc 80.4° (not the 104.5° regular 5-cell edge). === Truncating the 5-cells === [[File:Great (12) chords of unit thirds radius.png|thumb|400px|Truncating the 120-cell's 5-cells at ''one-third'' of their edge length produces a smaller 120-cell of ''one-half'' the radius, with vertices at {6} one-third intersection points of the 120° {{Radic|6}} chords (''not'' of the 104.5° {{Radic|5}} 5-cell edge chords). The green {6} hexagon is a 24-cell great hexagon in the resulting smaller-by-one-half 1200-point 4-polytopes. Because there are {12} such intersection points in each {12} central plane, there are two chiral ways to perform this truncation, which produce disjoint 1200-point 4-polytopes.]] A third simple way to truncate the 5-cell is at one-third of its edge length. This truncation of the 5-cell creates a 20-point, 10-cell semi-regular 4-polytope, known somewhat ambiguously as ''the'' [[w:Truncated_5-cell|truncated 5-cell]], with 5 truncated tetrahedron cells (like the bitruncated 5-cell's), and 5 regular tetrahedron cells (like the rectified 5-cell's). The 3-dimensional surface of the truncated 5-cell is an alternating honeycomb of 5 truncated tetrahedra and 5 regular tetrahedra. It resembles the smaller rectified 5-cell with truncated tetrahedra instead of octahedra, or the larger bitruncated 5-cell with half its truncated tetrahedra replaced by regular tetrahedra. When the regular 5-cell is truncated at ''one-third'' of its edge length, the radius and edge length of the the resulting truncated 5-cell are ''one-half'' the regular 5-cell's radius and edge length. When the 120 5-cells in a 120-cell of radius 2 are truncated at one-third of their edge length, the truncated 5-cells lie on a smaller 120-cell of radius 1. The edge length of the unit-radius truncated 5-cell is {{radic|5/8}}, one-half the unit-radius 5-cell's edge length of {{radic|5/2}}. The rectified 5-cell is characterized by the ratio between its edge and its radius, {{radic|5}} to {{radic|8}}, the way the regular 5-cell is characterized by the ratio {{radic|5}} to {{radic|2}}, and the rectified 5-cell is characterized by the ratio {{radic|5}} to {{radic|3}}. The 20-point truncated 5-cell is the convex common core of a stellated compound of four 5-cells (the four 5-cells' spatial intersection). The convex common core has half the radius of the convex hull of the compound. The four 5-cells are orthogonal (aligned on the four orthogonal axes), but none of their 20 vertices are antipodal. The 5-cells are ''not'' in dual position (not reciprocated at their common 3-sphere). The 5-cell edges do ''not'' intersect, but truncating the 120-cell's 5-cell edge chords at their one-third points truncates the 120-cell's other chords similarly. It is the 120-cell's 120° chords (of length {{Radic|6}} in a {{Radic|2}}-radius 120-cell, or {{Radic|3}} in a unit-radius 120-cell) which intersect each other at their one-third points. Four edges (one from each 5-cell) intersect orthogonally at just ''one'' of the two one-third intersection points on each of the 2400 120° chords that join vertices of two disjoint 5-cells. There are two chiral ways to perform this truncation of the 120-cell; they use the alternate intersection points on each edge, and produce disjoint 600-point 120-cells. The 52.25° edge chord of the truncated 5-cell (one-half the 5-cell's 104.5° edge chord) is not among the [[120-cell#Chords|chords of the 120-cell]], so the truncated 5-cell does not occur inscribed in the 120-cell; it occurs only in a compound of four 120-cells, and 480 regular 5-cells, and 120 truncated 5-cells. In the stellated compound of four orthogonal 5-cells (which does not occur in the 120-cell), the truncated 5-cell occurs in two places and two sizes: as both the convex hull of the 20 vertices, and the convex common core (of half the radius of the convex hull) of the 20 intersection points of four orthogonal 120° chords. == The perfection of Fuller's cyclic design == [[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 orthogonal axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]] This section is not an historical digression, but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry]]''}} After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session<ref>{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}</ref> where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.<ref>{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}</ref>|name=Snelson and Fuller}} A tensegrity icosahedron is an icosahedral geodesic sphere whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables. The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 10 of in Moxness's Hull #8 rhombicosidodecahedron, the real cell of the 11-cell. The Jessen's was named by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a [[W:Cubist|cubist]] shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point ''vector equilibrium'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point vector equilibrium, and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposite equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (a hexad), then backing away again while still rotating in the same directions. All this was overlaid with Fuller's own deep commentary, in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing "vector edge cube" (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}} Earlier, we noticed 10 Jessen's inscribed in each 60-point rhombicosidodecahedron central section of the 120-cell (each real hemi-icosahedron). Each rhombicosidodecahedron is a compound of 5 disjoint Jessen's, in two different ways, just the way the 120-cell is a compound of 5 disjoint 600-cells, in two different ways. In the rhombicosidodecahedron each regular icosahedron vertex has been replaced by the five vertices of a little pentagon face (a 120-cell face), and the regular icosahedron has been replaced by 5 disjoint (10 distinct) Jessen's icosahedra.{{Efn|name=compound of 5 cuboctahedra}} The 3 pairs of parallel 5-cell edges in each Jessen's lie a bit uncertainly, infinitesimally mobile and [[Kinematics of the cuboctahedron#Elastic-edge transformation|behaving like the struts of a tensegrity icosahedron]], so we can push any parallel pair of them apart or together infinitesimally, making each Jessen's icosahedron expand or contract infinitesimally. All 600 Jessen's, all 60 rhombicosidodecahedra, and the 120-cell itself expand or contract infinitesimally, together.{{Efn|name=tensegrity 120-cell}} Expansion and contraction are Boole Stott's operators of dimensional analogy, and that infinitesimal mobility is the infinite calculus of an inter-dimensional symmetry. The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three rectangular chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables. [[File:Jessen's √2 radius dimensions.png|thumb|400px|Moxness's 60-point section 8₃ rhombicosidodecahedron is a compound of 5 of this 12-point Jessen's icosahedron, shown here in a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed {{radic|1.5}} green cube, and its 8 equilateral triangle faces are 24-cell faces. This is a ''vertex figure'' of the 120-cell. The center point is also a vertex of the 120-cell.]] As a matter of convenience in this paper, we have used {{radic|2}}-radius metrics for 3-sphere polytopes, so e.g. the 5-cell edge is {{radic|5}}, where in unit-radius coordinates it would be {{Radic|5/2}}. Here we give two illustrations of the Jessen's using two different metrics: the 2-sphere Jessen's has a {{radic|5}} diameter, and the 3-sphere Jessen's has a {{radic|2}} radius. This reveals a curiously cyclic way in which our 2-sphere and 3-sphere metrics correspond. In the embedding into 4-space the characteristic root factors of the Jessen's seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord. We might have expected to find the 6-point hemi-icosahedron's 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the Jessen's are the 5-cell edges. A 5-cell face has its three {{radic|5}} edges in three different Jessen's icosahedra. The Jessen's is not a cell, but one of the 120-cell's vertex figures, like the [[600-cell#Icosahedra|120 regular icosahedron vertex figures in the 600-cell]]. That is why we find 600 Jessen's, of course. The center point in this Jessen's illustration is another ''vertex'' of the 120-cell, not the empty center of a cell.{{Efn|The 13 vertices of the illustration which include its center point lie in the curved 3-space of the 3-sphere, on the 120-cell's surface. In 4-space, this object is an [[W:Icosahedral pyramid|icosahedral pyramid]] with a Jessen's icosahedron as its base, and the apical center vertex as its apex. The center point in the illustration is a vertex of the 120-cell, and the center of the curved Jessen's, and the apex of the icosahedral pyramid, but it is not the center point in 4-space of a flat 3-dimensional Jessen's icosahedron. The center point of the base Jessen's icosahedron is a point inside the 120-cell, not a 120-cell vertex on its surface. It lies in the same 3-dimensional flat-slice hyperplane as the 12 vertices of the base Jessen's icosahedron, directly below the 13th 120-cell vertex.}} Each Jessen's includes the central apex vertex, {{radic|2}} radii, {{radic|2}} edges and {{radic|5}} chords of a vertex figure around the 120-cell vertex at its center. The {{radic|2}} face edges are 24-cell edges (also tesseract edges), and the inscribed green cube is the 24-cell's cube vertex figure. The 8 {{radic|2}} face triangles occur in 8 distinct 24-cells that meet at the apex vertex.{{Efn|Eight 24-cells meet at each vertex of a [[24-cell#Radially equilateral honeycomb|honeycomb of 24-cells]]: each one meets its opposite at that shared vertex, and the six others at a shared octahedral cell.{{Efn|In the 600-cell, which contains [[600-cell#Twenty-five 24-cells|25 24-cells]], 5 24-cells meet at each vertex. Each pair of 24-cells at the vertex meets at one of 200 distinct great hexagon central planes. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 8 other 24-cells. In the 120-cell, which contains 10 600-cells (5 disjoint 600-cells two different ways) and 225 24-cells (25 disjoint 24-cells), 8 24-cells meet at each vertex. Each 24-cell shares one of its great hexagons with 16 other 24-cells, and is completely disjoint from 208 other 24-cells. But since in the 120-cell the great hexagons lie in pairs in one of 200 {12} central planes (containing 400 great hexagons), each 24-cell shares one of its {12} central ''planes'' with .. other 24-cells.}}}} This Jessen's vertex figure includes 5-cell edges and 24-cell edges (which are also tesseract edges), so it is descriptive of the relationship between those regular 4-polytopes, but it does not include any 120-cell edges or 600-cell edges, so it has nothing to say, by itself, about the <math>H_4</math> polytopes. It is only a tiny fraction of the 120-cell's full vertex figure, which is a staggeringly complex star: 600 chords of 30 distinct lengths meet at each of the 600 vertices. The {{radic|5}} chords are 5-cell edges, connecting vertices in different 24-cells. The 3 pairs of parallel 5-cell edges in each Jessen's lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th pair of parallel 5-cell edges orthogonal to all of them, in fact three more orthogonal pairs, since 6 orthogonal planes (not just 4) intersect at a point in 4-space. The Jessen's situation is that it lies completely orthogonal to another Jessen's, the vertex figure of the antipodal vertex, and its 3 orthogonal planes (xy, yz, zx) lie completely orthogonal to its antipodal Jessen's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} These 6 pairs of parallel 5-cell edges form a 24-point 4-polytope, composed of two completely orthogonal 12-point Jessen's, inscribed in two completely orthogonal rhombicosidodecahedra. This 24-point 4-polytope is not a 24-cell: the 24-cell is not a compound of two 12-point Jessen's. But it turns out that two completely orthogonal 12-point Jessen's indirectly define a 24-point 24-cell. We shall see that their 4-space intersection is a 24-cell. This finding, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, brings Fuller's study of [[w:Tesseract#Radial_equilateral_symmetry|radially equilateral]] vector equilibrium polytopes to its completion in the 24-cell. Fuller began with the hexagon, the 6-point vector equilibrium in 2 dimensions, the only polygon with its radius equal to its edge length. He studied the cuboctahedron, the 12-point vector equilibrium in 3 dimensions, the only polyhedron with its radius equal to its edge length, in all its flexible guises. He discovered its stable equilibrium as the the Jessen's shadfish, with its cube of 6 open mouths and 90° dihedral angles between all its faces, the geometric center of [[WikiJournal Preprints/Kinematics of the cuboctahedron|the cuboctahedron's kinematic transformation]] through the regular polyhedra: tetrahedron, octahedron, Jessen's, regular icosahedron, and cuboctahedron. Fuller's study of kinematic Euclidean geometry did not reach the 4-polytopes, and the ultimate 24-point vector equilibrium in 4 dimensions, the 24-cell, the unique <math>F_4</math> symmetry found only in 4 dimensions. But Fuller led us up to it, through the kinematics of infinitesimal mobility, and that route to it is our clue to the infinite calculus of dimensional expansion and contraction. We observe this geometry, of two completely orthogonal 12-point Jessen's isomorphic to a 24-cell, only in the 120-cell. The 600-cell contains 12-point Jessen's, but no completely orthogonal pairs of them. The 24-cell individually, and the 25 24-cells in the 600-cell, are not occupied by a pair of 12-point Jessen's. The 24-point 24-cell is not, in fact, a compound of two 12-point Jessen's. While the 120-cell's ratio of disjoint 12-point Jessen's to disjoint 24-point 24-cells is <math>50/25 = 2/1</math>, the ratio of distinct 12-point Jessen's to distinct 24-point 24-cells is <math>600/225 = 8/3 </math>. We observe another geometry, of 24-cells in dual positions, only in the 120-cell. No two 24-cells in the 600-cell are in dual positions, but in the 120-cell with 225 distinct 24-cells (25 disjoint 24-cells), every 24-cell is in dual position to other 24-cells. The 24-cell is self-dual, and when two 24-cells of the same radius are in dual position, they are completely disjoint with respect to vertices, but they intersect at the midpoints of their 96 orthogonal edges. Since four orthogonal lines intersect at a point in 4-space, in addition to the midedge radius and the two intersecting edges there is a third intersecting edge through each point of contact: ''three'' 24-cells lie in dual positions to each other, with their orthogonal edges intersecting. Three ''pairs'' of 24-cells lie in orthogonal dual positions to each other, sharing no vertices, but the same 96 midedge points. We also observe this geometry, of 24-cells in dual positions, in the irregular {12} dodecagon central planes, which have two inscribed great {6} hexagons, offset from each other irregularly by a 15.5° arc on one side (a 120-cell edge chord) and a 44.5° arc on the other side. The 600-cell and the 24-cell contain only great {6} hexagon planes. The two inscribed great {6} hexagons in each {12} central plane belong to a pair of 24-cells in dual position. We observe inscribed 5-cells only in the 120-cell. The 600-cell has <math>5^2 = 25</math> distinct 24-cells inscribed in 120 vertices, and is a regular compound of <math>5</math> disjoint 24-cells in 10 different ways, but it has no inscribed 5-point 5-cells joining corresponding vertices of 5 of its 25 24-cells.{{Efn|The 600-cell does have inscribed 5-point great pentagons joining corresponding vertices of 5 of its 25 24-cells. The 600-cell has 2-dimensional pentads, but only the 120-cell has 4-dimensional pentads.}} The 120-cell has <math>5^2 \times 3^2 = 225</math> distinct 24-cells inscribed in 600 vertices, and is a regular compound of <math>5^2 = 25</math> disjoint 24-point 24-cells in 10 different ways, and it has 120 inscribed 5-cells joining corresponding vertices of 5 of its 225 24-cells. [[File:Great 5-cell √5 digons rectangle.png|thumb|400px|Three {{radic|5}} x {{radic|3}} rectangles (red) are found in 200 central planes of the radius {{radic|2}} 120-cell, and in its 600 Jessen's icosahedra, where 3 orthogonal rectangles comprise each 12-point Jessen's. Each central plane intersects {12} vertices in an irregular great dodecagon. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges (solid red), which form two opposing ''irregular'' great hexagons (truncated triangles) with the {{radic|5}} chords. The {12} central planes also contain four {{radic|6}} great triangles (green), inscribed in two {{radic|2}} ''regular'' great hexagons. 1200 smaller {{radic|5}} 5-cell ''face'' triangles (blue) occupy 600 other, non-central planes.]] The Jessen's eight {{radic|6}} triangle faces lie in eight great {6} hexagons in eight {12} central planes of the 120-cell. The Jessen's {{radic|5}} chords lie in great {4} rectangles ({{radic|5}} by {{radic|3}}) in orthogonal central planes of the Jessen's. These are ''also'' {12} central planes of the 120-cell. We can pick out the {{radic|5}} by {{radic|3}} rectangles in the {12} central plane chord diagrams (bounded by red dashed lines). The Jessen's vertex figure is bounded by eight {12} face planes, and divided by six orthogonal {12} central planes, and all 14 planes are {12} central planes of the 120-cell. The 5-cells' ''face'' planes are ''not'' central planes of the 120-cell. Recall that 10 distinct Jessen's are inscribed in each rhombicosidodecahedron, as two chiral sets of 5 completely disjoint Jessen's, such that two {{radic|5}} 5-cell edges meet at each vertex of the rhombicosidodecahedron. These are two of the four 5-cell edges that meet at each vertex of the 5-cell: edges of a 5-cell face, 20 of which are disjointly inscribed in each rhombicosidodecahedron. In each Jessen's the 6 {{radic|5}} reflex edges are disjoint, and in each rhombicosidodecahedron only two edges meet at each vertex, but in the 120-cell each {{radic|5}} chord meets three others, that lie in three other Jessen's. Each 5-cell face triangle has each edge in a distinct Jessen's, but the face triangle lies in just one rhombicosidodecahedron. The 1200 5-cell face triangles lie in opposing pairs, in one of 600 ''non-central'' hexagon ''face'' planes. Each of the 60 rhombicosidodecahedra is a compound of 10 Jessen's (5 disjoint Jessen's in two different ways), just the way the 120-cell is a compound of 10 600-cells (5 disjoint 600-cells in two different ways), and the 120-cell's dodecahedron cell is a compound of 10 600-cell tetrahedron cells (5 disjoint tetrahedra in two different ways). The 600 Jessen's in the 120-cell occur in bundles of 8 disjoint Jessen's, in 4 completely orthogonal pairs, each pair aligned with one of the four axes of the Cartesian coordinate system. Collectively they comprise 3 disjoint 24-cells in orthogonal dual position. They are [[24-cell#Clifford parallel polytopes|Clifford parallel 4-polytopes]], 3 completely disjoint 24-cells 90° apart, and two sets of 4 completely disjoint Jessen's 15.5° apart. Opposite triangle faces in a Jessen's occupy opposing positions in opposite great hexagons. In contrast, the two completely orthogonal Jessen's are completely disjoint, with completely orthogonal bounding planes that intersect only at one point, the center of the 120-cell. The corresponding {{radic|6}} triangle faces of two completely orthogonal Jessen's occupy completely orthogonal {12} central planes that share no vertices. If we look again at a single Jessen's, without considering its completely orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) Jessen's lies in 4-space, it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (vertex figure) Jessen's is part of a 16-point (8-cell) tesseract containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already found it was part of a 24-point (24-cell) 4-polytope, which contains 3 16-point (8-cell) tesseracts. Each 12-point (6 {{radic|5}} reflex edge) Jessen's is one of 10 concentric Jessen's in a rhombicosidodecahedron, two sets of 5 disjoint Jessen's rotated with respect to each other isoclinically by 12° x 12° = 15.5°, with a total of 60 disjoint {{radic|5}} edges. Each 12-point (24 {{radic|6}} edge) Jessen's is one of 8 concentric Jessen's in two 24-cells in dual positions, rotated with respect to each other isoclinically by 41.4° x 41.4° = 90°, with a total of 192 {{radic|6}} edges.{{Efn|There are 96 {{radic|6}} chords in each 24-cell, linking every other vertex under its 96 {{radic|2}} edges.}} The 24-point 24-cell has 4 Hopf fibrations of 4 hexagonal great circle fibers, so it is a complex of 16 great hexagons, generally not orthogonal to each other, but containing 3 sets of 4 orthogonal great hexagons. Three Borromean link great rectangles are inscribed in each great hexagon, and three tesseracts are inscribed in each 24-cell. Four of the 6 orthogonal [[w:Borromean_rings|Borromean link]] great rectangles in each completely orthogonal pair of Jessen's are inscribed in each tesseract. == Conclusion == Thus we see what the 11-cell really is: an unexpected seventh regular convex 4-polytope falling between the 600-cell and 120-cell, a quasi-regular compound of 600-cell and 5-cell (an icosahedron-tetrahedron analogue), as the 24-cell is an unexpected sixth regular convex polytope falling between the 8-cell and 600-cell, a quasi-regular compound of 8-cell and 16-cell (a cube-octahedron analogue). Like the 5-cell, the 11-cell is a far-side 4-polytope with its long edges spanning the near and far halves of the 3-sphere. Unlike the 5-cell, the 11-cell's left and right rotational instances are not the same object: they have distinct cell polyhedra, which are duals. The 11-cell is a real regular convex 4-polytope, not just an [[W:abstract polytope|abstract 4-polytope]], but not just a singleton regular convex 4-polytope, and not just a single kind of cell honeycomb on the 3-sphere.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} Though it is all those things singly, it never occurs singly, but its multiple instances in the 120-cell compound to all those things, and significantly more. The 11-cell (singular) is the 11-vertex (17 cell) non-uniform Blind 4-polytope, with 11 non-uniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. The abstract regular 11-point (11-cell) has a realization in Euclidean 4-space as this convex 4-polytope, with regular facets and regular triangle faces. The 11-cell (plural) is subsumed in the 120-cell, as all the regular convex 4-polytopes are. The compound of eleven 11-cells (the ..-cell) and Schoute's compound of five 24-cells (the 600-cell) is the quasi-regular 137-point (..-cell) 4-polytope, an object of further study. The 11-cells' realization in the 120-cell as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (4-simplex and 4-orthoplex), which are both inscribed in the 11-point (17-cell), 137-point (..-cell) and 600-point (120-cell), but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationship between the regular ''n''-polytopes of different ''n''. The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies. == Build with the blocks == <blockquote>"The best of truths is of no use unless it has become one's most personal inner experience."{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: "The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery."}}}}</blockquote> <blockquote>"Even the very wise cannot see all ends."{{Sfn|Tolkien|1954|loc=Gandalf}}</blockquote> No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to show how I came by my understanding of these objects, since I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, in my imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am. {{Regular convex 4-polytopes|columns=7|wiki=W:|radius={{radic|2}}|instance=2}} {{Regular convex 4-polytopes|columns=7|wiki=W:|radius=1}} == Acknowledgements == ... == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite web| url = https://www.youtube.com/watch?v=9sM44p385Ws| title = Vector Equilibrium | first = R. 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The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics. At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions. PDT treats a probability measure as the primary mathematical object and investigates: * invariant identities induced by reweighting, * composition and iteration of dilations, * fixed points and near-fixed behavior, * whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations). PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions. == Overview == PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is: * mathematically well-defined (positivity and normalization), * composable under iteration, * analyzable for invariants and fixed points. === Conceptual interpretation === A simplified conceptual flow of the PDT framework is: <pre> Baseline probability measure P ↓ Positive dilation field D(x) ↓ Reweighted probability measure P~ ↓ Observable statistical changes </pre> Repeated dilation may qualitatively behave as: <pre> Broad initial distribution ↓ Localized reweighting ↓ Probability concentration ↓ Emergent multiscale structure </pre> Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field. Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as: <pre> P₀ ↓ D₁ P₁ ↓ D₂ P₂ ↓ D₃ P₃ ↓ ⋯ </pre> where each dilation field reweights the probability structure generated by the previous step. Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. = Mathematical framework = == Definitions and notation == Let <math>(\Omega,\Sigma)</math> be a measurable space. * <math>P</math> denotes a probability measure on <math>(\Omega,\Sigma)</math>. * If <math>P</math> has a density <math>p</math> with respect to a reference measure <math>\mu</math>, then <math>dP=p\,d\mu</math>. * <math>D:\Omega\to(0,\infty)</math> is a measurable '''dilation field''' (a positive weight function). * <math>Z(P,D)</math> is the normalization constant: .<math> Z(P,D)=\int_\Omega D\,dP </math> * For an observable <math>f:\Omega\to\mathbb{R}</math> integrable under the relevant measure, <math> \mathbb{E}_P[f] = \int_\Omega f\,dP </math>. == PDT transformation (probability reweighting) == Given <math>P</math> and <math>D</math> with <math>0<Z(P,D)<\infty</math>, define the '''PDT transform''' <math>\widetilde{P}=\mathrm{PDT}(P;D)</math> by: <math> \widetilde{P}(A) = \frac{ \int_A D\,dP }{ \int_\Omega D\,dP } \quad\text{for all }A\in\Sigma </math> If <math>dP=p\,d\mu</math>, then <math>d\widetilde{P}=\widetilde{p}\,d\mu</math>, where <math> \widetilde{p}(x) = \frac{D(x)\,p(x)}{Z} </math> and <math> Z = \int_\Omega D(x)\,p(x)\,d\mu </math> '''Interpretation:''' the dilation field <math>D</math> shifts probability mass toward regions where <math>D</math> is larger, while renormalization keeps total probability equal to 1. PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures. Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations. A familiar physical example of a strictly positive factor is the Lorentz factor: <math> \gamma(v) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} </math> for <math> |v|<c </math> Lorentz contraction for a rod of rest length <math>L_0</math> moving at speed <math>v</math> is: <math> L(v)=\frac{L_0}{\gamma(v)} </math> To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on <math>\gamma</math>. == Worked finite example == Consider a finite probability space: <math> \Omega=\{a,b,c\} </math> with baseline probabilities: <math> P(a)=0.2,\quad P(b)=0.3,\quad P(c)=0.5 </math> Define a positive dilation field: <math> D(a)=1,\quad D(b)=2,\quad D(c)=4 </math> The normalization constant is: <math> Z=\sum_x D(x)P(x) </math> giving: <math> Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8 </math> The PDT-transformed probabilities become: <math> \widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071 </math> <math> \widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214 </math> <math> \widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714 </math> This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization. == Composition of dilations == An important structural property of sequential PDT transformations is that compose multiplicatively. Suppose two positive dilation fields: <math> D_1(x)>0 </math> and <math> D_2(x)>0 </math> are applied successively to a baseline probability measure <math>P</math>. The first dilation produces: <math> \widetilde{P}_1(A) = \frac{\int_A D_1\,dP} {\int_\Omega D_1\,dP} </math> Applying the second dilation field to <math>\widetilde{P}_1</math> gives: <math> \widetilde{P}_2(A) = \frac{\int_A D_2\,d\widetilde{P}_1} {\int_\Omega D_2\,d\widetilde{P}_1} </math> Substituting the first transformation into the second yields: <math> \widetilde{P}_2(A) = \frac{ \int_A D_2D_1\,dP }{ \int_\Omega D_2D_1\,dP } </math> This shows that sequential PDT transformations compose through multiplication of the dilation fields. This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application. == Fixed points and iterative dynamics == An important question in PDT concerns the long-term behavior of repeated PDT transformations. Given an initial probability measure: <math> P_0 </math> and a sequence of positive dilation fields: <math> D_1,D_2,D_3,\dots </math> successive PDT transformations generate a sequence of measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow P_3 \rightarrow \cdots </math> where each transformed measure is obtained by reweighting the previous one. A measure <math>P</math> is called a fixed point of a dilation field <math>D</math> if: <math> \widetilde{P}=P </math> under the PDT transformation. In the simplest case, this requires the dilation field to be constant almost everywhere with respect to <math>P</math>. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization. More generally, repeated compositions of nontrivial dilation fields may generate: * hierarchical probability structure; * multiscale statistical behavior; * attractor-like distributions; * approximately stable transformed measures. These questions connect PDT to broader areas of: * dynamical systems; * stochastic processes; * iterative renormalization methods; * probabilistic geometry. At present these iterative properties remain largely unexplored within the PDT framework. == Entropy and iterative probability flow == Repeated PDT transformations may alter the entropy structure of a probability measure. For a discrete probability distribution: <math> P=\{p_i\} </math> the Shannon entropy is: <math> H(P) = -\sum_i p_i \log p_i </math> Under iterative PDT transformation, successive transformed measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow \cdots </math> may exhibit changing entropy behavior depending on the structure of the dilation fields. For example: * strongly localized dilation fields may concentrate probability mass and reduce entropy; * broader or smoothing dilation fields may distribute probability more evenly and increase entropy; * iterative compositions may generate approximately stable entropy profiles. These questions connect PDT to: * information theory, * statistical mechanics, * stochastic dynamics, * and renormalization-style iterative systems. At present the entropy behavior of iterative PDT transformations remains an open area for investigation. == Toy experiment: entropy under repeated dilation == A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution. Let the initial probability distribution be: <math> P_0=(0.2,0.2,0.2,0.2,0.2) </math> and define a positive dilation field: <math> D=(1,1,2,4,8) </math> At each step, apply the PDT update: <math> P_{n+1}(i) = \frac{D(i)P_n(i)} {\sum_j D(j)P_n(j)} </math> The Shannon entropy is: <math> H(P_n) = -\sum_i P_n(i)\log P_n(i) </math> In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately: <math> H(P_0)\approx1.6094 </math> to: <math> H(P_{10})\approx0.00775 </math> The final distribution is approximately: <math> P_{10} \approx (0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437) </math> This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior. == Mathematical context == PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature. In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis. The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations. === Example entropy evolution === {| class="wikitable" ! Iteration !! Shannon entropy |- | 0 || 1.6094 |- | 1 || 1.2990 |- | 2 || 0.7790 |- | 3 || 0.4399 |- | 5 || 0.1500 |- | 10 || 0.0078 |} Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states. === Localized dilation fields === A useful class of PDT transformations is generated by localized positive dilation fields. Consider a one-dimensional finite configuration space with states indexed by: <math> x=0,1,2,\dots,N </math> and define a localized dilation field centered at <math>x_0</math>: <math> D(x) = \exp\!\left( \lambda \exp\!\left( -\frac{(x-x_0)^2}{2\sigma^2} \right) \right) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\sigma</math> controls the spatial width of the localized field. Narrow values of <math>\sigma</math> produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space. Under iterative PDT dynamics: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> the probability distribution may progressively concentrate near the center of the dilation field. === Example entropy evolution for localized fields === Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior: {| class="wikitable" ! Field width <math>\sigma</math> ! Final entropy after 10 iterations ! Maximum probability after 10 iterations |- | 1.5 || 0.0352 || 0.9950 |- | 3.0 || 0.8162 || 0.7141 |- | 6.0 || 1.5367 || 0.3595 |} [[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]] [[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]] These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction. == Comparative entropy-flow experiments == The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions. {| class="wikitable" |+ Comparative entropy-flow behavior under PDT field classes ! Field class ! Final entropy ! Entropy decrease ! Final max probability ! Qualitative behavior |- | Localized | 0.3104 | 3.4032 | 0.9275 | Strong probability concentration |- | Oscillatory | 1.5779 | 2.1357 | 0.3418 | Distributed oscillatory structure |- | Multi-peak | 0.2851 | 3.4284 | 0.9425 | Multiple concentration regions |- | Stochastic | 0.7744 | 2.9392 | 0.7413 | Fluctuating concentration behavior |} These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example. In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space. [[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]] The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Oscillatory dilation fields === Another useful class of PDT transformations is generated by oscillatory positive dilation fields. One example is: <math> D(x) = \exp(\lambda\sin(kx)) </math> where: * <math>\lambda>0</math> controls the strength of the oscillatory amplification; * <math>k</math> controls the spatial frequency of the oscillation. Because the exponential is always positive, the dilation field remains strictly positive for all states. Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space. Under repeated PDT transformation: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor. === Example oscillatory-field experiment === A finite-state experiment was performed using: * 41 discrete states; * an initially uniform probability distribution; * a positive oscillatory dilation field with three spatial oscillation cycles; * 10 successive PDT iterations. Representative entropy behavior was: {| class="wikitable" ! Iteration ! Shannon entropy |- | 0 || 3.7136 |- | 2 || 2.8699 |- | 5 || 2.3018 |- | 10 || 1.9335 |} Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space. After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state. This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures. The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Multi-peak localized dilation fields === A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space. One example is: <math> D(x) = \exp\!\left( \sum_k \lambda_k \exp\!\left( -\frac{(x-x_k)^2}{2\sigma_k^2} \right) \right) </math> where: * <math>x_k</math> are the locations of the dilation peaks; * <math>\lambda_k>0</math> control the amplification strength of each peak; * <math>\sigma_k</math> control the spatial width of each localized region. This construction generates a positive multimodal dilation landscape containing several competing amplification regions. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward multiple partially localized concentration regions. Unlike single localized dilation fields, multi-peak fields may generate: * competing attractor-like regions; * hierarchical probability concentration; * partially stabilized multimodal distributions; * multiscale probability structure. Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor: * dominance by a single peak; * coexistence of several concentration regions; * or slowly evolving metastable probability structures. === Conceptual interpretation === A qualitative iterative evolution may be visualized as: <pre> Broad initial distribution ↓ Multiple localized amplifications ↓ Competing concentration regions ↓ Emergent multimodal probability structure </pre> This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone. At present these behaviors remain exploratory computational observations within finite-state toy models. === Random and stochastic dilation fields === Another important class of PDT transformations arises when the dilation field itself varies stochastically. A simple stochastic dilation field may be written schematically as: <math> D_n(x) = \exp\!\left( \sigma \eta_n(x) \right) </math> where: * <math>\eta_n(x)</math> is a random field or stochastic fluctuation at iteration <math>n</math>; * <math>\sigma>0</math> controls the strength of the stochastic variation. Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D_n(x)P_n(x) }{ \sum_y D_n(y)P_n(y) } </math> the probability landscape itself fluctuates dynamically from one iteration to the next. Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate: * fluctuating concentration regions; * transient attractor-like structures; * noise-driven entropy evolution; * intermittent probability concentration; * metastable probabilistic configurations. === Conceptual interpretation === A qualitative stochastic evolution may be visualized as: <pre> Broad initial distribution ↓ Random localized amplification ↓ Fluctuating concentration regions ↓ Dynamic probabilistic structure </pre> Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit: * partial concentration, * persistent fluctuations, * stochastic stabilization, * or continuously evolving probabilistic structure. These ideas connect PDT to broader areas of: * stochastic processes; * random multiplicative systems; * statistical mechanics; * noise-driven dynamical systems; * probabilistic geometry. At present these behaviors remain exploratory computational possibilities within finite-state toy models. == Qualitative classes of iterative PDT behavior == Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation. The following table summarizes several representative classes explored within finite-state toy models. {| class="wikitable" ! Dilation-field class ! Typical iterative behavior ! Representative qualitative structure |- | Localized fields | Strong entropy reduction and concentration toward a dominant region | Single attractor-like concentration |- | Oscillatory fields | Distributed amplification with slower entropy reduction | Patterned multimodal structure |- | Multi-peak localized fields | Competition between several concentration regions | Hierarchical or metastable probability structure |- | Random and stochastic fields | Fluctuating amplification and noise-driven evolution | Dynamic probabilistic landscapes |} These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field. Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself. At present these qualitative behaviors remain exploratory computational observations within finite-state toy models. == Numerical simulation and iterative models == === Simulation model description === In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points. Two equivalent discrete implementations are common: * '''weighted evaluation''': retain all points and assign weights proportional to <math>D</math>; * '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to <math>D</math>. === Demonstration: reweighting mock galaxy catalogs === A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box. The demonstration pipeline is: # generate a baseline mock catalog; # define a positive dilation field over the configuration space; # perform PDT-style importance resampling; # compute the resulting two-point correlation function <math>\xi(r)</math>; # compare transformed and baseline catalogs. One example dilation field is: <math> D(x)=\exp(\lambda\phi(x)) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\phi(x)\ge0</math> is a nonnegative configuration-space field. An example seed-field construction is: <math> \phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right) </math> where <math>s_k</math> are seed locations and <math>\sigma</math> controls the width of the seed influence. The two-point correlation function may be estimated using the normalized Landy–Szalay estimator: <math> \xi(r) = \frac{DD(r)-2DR(r)+RR(r)}{RR(r)} </math> where <math>DD</math>, <math>DR</math>, and <math>RR</math> are normalized pair counts. {{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}} When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration. === Computational demonstrations === Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages. {{collapse top|Python demonstration placeholder}} <syntaxhighlight lang="python"> # Example implementations may be maintained separately # on GitHub, OSF, or supplementary Wikiversity pages. </syntaxhighlight> {{collapse bottom}} == Scope and Limitations == PDT is a mathematical framework for measure transformations. It does not claim: * a replacement theory for General Relativity or Quantum Mechanics; * empirical confirmation without explicit predictions and tests; * observational validation without independently reproducible analysis. The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations. == Speculative Extensions and Geometric Renormalization == ''This section is speculative and exploratory in nature.'' Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry. Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects. Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation. At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations. == Future directions == * develop canonical families of dilation fields and invariants; * clarify “structure-from-measure” diagnostics; * publish reproducible simulation notebooks and parameter sweeps; * compare multiple dilation families under shared evaluation criteria; * investigate connections between probabilistic geometry and curvature-dependent statistical measures. == Future Directions: Probability Element (PE) == A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics. The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution. This can be expressed in terms of a dimensionless ratio: <math>\eta = \frac{\sigma_P}{\sigma}</math> where: <math>\sigma_P</math> is a hypothesized minimal probability-resolution scale, <math>\sigma</math> is an effective distinguishability scale in probability-state space. === Conceptual motivation === Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry. === Illustrative toy model (not derived physics) === As a heuristic example, one may consider a modification to special relativistic time dilation of the form: <math>d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}</math> where: <math>v</math> is velocity, <math>c</math> is the speed of light, <math>\eta = \sigma_P / \sigma</math> encodes a proposed probability-resolution scale. This expression is constructed such that standard special relativity is recovered exactly in the limit <math>\eta \to 0</math>. === Status === The Probability Element concept is: Not part of standard Fisher information geometry not derived from quantum mechanics or general relativity not currently empirically established. It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale. === Open questions === Key open research directions include: Whether a consistent discrete formulation of probability geometry can be constructed. Whether a fundamental probability-resolution scale <math>\sigma_P</math> can be derived from known physical principles. Whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions. == Convergence behavior == Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations. === Qualitative convergence classes === Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior: * '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy. * '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration. * '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions. * '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior. === Entropy and convergence === In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time. The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations. === Attractor-like behavior === Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense. Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks. == Current limitations == PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation. Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms. Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks. == See also == * [[w:Buffon's needle problem|Buffon's needle problem]] * [[w:Probability measure|Probability measure]] * [[w:Importance sampling|Importance sampling]] * [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]] * [[w:Dynamical system|Dynamical systems]] * [[w:Entropy (information theory)|Entropy]] * [[w:Information theory|Information theory]] * [[w:Measure theory|Measure theory]] * [[w:Geometric probability|Geometric probability]] * [[w:Shannon entropy|Shannon entropy]] * [[w:Stochastic process|Stochastic process]] * [[w:Fixed point (mathematics)|Fixed point]] * [[w:Convergence (mathematics)|Convergence]] == Subpages == The following subpages develop mathematical extensions and specialized topics related to Probability Dilation Theory (PDT). * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows|Fisher Geometry and Dilation Flows]] – studies information geometry, Fisher distance, and geodesic properties of PDT trajectories. * [[Probability Dilation Theory/Logit Representation of PE|Logit Representation of PE]] – develops the log-odds representation of probability elements and exponential PDT flows. * [[Probability Dilation Theory/Convergence and Fixed Points|Convergence and Fixed Points]] – investigates invariant measures, attractors, and stability of iterative PDT transformations. * [[Probability Dilation Theory/Stochastic Dilation Fields|Stochastic Dilation Fields]] – studies random and time-dependent dilation fields, ergodicity, and stochastic measure evolution. * [[Probability Dilation Theory/Entropy Evolution|Entropy Evolution]] – examines Shannon entropy under repeated probability dilation. * [[Probability Dilation Theory/Wasserstein Geometry|Wasserstein Geometry]] – explores distances between probability measures and convergence in measure space. * [[Probability Dilation Theory/Measure-Theoretic Foundations|Measure-Theoretic Foundations]] – develops rigorous measure-theoretic aspects of PDT including normalization and existence conditions. * [[Probability Dilation Theory/Euler Methods and Continuous-Time PDT]] – investigates continuous probability flows and Euler approximations of PDT. * [[Probability Dilation Theory/Worked Example]] – canonical binary example illustrating PDT transformations and geometry. == Notation == Throughout PDT, the following notation is used: {| class="wikitable" ! Symbol ! Meaning |- | <math>P</math> | Probability measure |- | <math>P_n</math> | nth iterate of PDT |- | <math>T_D</math> | Probability dilation operator |- | <math>D(x)</math> | Dilation field |- | <math>Z(P,D)</math> | Normalization factor |- | <math>H(P)</math> | Shannon entropy |- | <math>d_F</math> | Fisher-Rao distance |- | <math>W_p</math> | Wasserstein distance |- | <math>\ell</math> | Logit coordinate |- | <math>PE</math> | Probability Element |} == Related probabilistic and geometric literature == Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works: * Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014. * Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19 * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997. * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005. * Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007. * Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656. == Copyright and licensing == Text and original figures © Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). Reuse permitted with attribution. smgufz92bz2cefsbx1uvkxozbqhpkp6 2816239 2816236 2026-06-18T19:31:33Z Howie2024 2995240 /* Subpages */ Inserting Iteration table. 2816239 wikitext text/x-wiki {{Research project}} {{Original research}} {{To be peer reviewed}} {{subst:proofread}} == Research abstract == '''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values. The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics. At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions. PDT treats a probability measure as the primary mathematical object and investigates: * invariant identities induced by reweighting, * composition and iteration of dilations, * fixed points and near-fixed behavior, * whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations). PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions. == Overview == PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is: * mathematically well-defined (positivity and normalization), * composable under iteration, * analyzable for invariants and fixed points. === Conceptual interpretation === A simplified conceptual flow of the PDT framework is: <pre> Baseline probability measure P ↓ Positive dilation field D(x) ↓ Reweighted probability measure P~ ↓ Observable statistical changes </pre> Repeated dilation may qualitatively behave as: <pre> Broad initial distribution ↓ Localized reweighting ↓ Probability concentration ↓ Emergent multiscale structure </pre> Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field. Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as: <pre> P₀ ↓ D₁ P₁ ↓ D₂ P₂ ↓ D₃ P₃ ↓ ⋯ </pre> where each dilation field reweights the probability structure generated by the previous step. Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. = Mathematical framework = == Definitions and notation == Let <math>(\Omega,\Sigma)</math> be a measurable space. * <math>P</math> denotes a probability measure on <math>(\Omega,\Sigma)</math>. * If <math>P</math> has a density <math>p</math> with respect to a reference measure <math>\mu</math>, then <math>dP=p\,d\mu</math>. * <math>D:\Omega\to(0,\infty)</math> is a measurable '''dilation field''' (a positive weight function). * <math>Z(P,D)</math> is the normalization constant: .<math> Z(P,D)=\int_\Omega D\,dP </math> * For an observable <math>f:\Omega\to\mathbb{R}</math> integrable under the relevant measure, <math> \mathbb{E}_P[f] = \int_\Omega f\,dP </math>. == PDT transformation (probability reweighting) == Given <math>P</math> and <math>D</math> with <math>0<Z(P,D)<\infty</math>, define the '''PDT transform''' <math>\widetilde{P}=\mathrm{PDT}(P;D)</math> by: <math> \widetilde{P}(A) = \frac{ \int_A D\,dP }{ \int_\Omega D\,dP } \quad\text{for all }A\in\Sigma </math> If <math>dP=p\,d\mu</math>, then <math>d\widetilde{P}=\widetilde{p}\,d\mu</math>, where <math> \widetilde{p}(x) = \frac{D(x)\,p(x)}{Z} </math> and <math> Z = \int_\Omega D(x)\,p(x)\,d\mu </math> '''Interpretation:''' the dilation field <math>D</math> shifts probability mass toward regions where <math>D</math> is larger, while renormalization keeps total probability equal to 1. PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures. Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations. A familiar physical example of a strictly positive factor is the Lorentz factor: <math> \gamma(v) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} </math> for <math> |v|<c </math> Lorentz contraction for a rod of rest length <math>L_0</math> moving at speed <math>v</math> is: <math> L(v)=\frac{L_0}{\gamma(v)} </math> To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on <math>\gamma</math>. == Worked finite example == Consider a finite probability space: <math> \Omega=\{a,b,c\} </math> with baseline probabilities: <math> P(a)=0.2,\quad P(b)=0.3,\quad P(c)=0.5 </math> Define a positive dilation field: <math> D(a)=1,\quad D(b)=2,\quad D(c)=4 </math> The normalization constant is: <math> Z=\sum_x D(x)P(x) </math> giving: <math> Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8 </math> The PDT-transformed probabilities become: <math> \widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071 </math> <math> \widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214 </math> <math> \widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714 </math> This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization. == Composition of dilations == An important structural property of sequential PDT transformations is that compose multiplicatively. Suppose two positive dilation fields: <math> D_1(x)>0 </math> and <math> D_2(x)>0 </math> are applied successively to a baseline probability measure <math>P</math>. The first dilation produces: <math> \widetilde{P}_1(A) = \frac{\int_A D_1\,dP} {\int_\Omega D_1\,dP} </math> Applying the second dilation field to <math>\widetilde{P}_1</math> gives: <math> \widetilde{P}_2(A) = \frac{\int_A D_2\,d\widetilde{P}_1} {\int_\Omega D_2\,d\widetilde{P}_1} </math> Substituting the first transformation into the second yields: <math> \widetilde{P}_2(A) = \frac{ \int_A D_2D_1\,dP }{ \int_\Omega D_2D_1\,dP } </math> This shows that sequential PDT transformations compose through multiplication of the dilation fields. This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application. == Fixed points and iterative dynamics == An important question in PDT concerns the long-term behavior of repeated PDT transformations. Given an initial probability measure: <math> P_0 </math> and a sequence of positive dilation fields: <math> D_1,D_2,D_3,\dots </math> successive PDT transformations generate a sequence of measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow P_3 \rightarrow \cdots </math> where each transformed measure is obtained by reweighting the previous one. A measure <math>P</math> is called a fixed point of a dilation field <math>D</math> if: <math> \widetilde{P}=P </math> under the PDT transformation. In the simplest case, this requires the dilation field to be constant almost everywhere with respect to <math>P</math>. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization. More generally, repeated compositions of nontrivial dilation fields may generate: * hierarchical probability structure; * multiscale statistical behavior; * attractor-like distributions; * approximately stable transformed measures. These questions connect PDT to broader areas of: * dynamical systems; * stochastic processes; * iterative renormalization methods; * probabilistic geometry. At present these iterative properties remain largely unexplored within the PDT framework. == Entropy and iterative probability flow == Repeated PDT transformations may alter the entropy structure of a probability measure. For a discrete probability distribution: <math> P=\{p_i\} </math> the Shannon entropy is: <math> H(P) = -\sum_i p_i \log p_i </math> Under iterative PDT transformation, successive transformed measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow \cdots </math> may exhibit changing entropy behavior depending on the structure of the dilation fields. For example: * strongly localized dilation fields may concentrate probability mass and reduce entropy; * broader or smoothing dilation fields may distribute probability more evenly and increase entropy; * iterative compositions may generate approximately stable entropy profiles. These questions connect PDT to: * information theory, * statistical mechanics, * stochastic dynamics, * and renormalization-style iterative systems. At present the entropy behavior of iterative PDT transformations remains an open area for investigation. == Toy experiment: entropy under repeated dilation == A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution. Let the initial probability distribution be: <math> P_0=(0.2,0.2,0.2,0.2,0.2) </math> and define a positive dilation field: <math> D=(1,1,2,4,8) </math> At each step, apply the PDT update: <math> P_{n+1}(i) = \frac{D(i)P_n(i)} {\sum_j D(j)P_n(j)} </math> The Shannon entropy is: <math> H(P_n) = -\sum_i P_n(i)\log P_n(i) </math> In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately: <math> H(P_0)\approx1.6094 </math> to: <math> H(P_{10})\approx0.00775 </math> The final distribution is approximately: <math> P_{10} \approx (0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437) </math> This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior. == Mathematical context == PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature. In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis. The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations. === Example entropy evolution === {| class="wikitable" ! Iteration !! Shannon entropy |- | 0 || 1.6094 |- | 1 || 1.2990 |- | 2 || 0.7790 |- | 3 || 0.4399 |- | 5 || 0.1500 |- | 10 || 0.0078 |} Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states. === Localized dilation fields === A useful class of PDT transformations is generated by localized positive dilation fields. Consider a one-dimensional finite configuration space with states indexed by: <math> x=0,1,2,\dots,N </math> and define a localized dilation field centered at <math>x_0</math>: <math> D(x) = \exp\!\left( \lambda \exp\!\left( -\frac{(x-x_0)^2}{2\sigma^2} \right) \right) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\sigma</math> controls the spatial width of the localized field. Narrow values of <math>\sigma</math> produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space. Under iterative PDT dynamics: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> the probability distribution may progressively concentrate near the center of the dilation field. === Example entropy evolution for localized fields === Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior: {| class="wikitable" ! Field width <math>\sigma</math> ! Final entropy after 10 iterations ! Maximum probability after 10 iterations |- | 1.5 || 0.0352 || 0.9950 |- | 3.0 || 0.8162 || 0.7141 |- | 6.0 || 1.5367 || 0.3595 |} [[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]] [[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]] These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction. == Comparative entropy-flow experiments == The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions. {| class="wikitable" |+ Comparative entropy-flow behavior under PDT field classes ! Field class ! Final entropy ! Entropy decrease ! Final max probability ! Qualitative behavior |- | Localized | 0.3104 | 3.4032 | 0.9275 | Strong probability concentration |- | Oscillatory | 1.5779 | 2.1357 | 0.3418 | Distributed oscillatory structure |- | Multi-peak | 0.2851 | 3.4284 | 0.9425 | Multiple concentration regions |- | Stochastic | 0.7744 | 2.9392 | 0.7413 | Fluctuating concentration behavior |} These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example. In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space. [[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]] The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Oscillatory dilation fields === Another useful class of PDT transformations is generated by oscillatory positive dilation fields. One example is: <math> D(x) = \exp(\lambda\sin(kx)) </math> where: * <math>\lambda>0</math> controls the strength of the oscillatory amplification; * <math>k</math> controls the spatial frequency of the oscillation. Because the exponential is always positive, the dilation field remains strictly positive for all states. Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space. Under repeated PDT transformation: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor. === Example oscillatory-field experiment === A finite-state experiment was performed using: * 41 discrete states; * an initially uniform probability distribution; * a positive oscillatory dilation field with three spatial oscillation cycles; * 10 successive PDT iterations. Representative entropy behavior was: {| class="wikitable" ! Iteration ! Shannon entropy |- | 0 || 3.7136 |- | 2 || 2.8699 |- | 5 || 2.3018 |- | 10 || 1.9335 |} Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space. After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state. This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures. The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Multi-peak localized dilation fields === A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space. One example is: <math> D(x) = \exp\!\left( \sum_k \lambda_k \exp\!\left( -\frac{(x-x_k)^2}{2\sigma_k^2} \right) \right) </math> where: * <math>x_k</math> are the locations of the dilation peaks; * <math>\lambda_k>0</math> control the amplification strength of each peak; * <math>\sigma_k</math> control the spatial width of each localized region. This construction generates a positive multimodal dilation landscape containing several competing amplification regions. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward multiple partially localized concentration regions. Unlike single localized dilation fields, multi-peak fields may generate: * competing attractor-like regions; * hierarchical probability concentration; * partially stabilized multimodal distributions; * multiscale probability structure. Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor: * dominance by a single peak; * coexistence of several concentration regions; * or slowly evolving metastable probability structures. === Conceptual interpretation === A qualitative iterative evolution may be visualized as: <pre> Broad initial distribution ↓ Multiple localized amplifications ↓ Competing concentration regions ↓ Emergent multimodal probability structure </pre> This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone. At present these behaviors remain exploratory computational observations within finite-state toy models. === Random and stochastic dilation fields === Another important class of PDT transformations arises when the dilation field itself varies stochastically. A simple stochastic dilation field may be written schematically as: <math> D_n(x) = \exp\!\left( \sigma \eta_n(x) \right) </math> where: * <math>\eta_n(x)</math> is a random field or stochastic fluctuation at iteration <math>n</math>; * <math>\sigma>0</math> controls the strength of the stochastic variation. Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D_n(x)P_n(x) }{ \sum_y D_n(y)P_n(y) } </math> the probability landscape itself fluctuates dynamically from one iteration to the next. Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate: * fluctuating concentration regions; * transient attractor-like structures; * noise-driven entropy evolution; * intermittent probability concentration; * metastable probabilistic configurations. === Conceptual interpretation === A qualitative stochastic evolution may be visualized as: <pre> Broad initial distribution ↓ Random localized amplification ↓ Fluctuating concentration regions ↓ Dynamic probabilistic structure </pre> Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit: * partial concentration, * persistent fluctuations, * stochastic stabilization, * or continuously evolving probabilistic structure. These ideas connect PDT to broader areas of: * stochastic processes; * random multiplicative systems; * statistical mechanics; * noise-driven dynamical systems; * probabilistic geometry. At present these behaviors remain exploratory computational possibilities within finite-state toy models. == Qualitative classes of iterative PDT behavior == Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation. The following table summarizes several representative classes explored within finite-state toy models. {| class="wikitable" ! Dilation-field class ! Typical iterative behavior ! Representative qualitative structure |- | Localized fields | Strong entropy reduction and concentration toward a dominant region | Single attractor-like concentration |- | Oscillatory fields | Distributed amplification with slower entropy reduction | Patterned multimodal structure |- | Multi-peak localized fields | Competition between several concentration regions | Hierarchical or metastable probability structure |- | Random and stochastic fields | Fluctuating amplification and noise-driven evolution | Dynamic probabilistic landscapes |} These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field. Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself. At present these qualitative behaviors remain exploratory computational observations within finite-state toy models. == Numerical simulation and iterative models == === Simulation model description === In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points. Two equivalent discrete implementations are common: * '''weighted evaluation''': retain all points and assign weights proportional to <math>D</math>; * '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to <math>D</math>. === Demonstration: reweighting mock galaxy catalogs === A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box. The demonstration pipeline is: # generate a baseline mock catalog; # define a positive dilation field over the configuration space; # perform PDT-style importance resampling; # compute the resulting two-point correlation function <math>\xi(r)</math>; # compare transformed and baseline catalogs. One example dilation field is: <math> D(x)=\exp(\lambda\phi(x)) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\phi(x)\ge0</math> is a nonnegative configuration-space field. An example seed-field construction is: <math> \phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right) </math> where <math>s_k</math> are seed locations and <math>\sigma</math> controls the width of the seed influence. The two-point correlation function may be estimated using the normalized Landy–Szalay estimator: <math> \xi(r) = \frac{DD(r)-2DR(r)+RR(r)}{RR(r)} </math> where <math>DD</math>, <math>DR</math>, and <math>RR</math> are normalized pair counts. {{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}} When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration. === Computational demonstrations === Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages. {{collapse top|Python demonstration placeholder}} <syntaxhighlight lang="python"> # Example implementations may be maintained separately # on GitHub, OSF, or supplementary Wikiversity pages. </syntaxhighlight> {{collapse bottom}} == Scope and Limitations == PDT is a mathematical framework for measure transformations. It does not claim: * a replacement theory for General Relativity or Quantum Mechanics; * empirical confirmation without explicit predictions and tests; * observational validation without independently reproducible analysis. The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations. == Speculative Extensions and Geometric Renormalization == ''This section is speculative and exploratory in nature.'' Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry. Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects. Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation. At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations. == Future directions == * develop canonical families of dilation fields and invariants; * clarify “structure-from-measure” diagnostics; * publish reproducible simulation notebooks and parameter sweeps; * compare multiple dilation families under shared evaluation criteria; * investigate connections between probabilistic geometry and curvature-dependent statistical measures. == Future Directions: Probability Element (PE) == A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics. The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution. This can be expressed in terms of a dimensionless ratio: <math>\eta = \frac{\sigma_P}{\sigma}</math> where: <math>\sigma_P</math> is a hypothesized minimal probability-resolution scale, <math>\sigma</math> is an effective distinguishability scale in probability-state space. === Conceptual motivation === Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry. === Illustrative toy model (not derived physics) === As a heuristic example, one may consider a modification to special relativistic time dilation of the form: <math>d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}</math> where: <math>v</math> is velocity, <math>c</math> is the speed of light, <math>\eta = \sigma_P / \sigma</math> encodes a proposed probability-resolution scale. This expression is constructed such that standard special relativity is recovered exactly in the limit <math>\eta \to 0</math>. === Status === The Probability Element concept is: Not part of standard Fisher information geometry not derived from quantum mechanics or general relativity not currently empirically established. It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale. === Open questions === Key open research directions include: Whether a consistent discrete formulation of probability geometry can be constructed. Whether a fundamental probability-resolution scale <math>\sigma_P</math> can be derived from known physical principles. Whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions. == Convergence behavior == Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations. === Qualitative convergence classes === Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior: * '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy. * '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration. * '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions. * '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior. === Entropy and convergence === In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time. The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations. === Attractor-like behavior === Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense. Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks. == Current limitations == PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation. Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms. Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks. == See also == * [[w:Buffon's needle problem|Buffon's needle problem]] * [[w:Probability measure|Probability measure]] * [[w:Importance sampling|Importance sampling]] * [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]] * [[w:Dynamical system|Dynamical systems]] * [[w:Entropy (information theory)|Entropy]] * [[w:Information theory|Information theory]] * [[w:Measure theory|Measure theory]] * [[w:Geometric probability|Geometric probability]] * [[w:Shannon entropy|Shannon entropy]] * [[w:Stochastic process|Stochastic process]] * [[w:Fixed point (mathematics)|Fixed point]] * [[w:Convergence (mathematics)|Convergence]] == Subpages == The following subpages develop mathematical extensions and specialized topics related to Probability Dilation Theory (PDT). * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows|Fisher Geometry and Dilation Flows]] – studies information geometry, Fisher distance, and geodesic properties of PDT trajectories. * [[Probability Dilation Theory/Logit Representation of PE|Logit Representation of PE]] – develops the log-odds representation of probability elements and exponential PDT flows. * [[Probability Dilation Theory/Convergence and Fixed Points|Convergence and Fixed Points]] – investigates invariant measures, attractors, and stability of iterative PDT transformations. * [[Probability Dilation Theory/Stochastic Dilation Fields|Stochastic Dilation Fields]] – studies random and time-dependent dilation fields, ergodicity, and stochastic measure evolution. * [[Probability Dilation Theory/Entropy Evolution|Entropy Evolution]] – examines Shannon entropy under repeated probability dilation. * [[Probability Dilation Theory/Wasserstein Geometry|Wasserstein Geometry]] – explores distances between probability measures and convergence in measure space. * [[Probability Dilation Theory/Measure-Theoretic Foundations|Measure-Theoretic Foundations]] – develops rigorous measure-theoretic aspects of PDT including normalization and existence conditions. * [[Probability Dilation Theory/Euler Methods and Continuous-Time PDT]] – investigates continuous probability flows and Euler approximations of PDT. * [[Probability Dilation Theory/Worked Example]] – canonical binary example illustrating PDT transformations and geometry. == Numerical Iteration Table == The canonical PDT example may be iterated repeatedly using the dilation field <math> D=(2,1). </math> Starting from <math> P_0=(0.30,0.70), </math> successive PDT iterations produce the following approximate values. {| class="wikitable" ! Iteration ! First-State Probability ! Second-State Probability | ! Shannon Entropy | | ----------------- | | 0 | | 0.300 | | 0.700 | | 0.611 | | - | | 1 | | 0.462 | | 0.538 | | 0.690 | | - | | 2 | | 0.632 | | 0.368 | | 0.658 | | - | | 3 | | 0.774 | | 0.226 | | 0.534 | | - | | 4 | | 0.873 | | 0.127 | | 0.381 | | - | | 5 | | 0.932 | | 0.068 | | 0.249 | | } | The entropy initially increases as the distribution moves closer to uniformity. After passing near the maximum-entropy state, entropy decreases as probability becomes increasingly concentrated in the first state. This behavior illustrates that entropy evolution under PDT need not be monotonic. == Notation == Throughout PDT, the following notation is used: {| class="wikitable" ! Symbol ! Meaning |- | <math>P</math> | Probability measure |- | <math>P_n</math> | nth iterate of PDT |- | <math>T_D</math> | Probability dilation operator |- | <math>D(x)</math> | Dilation field |- | <math>Z(P,D)</math> | Normalization factor |- | <math>H(P)</math> | Shannon entropy |- | <math>d_F</math> | Fisher-Rao distance |- | <math>W_p</math> | Wasserstein distance |- | <math>\ell</math> | Logit coordinate |- | <math>PE</math> | Probability Element |} == Related probabilistic and geometric literature == Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works: * Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014. * Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19 * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997. * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005. * Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007. * Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656. == Copyright and licensing == Text and original figures © Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). Reuse permitted with attribution. gg8mljxpa3tqjk9gnd968iix9xtidgo 2816240 2816239 2026-06-18T19:37:14Z Howie2024 2995240 /* Subpages */ Future Research Directions subpage 2816240 wikitext text/x-wiki {{Research project}} {{Original research}} {{To be peer reviewed}} {{subst:proofread}} == Research abstract == '''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values. The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics. At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions. PDT treats a probability measure as the primary mathematical object and investigates: * invariant identities induced by reweighting, * composition and iteration of dilations, * fixed points and near-fixed behavior, * whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations). PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions. == Overview == PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is: * mathematically well-defined (positivity and normalization), * composable under iteration, * analyzable for invariants and fixed points. === Conceptual interpretation === A simplified conceptual flow of the PDT framework is: <pre> Baseline probability measure P ↓ Positive dilation field D(x) ↓ Reweighted probability measure P~ ↓ Observable statistical changes </pre> Repeated dilation may qualitatively behave as: <pre> Broad initial distribution ↓ Localized reweighting ↓ Probability concentration ↓ Emergent multiscale structure </pre> Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field. Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as: <pre> P₀ ↓ D₁ P₁ ↓ D₂ P₂ ↓ D₃ P₃ ↓ ⋯ </pre> where each dilation field reweights the probability structure generated by the previous step. Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. = Mathematical framework = == Definitions and notation == Let <math>(\Omega,\Sigma)</math> be a measurable space. * <math>P</math> denotes a probability measure on <math>(\Omega,\Sigma)</math>. * If <math>P</math> has a density <math>p</math> with respect to a reference measure <math>\mu</math>, then <math>dP=p\,d\mu</math>. * <math>D:\Omega\to(0,\infty)</math> is a measurable '''dilation field''' (a positive weight function). * <math>Z(P,D)</math> is the normalization constant: .<math> Z(P,D)=\int_\Omega D\,dP </math> * For an observable <math>f:\Omega\to\mathbb{R}</math> integrable under the relevant measure, <math> \mathbb{E}_P[f] = \int_\Omega f\,dP </math>. == PDT transformation (probability reweighting) == Given <math>P</math> and <math>D</math> with <math>0<Z(P,D)<\infty</math>, define the '''PDT transform''' <math>\widetilde{P}=\mathrm{PDT}(P;D)</math> by: <math> \widetilde{P}(A) = \frac{ \int_A D\,dP }{ \int_\Omega D\,dP } \quad\text{for all }A\in\Sigma </math> If <math>dP=p\,d\mu</math>, then <math>d\widetilde{P}=\widetilde{p}\,d\mu</math>, where <math> \widetilde{p}(x) = \frac{D(x)\,p(x)}{Z} </math> and <math> Z = \int_\Omega D(x)\,p(x)\,d\mu </math> '''Interpretation:''' the dilation field <math>D</math> shifts probability mass toward regions where <math>D</math> is larger, while renormalization keeps total probability equal to 1. PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures. Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations. A familiar physical example of a strictly positive factor is the Lorentz factor: <math> \gamma(v) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} </math> for <math> |v|<c </math> Lorentz contraction for a rod of rest length <math>L_0</math> moving at speed <math>v</math> is: <math> L(v)=\frac{L_0}{\gamma(v)} </math> To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on <math>\gamma</math>. == Worked finite example == Consider a finite probability space: <math> \Omega=\{a,b,c\} </math> with baseline probabilities: <math> P(a)=0.2,\quad P(b)=0.3,\quad P(c)=0.5 </math> Define a positive dilation field: <math> D(a)=1,\quad D(b)=2,\quad D(c)=4 </math> The normalization constant is: <math> Z=\sum_x D(x)P(x) </math> giving: <math> Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8 </math> The PDT-transformed probabilities become: <math> \widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071 </math> <math> \widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214 </math> <math> \widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714 </math> This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization. == Composition of dilations == An important structural property of sequential PDT transformations is that compose multiplicatively. Suppose two positive dilation fields: <math> D_1(x)>0 </math> and <math> D_2(x)>0 </math> are applied successively to a baseline probability measure <math>P</math>. The first dilation produces: <math> \widetilde{P}_1(A) = \frac{\int_A D_1\,dP} {\int_\Omega D_1\,dP} </math> Applying the second dilation field to <math>\widetilde{P}_1</math> gives: <math> \widetilde{P}_2(A) = \frac{\int_A D_2\,d\widetilde{P}_1} {\int_\Omega D_2\,d\widetilde{P}_1} </math> Substituting the first transformation into the second yields: <math> \widetilde{P}_2(A) = \frac{ \int_A D_2D_1\,dP }{ \int_\Omega D_2D_1\,dP } </math> This shows that sequential PDT transformations compose through multiplication of the dilation fields. This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application. == Fixed points and iterative dynamics == An important question in PDT concerns the long-term behavior of repeated PDT transformations. Given an initial probability measure: <math> P_0 </math> and a sequence of positive dilation fields: <math> D_1,D_2,D_3,\dots </math> successive PDT transformations generate a sequence of measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow P_3 \rightarrow \cdots </math> where each transformed measure is obtained by reweighting the previous one. A measure <math>P</math> is called a fixed point of a dilation field <math>D</math> if: <math> \widetilde{P}=P </math> under the PDT transformation. In the simplest case, this requires the dilation field to be constant almost everywhere with respect to <math>P</math>. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization. More generally, repeated compositions of nontrivial dilation fields may generate: * hierarchical probability structure; * multiscale statistical behavior; * attractor-like distributions; * approximately stable transformed measures. These questions connect PDT to broader areas of: * dynamical systems; * stochastic processes; * iterative renormalization methods; * probabilistic geometry. At present these iterative properties remain largely unexplored within the PDT framework. == Entropy and iterative probability flow == Repeated PDT transformations may alter the entropy structure of a probability measure. For a discrete probability distribution: <math> P=\{p_i\} </math> the Shannon entropy is: <math> H(P) = -\sum_i p_i \log p_i </math> Under iterative PDT transformation, successive transformed measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow \cdots </math> may exhibit changing entropy behavior depending on the structure of the dilation fields. For example: * strongly localized dilation fields may concentrate probability mass and reduce entropy; * broader or smoothing dilation fields may distribute probability more evenly and increase entropy; * iterative compositions may generate approximately stable entropy profiles. These questions connect PDT to: * information theory, * statistical mechanics, * stochastic dynamics, * and renormalization-style iterative systems. At present the entropy behavior of iterative PDT transformations remains an open area for investigation. == Toy experiment: entropy under repeated dilation == A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution. Let the initial probability distribution be: <math> P_0=(0.2,0.2,0.2,0.2,0.2) </math> and define a positive dilation field: <math> D=(1,1,2,4,8) </math> At each step, apply the PDT update: <math> P_{n+1}(i) = \frac{D(i)P_n(i)} {\sum_j D(j)P_n(j)} </math> The Shannon entropy is: <math> H(P_n) = -\sum_i P_n(i)\log P_n(i) </math> In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately: <math> H(P_0)\approx1.6094 </math> to: <math> H(P_{10})\approx0.00775 </math> The final distribution is approximately: <math> P_{10} \approx (0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437) </math> This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior. == Mathematical context == PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature. In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis. The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations. === Example entropy evolution === {| class="wikitable" ! Iteration !! Shannon entropy |- | 0 || 1.6094 |- | 1 || 1.2990 |- | 2 || 0.7790 |- | 3 || 0.4399 |- | 5 || 0.1500 |- | 10 || 0.0078 |} Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states. === Localized dilation fields === A useful class of PDT transformations is generated by localized positive dilation fields. Consider a one-dimensional finite configuration space with states indexed by: <math> x=0,1,2,\dots,N </math> and define a localized dilation field centered at <math>x_0</math>: <math> D(x) = \exp\!\left( \lambda \exp\!\left( -\frac{(x-x_0)^2}{2\sigma^2} \right) \right) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\sigma</math> controls the spatial width of the localized field. Narrow values of <math>\sigma</math> produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space. Under iterative PDT dynamics: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> the probability distribution may progressively concentrate near the center of the dilation field. === Example entropy evolution for localized fields === Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior: {| class="wikitable" ! Field width <math>\sigma</math> ! Final entropy after 10 iterations ! Maximum probability after 10 iterations |- | 1.5 || 0.0352 || 0.9950 |- | 3.0 || 0.8162 || 0.7141 |- | 6.0 || 1.5367 || 0.3595 |} [[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]] [[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]] These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction. == Comparative entropy-flow experiments == The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions. {| class="wikitable" |+ Comparative entropy-flow behavior under PDT field classes ! Field class ! Final entropy ! Entropy decrease ! Final max probability ! Qualitative behavior |- | Localized | 0.3104 | 3.4032 | 0.9275 | Strong probability concentration |- | Oscillatory | 1.5779 | 2.1357 | 0.3418 | Distributed oscillatory structure |- | Multi-peak | 0.2851 | 3.4284 | 0.9425 | Multiple concentration regions |- | Stochastic | 0.7744 | 2.9392 | 0.7413 | Fluctuating concentration behavior |} These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example. In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space. [[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]] The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Oscillatory dilation fields === Another useful class of PDT transformations is generated by oscillatory positive dilation fields. One example is: <math> D(x) = \exp(\lambda\sin(kx)) </math> where: * <math>\lambda>0</math> controls the strength of the oscillatory amplification; * <math>k</math> controls the spatial frequency of the oscillation. Because the exponential is always positive, the dilation field remains strictly positive for all states. Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space. Under repeated PDT transformation: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor. === Example oscillatory-field experiment === A finite-state experiment was performed using: * 41 discrete states; * an initially uniform probability distribution; * a positive oscillatory dilation field with three spatial oscillation cycles; * 10 successive PDT iterations. Representative entropy behavior was: {| class="wikitable" ! Iteration ! Shannon entropy |- | 0 || 3.7136 |- | 2 || 2.8699 |- | 5 || 2.3018 |- | 10 || 1.9335 |} Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space. After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state. This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures. The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Multi-peak localized dilation fields === A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space. One example is: <math> D(x) = \exp\!\left( \sum_k \lambda_k \exp\!\left( -\frac{(x-x_k)^2}{2\sigma_k^2} \right) \right) </math> where: * <math>x_k</math> are the locations of the dilation peaks; * <math>\lambda_k>0</math> control the amplification strength of each peak; * <math>\sigma_k</math> control the spatial width of each localized region. This construction generates a positive multimodal dilation landscape containing several competing amplification regions. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward multiple partially localized concentration regions. Unlike single localized dilation fields, multi-peak fields may generate: * competing attractor-like regions; * hierarchical probability concentration; * partially stabilized multimodal distributions; * multiscale probability structure. Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor: * dominance by a single peak; * coexistence of several concentration regions; * or slowly evolving metastable probability structures. === Conceptual interpretation === A qualitative iterative evolution may be visualized as: <pre> Broad initial distribution ↓ Multiple localized amplifications ↓ Competing concentration regions ↓ Emergent multimodal probability structure </pre> This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone. At present these behaviors remain exploratory computational observations within finite-state toy models. === Random and stochastic dilation fields === Another important class of PDT transformations arises when the dilation field itself varies stochastically. A simple stochastic dilation field may be written schematically as: <math> D_n(x) = \exp\!\left( \sigma \eta_n(x) \right) </math> where: * <math>\eta_n(x)</math> is a random field or stochastic fluctuation at iteration <math>n</math>; * <math>\sigma>0</math> controls the strength of the stochastic variation. Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D_n(x)P_n(x) }{ \sum_y D_n(y)P_n(y) } </math> the probability landscape itself fluctuates dynamically from one iteration to the next. Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate: * fluctuating concentration regions; * transient attractor-like structures; * noise-driven entropy evolution; * intermittent probability concentration; * metastable probabilistic configurations. === Conceptual interpretation === A qualitative stochastic evolution may be visualized as: <pre> Broad initial distribution ↓ Random localized amplification ↓ Fluctuating concentration regions ↓ Dynamic probabilistic structure </pre> Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit: * partial concentration, * persistent fluctuations, * stochastic stabilization, * or continuously evolving probabilistic structure. These ideas connect PDT to broader areas of: * stochastic processes; * random multiplicative systems; * statistical mechanics; * noise-driven dynamical systems; * probabilistic geometry. At present these behaviors remain exploratory computational possibilities within finite-state toy models. == Qualitative classes of iterative PDT behavior == Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation. The following table summarizes several representative classes explored within finite-state toy models. {| class="wikitable" ! Dilation-field class ! Typical iterative behavior ! Representative qualitative structure |- | Localized fields | Strong entropy reduction and concentration toward a dominant region | Single attractor-like concentration |- | Oscillatory fields | Distributed amplification with slower entropy reduction | Patterned multimodal structure |- | Multi-peak localized fields | Competition between several concentration regions | Hierarchical or metastable probability structure |- | Random and stochastic fields | Fluctuating amplification and noise-driven evolution | Dynamic probabilistic landscapes |} These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field. Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself. At present these qualitative behaviors remain exploratory computational observations within finite-state toy models. == Numerical simulation and iterative models == === Simulation model description === In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points. Two equivalent discrete implementations are common: * '''weighted evaluation''': retain all points and assign weights proportional to <math>D</math>; * '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to <math>D</math>. === Demonstration: reweighting mock galaxy catalogs === A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box. The demonstration pipeline is: # generate a baseline mock catalog; # define a positive dilation field over the configuration space; # perform PDT-style importance resampling; # compute the resulting two-point correlation function <math>\xi(r)</math>; # compare transformed and baseline catalogs. One example dilation field is: <math> D(x)=\exp(\lambda\phi(x)) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\phi(x)\ge0</math> is a nonnegative configuration-space field. An example seed-field construction is: <math> \phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right) </math> where <math>s_k</math> are seed locations and <math>\sigma</math> controls the width of the seed influence. The two-point correlation function may be estimated using the normalized Landy–Szalay estimator: <math> \xi(r) = \frac{DD(r)-2DR(r)+RR(r)}{RR(r)} </math> where <math>DD</math>, <math>DR</math>, and <math>RR</math> are normalized pair counts. {{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}} When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration. === Computational demonstrations === Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages. {{collapse top|Python demonstration placeholder}} <syntaxhighlight lang="python"> # Example implementations may be maintained separately # on GitHub, OSF, or supplementary Wikiversity pages. </syntaxhighlight> {{collapse bottom}} == Scope and Limitations == PDT is a mathematical framework for measure transformations. It does not claim: * a replacement theory for General Relativity or Quantum Mechanics; * empirical confirmation without explicit predictions and tests; * observational validation without independently reproducible analysis. The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations. == Speculative Extensions and Geometric Renormalization == ''This section is speculative and exploratory in nature.'' Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry. Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects. Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation. At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations. == Future directions == * develop canonical families of dilation fields and invariants; * clarify “structure-from-measure” diagnostics; * publish reproducible simulation notebooks and parameter sweeps; * compare multiple dilation families under shared evaluation criteria; * investigate connections between probabilistic geometry and curvature-dependent statistical measures. == Future Directions: Probability Element (PE) == A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics. The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution. This can be expressed in terms of a dimensionless ratio: <math>\eta = \frac{\sigma_P}{\sigma}</math> where: <math>\sigma_P</math> is a hypothesized minimal probability-resolution scale, <math>\sigma</math> is an effective distinguishability scale in probability-state space. === Conceptual motivation === Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry. === Illustrative toy model (not derived physics) === As a heuristic example, one may consider a modification to special relativistic time dilation of the form: <math>d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}</math> where: <math>v</math> is velocity, <math>c</math> is the speed of light, <math>\eta = \sigma_P / \sigma</math> encodes a proposed probability-resolution scale. This expression is constructed such that standard special relativity is recovered exactly in the limit <math>\eta \to 0</math>. === Status === The Probability Element concept is: Not part of standard Fisher information geometry not derived from quantum mechanics or general relativity not currently empirically established. It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale. === Open questions === Key open research directions include: Whether a consistent discrete formulation of probability geometry can be constructed. Whether a fundamental probability-resolution scale <math>\sigma_P</math> can be derived from known physical principles. Whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions. == Convergence behavior == Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations. === Qualitative convergence classes === Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior: * '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy. * '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration. * '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions. * '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior. === Entropy and convergence === In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time. The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations. === Attractor-like behavior === Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense. Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks. == Current limitations == PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation. Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms. Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks. == See also == * [[w:Buffon's needle problem|Buffon's needle problem]] * [[w:Probability measure|Probability measure]] * [[w:Importance sampling|Importance sampling]] * [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]] * [[w:Dynamical system|Dynamical systems]] * [[w:Entropy (information theory)|Entropy]] * [[w:Information theory|Information theory]] * [[w:Measure theory|Measure theory]] * [[w:Geometric probability|Geometric probability]] * [[w:Shannon entropy|Shannon entropy]] * [[w:Stochastic process|Stochastic process]] * [[w:Fixed point (mathematics)|Fixed point]] * [[w:Convergence (mathematics)|Convergence]] == Subpages == The following subpages develop mathematical extensions and specialized topics related to Probability Dilation Theory (PDT). * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows|Fisher Geometry and Dilation Flows]] – studies information geometry, Fisher distance, and geodesic properties of PDT trajectories. * [[Probability Dilation Theory/Logit Representation of PE|Logit Representation of PE]] – develops the log-odds representation of probability elements and exponential PDT flows. * [[Probability Dilation Theory/Convergence and Fixed Points|Convergence and Fixed Points]] – investigates invariant measures, attractors, and stability of iterative PDT transformations. * [[Probability Dilation Theory/Stochastic Dilation Fields|Stochastic Dilation Fields]] – studies random and time-dependent dilation fields, ergodicity, and stochastic measure evolution. * [[Probability Dilation Theory/Entropy Evolution|Entropy Evolution]] – examines Shannon entropy under repeated probability dilation. * [[Probability Dilation Theory/Wasserstein Geometry|Wasserstein Geometry]] – explores distances between probability measures and convergence in measure space. * [[Probability Dilation Theory/Measure-Theoretic Foundations|Measure-Theoretic Foundations]] – develops rigorous measure-theoretic aspects of PDT including normalization and existence conditions. * [[Probability Dilation Theory/Euler Methods and Continuous-Time PDT]] – investigates continuous probability flows and Euler approximations of PDT. * [[Probability Dilation Theory/Worked Example]] – canonical binary example illustrating PDT transformations and geometry. Probability Dilation Theory/Future Research Directions == Numerical Iteration Table == The canonical PDT example may be iterated repeatedly using the dilation field <math> D=(2,1). </math> Starting from <math> P_0=(0.30,0.70), </math> successive PDT iterations produce the following approximate values. {| class="wikitable" ! Iteration ! First-State Probability ! Second-State Probability | ! Shannon Entropy | | ----------------- | | 0 | | 0.300 | | 0.700 | | 0.611 | | - | | 1 | | 0.462 | | 0.538 | | 0.690 | | - | | 2 | | 0.632 | | 0.368 | | 0.658 | | - | | 3 | | 0.774 | | 0.226 | | 0.534 | | - | | 4 | | 0.873 | | 0.127 | | 0.381 | | - | | 5 | | 0.932 | | 0.068 | | 0.249 | | } | The entropy initially increases as the distribution moves closer to uniformity. After passing near the maximum-entropy state, entropy decreases as probability becomes increasingly concentrated in the first state. This behavior illustrates that entropy evolution under PDT need not be monotonic. == Notation == Throughout PDT, the following notation is used: {| class="wikitable" ! Symbol ! Meaning |- | <math>P</math> | Probability measure |- | <math>P_n</math> | nth iterate of PDT |- | <math>T_D</math> | Probability dilation operator |- | <math>D(x)</math> | Dilation field |- | <math>Z(P,D)</math> | Normalization factor |- | <math>H(P)</math> | Shannon entropy |- | <math>d_F</math> | Fisher-Rao distance |- | <math>W_p</math> | Wasserstein distance |- | <math>\ell</math> | Logit coordinate |- | <math>PE</math> | Probability Element |} == Related probabilistic and geometric literature == Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works: * Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014. * Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19 * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997. * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005. * Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007. * Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656. == Copyright and licensing == Text and original figures © Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). Reuse permitted with attribution. hkdb3r0no0ljwp6hevwrbsv4ja73lpc 2816257 2816240 2026-06-19T00:03:10Z Howie2024 2995240 /* Subpages */ Probability Dilation Theory/Decoherence Analogy and Simulation 2816257 wikitext text/x-wiki {{Research project}} {{Original research}} {{To be peer reviewed}} {{subst:proofread}} == Research abstract == '''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values. The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics. At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions. PDT treats a probability measure as the primary mathematical object and investigates: * invariant identities induced by reweighting, * composition and iteration of dilations, * fixed points and near-fixed behavior, * whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations). PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions. == Overview == PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is: * mathematically well-defined (positivity and normalization), * composable under iteration, * analyzable for invariants and fixed points. === Conceptual interpretation === A simplified conceptual flow of the PDT framework is: <pre> Baseline probability measure P ↓ Positive dilation field D(x) ↓ Reweighted probability measure P~ ↓ Observable statistical changes </pre> Repeated dilation may qualitatively behave as: <pre> Broad initial distribution ↓ Localized reweighting ↓ Probability concentration ↓ Emergent multiscale structure </pre> Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field. Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as: <pre> P₀ ↓ D₁ P₁ ↓ D₂ P₂ ↓ D₃ P₃ ↓ ⋯ </pre> where each dilation field reweights the probability structure generated by the previous step. Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. = Mathematical framework = == Definitions and notation == Let <math>(\Omega,\Sigma)</math> be a measurable space. * <math>P</math> denotes a probability measure on <math>(\Omega,\Sigma)</math>. * If <math>P</math> has a density <math>p</math> with respect to a reference measure <math>\mu</math>, then <math>dP=p\,d\mu</math>. * <math>D:\Omega\to(0,\infty)</math> is a measurable '''dilation field''' (a positive weight function). * <math>Z(P,D)</math> is the normalization constant: .<math> Z(P,D)=\int_\Omega D\,dP </math> * For an observable <math>f:\Omega\to\mathbb{R}</math> integrable under the relevant measure, <math> \mathbb{E}_P[f] = \int_\Omega f\,dP </math>. == PDT transformation (probability reweighting) == Given <math>P</math> and <math>D</math> with <math>0<Z(P,D)<\infty</math>, define the '''PDT transform''' <math>\widetilde{P}=\mathrm{PDT}(P;D)</math> by: <math> \widetilde{P}(A) = \frac{ \int_A D\,dP }{ \int_\Omega D\,dP } \quad\text{for all }A\in\Sigma </math> If <math>dP=p\,d\mu</math>, then <math>d\widetilde{P}=\widetilde{p}\,d\mu</math>, where <math> \widetilde{p}(x) = \frac{D(x)\,p(x)}{Z} </math> and <math> Z = \int_\Omega D(x)\,p(x)\,d\mu </math> '''Interpretation:''' the dilation field <math>D</math> shifts probability mass toward regions where <math>D</math> is larger, while renormalization keeps total probability equal to 1. PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures. Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations. A familiar physical example of a strictly positive factor is the Lorentz factor: <math> \gamma(v) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} </math> for <math> |v|<c </math> Lorentz contraction for a rod of rest length <math>L_0</math> moving at speed <math>v</math> is: <math> L(v)=\frac{L_0}{\gamma(v)} </math> To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on <math>\gamma</math>. == Worked finite example == Consider a finite probability space: <math> \Omega=\{a,b,c\} </math> with baseline probabilities: <math> P(a)=0.2,\quad P(b)=0.3,\quad P(c)=0.5 </math> Define a positive dilation field: <math> D(a)=1,\quad D(b)=2,\quad D(c)=4 </math> The normalization constant is: <math> Z=\sum_x D(x)P(x) </math> giving: <math> Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8 </math> The PDT-transformed probabilities become: <math> \widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071 </math> <math> \widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214 </math> <math> \widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714 </math> This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization. == Composition of dilations == An important structural property of sequential PDT transformations is that compose multiplicatively. Suppose two positive dilation fields: <math> D_1(x)>0 </math> and <math> D_2(x)>0 </math> are applied successively to a baseline probability measure <math>P</math>. The first dilation produces: <math> \widetilde{P}_1(A) = \frac{\int_A D_1\,dP} {\int_\Omega D_1\,dP} </math> Applying the second dilation field to <math>\widetilde{P}_1</math> gives: <math> \widetilde{P}_2(A) = \frac{\int_A D_2\,d\widetilde{P}_1} {\int_\Omega D_2\,d\widetilde{P}_1} </math> Substituting the first transformation into the second yields: <math> \widetilde{P}_2(A) = \frac{ \int_A D_2D_1\,dP }{ \int_\Omega D_2D_1\,dP } </math> This shows that sequential PDT transformations compose through multiplication of the dilation fields. This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application. == Fixed points and iterative dynamics == An important question in PDT concerns the long-term behavior of repeated PDT transformations. Given an initial probability measure: <math> P_0 </math> and a sequence of positive dilation fields: <math> D_1,D_2,D_3,\dots </math> successive PDT transformations generate a sequence of measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow P_3 \rightarrow \cdots </math> where each transformed measure is obtained by reweighting the previous one. A measure <math>P</math> is called a fixed point of a dilation field <math>D</math> if: <math> \widetilde{P}=P </math> under the PDT transformation. In the simplest case, this requires the dilation field to be constant almost everywhere with respect to <math>P</math>. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization. More generally, repeated compositions of nontrivial dilation fields may generate: * hierarchical probability structure; * multiscale statistical behavior; * attractor-like distributions; * approximately stable transformed measures. These questions connect PDT to broader areas of: * dynamical systems; * stochastic processes; * iterative renormalization methods; * probabilistic geometry. At present these iterative properties remain largely unexplored within the PDT framework. == Entropy and iterative probability flow == Repeated PDT transformations may alter the entropy structure of a probability measure. For a discrete probability distribution: <math> P=\{p_i\} </math> the Shannon entropy is: <math> H(P) = -\sum_i p_i \log p_i </math> Under iterative PDT transformation, successive transformed measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow \cdots </math> may exhibit changing entropy behavior depending on the structure of the dilation fields. For example: * strongly localized dilation fields may concentrate probability mass and reduce entropy; * broader or smoothing dilation fields may distribute probability more evenly and increase entropy; * iterative compositions may generate approximately stable entropy profiles. These questions connect PDT to: * information theory, * statistical mechanics, * stochastic dynamics, * and renormalization-style iterative systems. At present the entropy behavior of iterative PDT transformations remains an open area for investigation. == Toy experiment: entropy under repeated dilation == A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution. Let the initial probability distribution be: <math> P_0=(0.2,0.2,0.2,0.2,0.2) </math> and define a positive dilation field: <math> D=(1,1,2,4,8) </math> At each step, apply the PDT update: <math> P_{n+1}(i) = \frac{D(i)P_n(i)} {\sum_j D(j)P_n(j)} </math> The Shannon entropy is: <math> H(P_n) = -\sum_i P_n(i)\log P_n(i) </math> In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately: <math> H(P_0)\approx1.6094 </math> to: <math> H(P_{10})\approx0.00775 </math> The final distribution is approximately: <math> P_{10} \approx (0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437) </math> This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior. == Mathematical context == PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature. In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis. The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations. === Example entropy evolution === {| class="wikitable" ! Iteration !! Shannon entropy |- | 0 || 1.6094 |- | 1 || 1.2990 |- | 2 || 0.7790 |- | 3 || 0.4399 |- | 5 || 0.1500 |- | 10 || 0.0078 |} Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states. === Localized dilation fields === A useful class of PDT transformations is generated by localized positive dilation fields. Consider a one-dimensional finite configuration space with states indexed by: <math> x=0,1,2,\dots,N </math> and define a localized dilation field centered at <math>x_0</math>: <math> D(x) = \exp\!\left( \lambda \exp\!\left( -\frac{(x-x_0)^2}{2\sigma^2} \right) \right) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\sigma</math> controls the spatial width of the localized field. Narrow values of <math>\sigma</math> produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space. Under iterative PDT dynamics: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> the probability distribution may progressively concentrate near the center of the dilation field. === Example entropy evolution for localized fields === Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior: {| class="wikitable" ! Field width <math>\sigma</math> ! Final entropy after 10 iterations ! Maximum probability after 10 iterations |- | 1.5 || 0.0352 || 0.9950 |- | 3.0 || 0.8162 || 0.7141 |- | 6.0 || 1.5367 || 0.3595 |} [[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]] [[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]] These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction. == Comparative entropy-flow experiments == The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions. {| class="wikitable" |+ Comparative entropy-flow behavior under PDT field classes ! Field class ! Final entropy ! Entropy decrease ! Final max probability ! Qualitative behavior |- | Localized | 0.3104 | 3.4032 | 0.9275 | Strong probability concentration |- | Oscillatory | 1.5779 | 2.1357 | 0.3418 | Distributed oscillatory structure |- | Multi-peak | 0.2851 | 3.4284 | 0.9425 | Multiple concentration regions |- | Stochastic | 0.7744 | 2.9392 | 0.7413 | Fluctuating concentration behavior |} These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example. In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space. [[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]] The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Oscillatory dilation fields === Another useful class of PDT transformations is generated by oscillatory positive dilation fields. One example is: <math> D(x) = \exp(\lambda\sin(kx)) </math> where: * <math>\lambda>0</math> controls the strength of the oscillatory amplification; * <math>k</math> controls the spatial frequency of the oscillation. Because the exponential is always positive, the dilation field remains strictly positive for all states. Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space. Under repeated PDT transformation: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor. === Example oscillatory-field experiment === A finite-state experiment was performed using: * 41 discrete states; * an initially uniform probability distribution; * a positive oscillatory dilation field with three spatial oscillation cycles; * 10 successive PDT iterations. Representative entropy behavior was: {| class="wikitable" ! Iteration ! Shannon entropy |- | 0 || 3.7136 |- | 2 || 2.8699 |- | 5 || 2.3018 |- | 10 || 1.9335 |} Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space. After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state. This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures. The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Multi-peak localized dilation fields === A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space. One example is: <math> D(x) = \exp\!\left( \sum_k \lambda_k \exp\!\left( -\frac{(x-x_k)^2}{2\sigma_k^2} \right) \right) </math> where: * <math>x_k</math> are the locations of the dilation peaks; * <math>\lambda_k>0</math> control the amplification strength of each peak; * <math>\sigma_k</math> control the spatial width of each localized region. This construction generates a positive multimodal dilation landscape containing several competing amplification regions. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward multiple partially localized concentration regions. Unlike single localized dilation fields, multi-peak fields may generate: * competing attractor-like regions; * hierarchical probability concentration; * partially stabilized multimodal distributions; * multiscale probability structure. Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor: * dominance by a single peak; * coexistence of several concentration regions; * or slowly evolving metastable probability structures. === Conceptual interpretation === A qualitative iterative evolution may be visualized as: <pre> Broad initial distribution ↓ Multiple localized amplifications ↓ Competing concentration regions ↓ Emergent multimodal probability structure </pre> This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone. At present these behaviors remain exploratory computational observations within finite-state toy models. === Random and stochastic dilation fields === Another important class of PDT transformations arises when the dilation field itself varies stochastically. A simple stochastic dilation field may be written schematically as: <math> D_n(x) = \exp\!\left( \sigma \eta_n(x) \right) </math> where: * <math>\eta_n(x)</math> is a random field or stochastic fluctuation at iteration <math>n</math>; * <math>\sigma>0</math> controls the strength of the stochastic variation. Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D_n(x)P_n(x) }{ \sum_y D_n(y)P_n(y) } </math> the probability landscape itself fluctuates dynamically from one iteration to the next. Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate: * fluctuating concentration regions; * transient attractor-like structures; * noise-driven entropy evolution; * intermittent probability concentration; * metastable probabilistic configurations. === Conceptual interpretation === A qualitative stochastic evolution may be visualized as: <pre> Broad initial distribution ↓ Random localized amplification ↓ Fluctuating concentration regions ↓ Dynamic probabilistic structure </pre> Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit: * partial concentration, * persistent fluctuations, * stochastic stabilization, * or continuously evolving probabilistic structure. These ideas connect PDT to broader areas of: * stochastic processes; * random multiplicative systems; * statistical mechanics; * noise-driven dynamical systems; * probabilistic geometry. At present these behaviors remain exploratory computational possibilities within finite-state toy models. == Qualitative classes of iterative PDT behavior == Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation. The following table summarizes several representative classes explored within finite-state toy models. {| class="wikitable" ! Dilation-field class ! Typical iterative behavior ! Representative qualitative structure |- | Localized fields | Strong entropy reduction and concentration toward a dominant region | Single attractor-like concentration |- | Oscillatory fields | Distributed amplification with slower entropy reduction | Patterned multimodal structure |- | Multi-peak localized fields | Competition between several concentration regions | Hierarchical or metastable probability structure |- | Random and stochastic fields | Fluctuating amplification and noise-driven evolution | Dynamic probabilistic landscapes |} These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field. Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself. At present these qualitative behaviors remain exploratory computational observations within finite-state toy models. == Numerical simulation and iterative models == === Simulation model description === In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points. Two equivalent discrete implementations are common: * '''weighted evaluation''': retain all points and assign weights proportional to <math>D</math>; * '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to <math>D</math>. === Demonstration: reweighting mock galaxy catalogs === A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box. The demonstration pipeline is: # generate a baseline mock catalog; # define a positive dilation field over the configuration space; # perform PDT-style importance resampling; # compute the resulting two-point correlation function <math>\xi(r)</math>; # compare transformed and baseline catalogs. One example dilation field is: <math> D(x)=\exp(\lambda\phi(x)) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\phi(x)\ge0</math> is a nonnegative configuration-space field. An example seed-field construction is: <math> \phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right) </math> where <math>s_k</math> are seed locations and <math>\sigma</math> controls the width of the seed influence. The two-point correlation function may be estimated using the normalized Landy–Szalay estimator: <math> \xi(r) = \frac{DD(r)-2DR(r)+RR(r)}{RR(r)} </math> where <math>DD</math>, <math>DR</math>, and <math>RR</math> are normalized pair counts. {{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}} When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration. === Computational demonstrations === Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages. {{collapse top|Python demonstration placeholder}} <syntaxhighlight lang="python"> # Example implementations may be maintained separately # on GitHub, OSF, or supplementary Wikiversity pages. </syntaxhighlight> {{collapse bottom}} == Scope and Limitations == PDT is a mathematical framework for measure transformations. It does not claim: * a replacement theory for General Relativity or Quantum Mechanics; * empirical confirmation without explicit predictions and tests; * observational validation without independently reproducible analysis. The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations. == Speculative Extensions and Geometric Renormalization == ''This section is speculative and exploratory in nature.'' Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry. Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects. Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation. At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations. == Future directions == * develop canonical families of dilation fields and invariants; * clarify “structure-from-measure” diagnostics; * publish reproducible simulation notebooks and parameter sweeps; * compare multiple dilation families under shared evaluation criteria; * investigate connections between probabilistic geometry and curvature-dependent statistical measures. == Future Directions: Probability Element (PE) == A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics. The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution. This can be expressed in terms of a dimensionless ratio: <math>\eta = \frac{\sigma_P}{\sigma}</math> where: <math>\sigma_P</math> is a hypothesized minimal probability-resolution scale, <math>\sigma</math> is an effective distinguishability scale in probability-state space. === Conceptual motivation === Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry. === Illustrative toy model (not derived physics) === As a heuristic example, one may consider a modification to special relativistic time dilation of the form: <math>d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}</math> where: <math>v</math> is velocity, <math>c</math> is the speed of light, <math>\eta = \sigma_P / \sigma</math> encodes a proposed probability-resolution scale. This expression is constructed such that standard special relativity is recovered exactly in the limit <math>\eta \to 0</math>. === Status === The Probability Element concept is: Not part of standard Fisher information geometry not derived from quantum mechanics or general relativity not currently empirically established. It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale. === Open questions === Key open research directions include: Whether a consistent discrete formulation of probability geometry can be constructed. Whether a fundamental probability-resolution scale <math>\sigma_P</math> can be derived from known physical principles. Whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions. == Convergence behavior == Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations. === Qualitative convergence classes === Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior: * '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy. * '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration. * '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions. * '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior. === Entropy and convergence === In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time. The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations. === Attractor-like behavior === Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense. Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks. == Current limitations == PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation. Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms. Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks. == See also == * [[w:Buffon's needle problem|Buffon's needle problem]] * [[w:Probability measure|Probability measure]] * [[w:Importance sampling|Importance sampling]] * [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]] * [[w:Dynamical system|Dynamical systems]] * [[w:Entropy (information theory)|Entropy]] * [[w:Information theory|Information theory]] * [[w:Measure theory|Measure theory]] * [[w:Geometric probability|Geometric probability]] * [[w:Shannon entropy|Shannon entropy]] * [[w:Stochastic process|Stochastic process]] * [[w:Fixed point (mathematics)|Fixed point]] * [[w:Convergence (mathematics)|Convergence]] == Subpages == The following subpages develop mathematical extensions and specialized topics related to Probability Dilation Theory (PDT). * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows|Fisher Geometry and Dilation Flows]] – studies information geometry, Fisher distance, and geodesic properties of PDT trajectories. * [[Probability Dilation Theory/Logit Representation of PE|Logit Representation of PE]] – develops the log-odds representation of probability elements and exponential PDT flows. * [[Probability Dilation Theory/Convergence and Fixed Points|Convergence and Fixed Points]] – investigates invariant measures, attractors, and stability of iterative PDT transformations. * [[Probability Dilation Theory/Stochastic Dilation Fields|Stochastic Dilation Fields]] – studies random and time-dependent dilation fields, ergodicity, and stochastic measure evolution. * [[Probability Dilation Theory/Entropy Evolution|Entropy Evolution]] – examines Shannon entropy under repeated probability dilation. * [[Probability Dilation Theory/Wasserstein Geometry|Wasserstein Geometry]] – explores distances between probability measures and convergence in measure space. * [[Probability Dilation Theory/Measure-Theoretic Foundations|Measure-Theoretic Foundations]] – develops rigorous measure-theoretic aspects of PDT including normalization and existence conditions. * [[Probability Dilation Theory/Euler Methods and Continuous-Time PDT]] – investigates continuous probability flows and Euler approximations of PDT. * [[Probability Dilation Theory/Worked Example]] – canonical binary example illustrating PDT transformations and geometry. ==Probability Dilation Theory/Future Research Directions== = Probability Dilation Theory / Decoherence Analogy and Simulation = ''This page is a subpage of [[Probability Dilation Theory]] and develops a speculative but mathematically well‑defined analogy between iterative probability dilation and quantum decoherence. It includes definitions, motivation, and a reproducible Python simulation.'' == 1. Overview == This page explores how Probability Dilation Theory (PDT) can be used to construct toy models that resemble certain statistical aspects of quantum decoherence. This work is: mathematical, not physical exploratory, not authoritative analogical, not a claim about quantum mechanics open for critique and improvement The goal is to provide a clear, reproducible framework for studying how iterative reweighting of probability measures can mimic the probability‑flow behavior seen in decoherence processes. == 2. Background: PDT and Decoherence == === 2.1 Probability Dilation Theory (PDT) === PDT studies how a probability measure :<math>P</math> is transformed by a positive dilation field :<math>D(x) > 0</math> via the operator :<math>P_D(A) = \frac{\int_A D(x)\, dP(x)}{\int_X D(x)\, dP(x)}.</math> Iterating this operator produces a sequence :<math>P_{n+1} = \mathcal{D}(P_n)</math> which may converge to attractors, fixed points, or stable distributions. === 2.2 Decoherence (informal summary) === In quantum mechanics, decoherence describes how a system loses phase coherence due to interaction with an environment. A density matrix :<math>\rho</math> evolves under a completely positive trace‑preserving (CPTP) map :<math>\mathcal{E}(\rho)</math> that typically: leaves diagonal probabilities unchanged or slowly biased exponentially suppresses off‑diagonal terms (coherences) This produces an effectively classical mixture. === 2.3 Why compare them? === Although PDT is purely classical, the diagonal part of many decoherence channels behaves like a dilation step, and the off‑diagonal decay can be modeled as a simple contraction. This makes PDT a useful toy model for exploring: probability concentration pointer‑state attractors emergent classicality iterative reweighting dynamics No physical claims are made. == 3. A Minimal Toy Model == We consider a two‑state system with classical probabilities :<math>P_n = (p_0^{(n)}, p_1^{(n)})</math> and a coherence magnitude :<math>r_n = |c_n|.</math> The model consists of two coupled updates: === 3.1 PDT dilation on probabilities === Given a dilation field :<math>D = (D_0, D_1)</math> we update :<math>p_i^{(n+1)} = \frac{D_i\, p_i^{(n)}}{D_0 p_0^{(n)} + D_1 p_1^{(n)}}.</math> This biases the system toward states with larger <math>D_i</math>. === 3.2 Coherence decay === We model decoherence by :<math>r_{n+1} = \alpha\, r_n</math> with :<math>0 \le \alpha < 1.</math> This is not quantum mechanical — it is a simple exponential decay rule. === 3.3 Combined update === Each iteration applies: PDT dilation on the diagonal Exponential decay on the coherence magnitude This produces a system that: flows toward a classical attractor loses coherence over time resembles decoherence in its statistical behavior == 4. Numerical Simulation (Python) == Below is a complete, runnable Python script that simulates the toy model and produces probability and coherence plots. <syntaxhighlight lang="python"> import numpy as np import matplotlib.pyplot as plt def simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=20 ): p0 = p0_init p1 = 1.0 - p0_init coh = coherence_init p0_hist = [p0] p1_hist = [p1] coh_hist = [coh] for n in range(steps): # PDT dilation step Z = D0 * p0 + D1 * p1 p0 = (D0 * p0) / Z p1 = (D1 * p1) / Z Coherence decay coh = alpha * coh p0_hist.append(p0) p1_hist.append(p1) coh_hist.append(coh) return np.array(p0_hist), np.array(p1_hist), np.array(coh_hist) if name == "main": p0_hist, p1_hist, coh_hist = simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=25 ) t = np.arange(len(p0_hist)) fig, ax1 = plt.subplots() ax1.set_xlabel("Iteration") ax1.set_ylabel("Probability", color="tab:blue") ax1.plot(t, p0_hist, label="p0", color="tab:blue") ax1.plot(t, p1_hist, label="p1", color="tab:cyan", linestyle="--") ax1.tick_params(axis="y", labelcolor="tab:blue") ax1.legend(loc="upper left") ax2 = ax1.twinx() ax2.set_ylabel("Coherence |c|", color="tab:red") ax2.plot(t, coh_hist, label="|c|", color="tab:red") ax2.tick_params(axis="y", labelcolor="tab:red") fig.tight_layout() plt.title("Iterative Dilation + Decoherence Toy Model") plt.show() </syntaxhighlight> == 5. Interpretation == === 5.1 Probability flow === If <math>D_0 > D_1</math>, the system flows toward :<math>p_0 \to 1.</math> This is analogous to a pointer state in decoherence. === 5.2 Coherence decay === The coherence magnitude :<math>r_n</math> decays exponentially, mimicking the suppression of off‑diagonal terms in a density matrix. === 5.3 Combined effect === The system becomes: more classical (probabilities concentrate) less coherent (off‑diagonal terms vanish) This mirrors the qualitative behavior of decoherence. == 6. Limitations == This is not a quantum model. No physical claims are made about dilation fields. The analogy is structural, not ontological. Decoherence involves entanglement; PDT does not. The model is intended for intuition and exploration only. == 7. Open Questions == Can more general decoherence channels be embedded in PDT? What are the fixed points of multi‑state dilation systems? Can continuous‑time dilation flows be defined? Are there PDE analogues of iterative dilation? Can this framework be useful in machine learning or statistical physics? == 8. See Also == [[Probability Dilation Theory]] [[Quantum decoherence]] [[Density matrix]] [[Bayesian updating]] == 9. Invitation for Collaboration == This page is part of an ongoing exploration of PDT and its possible mathematical analogies. Feedback, critique, and contributions from mathematicians, physicists, and computer scientists are welcome. == Numerical Iteration Table == The canonical PDT example may be iterated repeatedly using the dilation field <math> D=(2,1). </math> Starting from <math> P_0=(0.30,0.70), </math> successive PDT iterations produce the following approximate values. {| class="wikitable" ! Iteration ! First-State Probability ! Second-State Probability | ! Shannon Entropy | | ----------------- | | 0 | | 0.300 | | 0.700 | | 0.611 | | - | | 1 | | 0.462 | | 0.538 | | 0.690 | | - | | 2 | | 0.632 | | 0.368 | | 0.658 | | - | | 3 | | 0.774 | | 0.226 | | 0.534 | | - | | 4 | | 0.873 | | 0.127 | | 0.381 | | - | | 5 | | 0.932 | | 0.068 | | 0.249 | | } | The entropy initially increases as the distribution moves closer to uniformity. After passing near the maximum-entropy state, entropy decreases as probability becomes increasingly concentrated in the first state. This behavior illustrates that entropy evolution under PDT need not be monotonic. == Notation == Throughout PDT, the following notation is used: {| class="wikitable" ! Symbol ! Meaning |- | <math>P</math> | Probability measure |- | <math>P_n</math> | nth iterate of PDT |- | <math>T_D</math> | Probability dilation operator |- | <math>D(x)</math> | Dilation field |- | <math>Z(P,D)</math> | Normalization factor |- | <math>H(P)</math> | Shannon entropy |- | <math>d_F</math> | Fisher-Rao distance |- | <math>W_p</math> | Wasserstein distance |- | <math>\ell</math> | Logit coordinate |- | <math>PE</math> | Probability Element |} == Related probabilistic and geometric literature == Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works: * Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014. * Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19 * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997. * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005. * Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007. * Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656. == Copyright and licensing == Text and original figures © Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). Reuse permitted with attribution. dbcufyhsa7hgjxcx73ideufav8893us 2816258 2816257 2026-06-19T00:04:45Z Howie2024 2995240 /* Probability Dilation Theory/Future Research Directions */ 2816258 wikitext text/x-wiki {{Research project}} {{Original research}} {{To be peer reviewed}} {{subst:proofread}} == Research abstract == '''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values. The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics. At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions. PDT treats a probability measure as the primary mathematical object and investigates: * invariant identities induced by reweighting, * composition and iteration of dilations, * fixed points and near-fixed behavior, * whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations). PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions. == Overview == PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is: * mathematically well-defined (positivity and normalization), * composable under iteration, * analyzable for invariants and fixed points. === Conceptual interpretation === A simplified conceptual flow of the PDT framework is: <pre> Baseline probability measure P ↓ Positive dilation field D(x) ↓ Reweighted probability measure P~ ↓ Observable statistical changes </pre> Repeated dilation may qualitatively behave as: <pre> Broad initial distribution ↓ Localized reweighting ↓ Probability concentration ↓ Emergent multiscale structure </pre> Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field. Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as: <pre> P₀ ↓ D₁ P₁ ↓ D₂ P₂ ↓ D₃ P₃ ↓ ⋯ </pre> where each dilation field reweights the probability structure generated by the previous step. Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. = Mathematical framework = == Definitions and notation == Let <math>(\Omega,\Sigma)</math> be a measurable space. * <math>P</math> denotes a probability measure on <math>(\Omega,\Sigma)</math>. * If <math>P</math> has a density <math>p</math> with respect to a reference measure <math>\mu</math>, then <math>dP=p\,d\mu</math>. * <math>D:\Omega\to(0,\infty)</math> is a measurable '''dilation field''' (a positive weight function). * <math>Z(P,D)</math> is the normalization constant: .<math> Z(P,D)=\int_\Omega D\,dP </math> * For an observable <math>f:\Omega\to\mathbb{R}</math> integrable under the relevant measure, <math> \mathbb{E}_P[f] = \int_\Omega f\,dP </math>. == PDT transformation (probability reweighting) == Given <math>P</math> and <math>D</math> with <math>0<Z(P,D)<\infty</math>, define the '''PDT transform''' <math>\widetilde{P}=\mathrm{PDT}(P;D)</math> by: <math> \widetilde{P}(A) = \frac{ \int_A D\,dP }{ \int_\Omega D\,dP } \quad\text{for all }A\in\Sigma </math> If <math>dP=p\,d\mu</math>, then <math>d\widetilde{P}=\widetilde{p}\,d\mu</math>, where <math> \widetilde{p}(x) = \frac{D(x)\,p(x)}{Z} </math> and <math> Z = \int_\Omega D(x)\,p(x)\,d\mu </math> '''Interpretation:''' the dilation field <math>D</math> shifts probability mass toward regions where <math>D</math> is larger, while renormalization keeps total probability equal to 1. PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures. Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations. A familiar physical example of a strictly positive factor is the Lorentz factor: <math> \gamma(v) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} </math> for <math> |v|<c </math> Lorentz contraction for a rod of rest length <math>L_0</math> moving at speed <math>v</math> is: <math> L(v)=\frac{L_0}{\gamma(v)} </math> To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on <math>\gamma</math>. == Worked finite example == Consider a finite probability space: <math> \Omega=\{a,b,c\} </math> with baseline probabilities: <math> P(a)=0.2,\quad P(b)=0.3,\quad P(c)=0.5 </math> Define a positive dilation field: <math> D(a)=1,\quad D(b)=2,\quad D(c)=4 </math> The normalization constant is: <math> Z=\sum_x D(x)P(x) </math> giving: <math> Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8 </math> The PDT-transformed probabilities become: <math> \widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071 </math> <math> \widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214 </math> <math> \widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714 </math> This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization. == Composition of dilations == An important structural property of sequential PDT transformations is that compose multiplicatively. Suppose two positive dilation fields: <math> D_1(x)>0 </math> and <math> D_2(x)>0 </math> are applied successively to a baseline probability measure <math>P</math>. The first dilation produces: <math> \widetilde{P}_1(A) = \frac{\int_A D_1\,dP} {\int_\Omega D_1\,dP} </math> Applying the second dilation field to <math>\widetilde{P}_1</math> gives: <math> \widetilde{P}_2(A) = \frac{\int_A D_2\,d\widetilde{P}_1} {\int_\Omega D_2\,d\widetilde{P}_1} </math> Substituting the first transformation into the second yields: <math> \widetilde{P}_2(A) = \frac{ \int_A D_2D_1\,dP }{ \int_\Omega D_2D_1\,dP } </math> This shows that sequential PDT transformations compose through multiplication of the dilation fields. This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application. == Fixed points and iterative dynamics == An important question in PDT concerns the long-term behavior of repeated PDT transformations. Given an initial probability measure: <math> P_0 </math> and a sequence of positive dilation fields: <math> D_1,D_2,D_3,\dots </math> successive PDT transformations generate a sequence of measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow P_3 \rightarrow \cdots </math> where each transformed measure is obtained by reweighting the previous one. A measure <math>P</math> is called a fixed point of a dilation field <math>D</math> if: <math> \widetilde{P}=P </math> under the PDT transformation. In the simplest case, this requires the dilation field to be constant almost everywhere with respect to <math>P</math>. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization. More generally, repeated compositions of nontrivial dilation fields may generate: * hierarchical probability structure; * multiscale statistical behavior; * attractor-like distributions; * approximately stable transformed measures. These questions connect PDT to broader areas of: * dynamical systems; * stochastic processes; * iterative renormalization methods; * probabilistic geometry. At present these iterative properties remain largely unexplored within the PDT framework. == Entropy and iterative probability flow == Repeated PDT transformations may alter the entropy structure of a probability measure. For a discrete probability distribution: <math> P=\{p_i\} </math> the Shannon entropy is: <math> H(P) = -\sum_i p_i \log p_i </math> Under iterative PDT transformation, successive transformed measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow \cdots </math> may exhibit changing entropy behavior depending on the structure of the dilation fields. For example: * strongly localized dilation fields may concentrate probability mass and reduce entropy; * broader or smoothing dilation fields may distribute probability more evenly and increase entropy; * iterative compositions may generate approximately stable entropy profiles. These questions connect PDT to: * information theory, * statistical mechanics, * stochastic dynamics, * and renormalization-style iterative systems. At present the entropy behavior of iterative PDT transformations remains an open area for investigation. == Toy experiment: entropy under repeated dilation == A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution. Let the initial probability distribution be: <math> P_0=(0.2,0.2,0.2,0.2,0.2) </math> and define a positive dilation field: <math> D=(1,1,2,4,8) </math> At each step, apply the PDT update: <math> P_{n+1}(i) = \frac{D(i)P_n(i)} {\sum_j D(j)P_n(j)} </math> The Shannon entropy is: <math> H(P_n) = -\sum_i P_n(i)\log P_n(i) </math> In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately: <math> H(P_0)\approx1.6094 </math> to: <math> H(P_{10})\approx0.00775 </math> The final distribution is approximately: <math> P_{10} \approx (0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437) </math> This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior. == Mathematical context == PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature. In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis. The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations. === Example entropy evolution === {| class="wikitable" ! Iteration !! Shannon entropy |- | 0 || 1.6094 |- | 1 || 1.2990 |- | 2 || 0.7790 |- | 3 || 0.4399 |- | 5 || 0.1500 |- | 10 || 0.0078 |} Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states. === Localized dilation fields === A useful class of PDT transformations is generated by localized positive dilation fields. Consider a one-dimensional finite configuration space with states indexed by: <math> x=0,1,2,\dots,N </math> and define a localized dilation field centered at <math>x_0</math>: <math> D(x) = \exp\!\left( \lambda \exp\!\left( -\frac{(x-x_0)^2}{2\sigma^2} \right) \right) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\sigma</math> controls the spatial width of the localized field. Narrow values of <math>\sigma</math> produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space. Under iterative PDT dynamics: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> the probability distribution may progressively concentrate near the center of the dilation field. === Example entropy evolution for localized fields === Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior: {| class="wikitable" ! Field width <math>\sigma</math> ! Final entropy after 10 iterations ! Maximum probability after 10 iterations |- | 1.5 || 0.0352 || 0.9950 |- | 3.0 || 0.8162 || 0.7141 |- | 6.0 || 1.5367 || 0.3595 |} [[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]] [[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]] These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction. == Comparative entropy-flow experiments == The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions. {| class="wikitable" |+ Comparative entropy-flow behavior under PDT field classes ! Field class ! Final entropy ! Entropy decrease ! Final max probability ! Qualitative behavior |- | Localized | 0.3104 | 3.4032 | 0.9275 | Strong probability concentration |- | Oscillatory | 1.5779 | 2.1357 | 0.3418 | Distributed oscillatory structure |- | Multi-peak | 0.2851 | 3.4284 | 0.9425 | Multiple concentration regions |- | Stochastic | 0.7744 | 2.9392 | 0.7413 | Fluctuating concentration behavior |} These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example. In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space. [[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]] The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Oscillatory dilation fields === Another useful class of PDT transformations is generated by oscillatory positive dilation fields. One example is: <math> D(x) = \exp(\lambda\sin(kx)) </math> where: * <math>\lambda>0</math> controls the strength of the oscillatory amplification; * <math>k</math> controls the spatial frequency of the oscillation. Because the exponential is always positive, the dilation field remains strictly positive for all states. Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space. Under repeated PDT transformation: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor. === Example oscillatory-field experiment === A finite-state experiment was performed using: * 41 discrete states; * an initially uniform probability distribution; * a positive oscillatory dilation field with three spatial oscillation cycles; * 10 successive PDT iterations. Representative entropy behavior was: {| class="wikitable" ! Iteration ! Shannon entropy |- | 0 || 3.7136 |- | 2 || 2.8699 |- | 5 || 2.3018 |- | 10 || 1.9335 |} Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space. After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state. This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures. The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Multi-peak localized dilation fields === A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space. One example is: <math> D(x) = \exp\!\left( \sum_k \lambda_k \exp\!\left( -\frac{(x-x_k)^2}{2\sigma_k^2} \right) \right) </math> where: * <math>x_k</math> are the locations of the dilation peaks; * <math>\lambda_k>0</math> control the amplification strength of each peak; * <math>\sigma_k</math> control the spatial width of each localized region. This construction generates a positive multimodal dilation landscape containing several competing amplification regions. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward multiple partially localized concentration regions. Unlike single localized dilation fields, multi-peak fields may generate: * competing attractor-like regions; * hierarchical probability concentration; * partially stabilized multimodal distributions; * multiscale probability structure. Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor: * dominance by a single peak; * coexistence of several concentration regions; * or slowly evolving metastable probability structures. === Conceptual interpretation === A qualitative iterative evolution may be visualized as: <pre> Broad initial distribution ↓ Multiple localized amplifications ↓ Competing concentration regions ↓ Emergent multimodal probability structure </pre> This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone. At present these behaviors remain exploratory computational observations within finite-state toy models. === Random and stochastic dilation fields === Another important class of PDT transformations arises when the dilation field itself varies stochastically. A simple stochastic dilation field may be written schematically as: <math> D_n(x) = \exp\!\left( \sigma \eta_n(x) \right) </math> where: * <math>\eta_n(x)</math> is a random field or stochastic fluctuation at iteration <math>n</math>; * <math>\sigma>0</math> controls the strength of the stochastic variation. Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D_n(x)P_n(x) }{ \sum_y D_n(y)P_n(y) } </math> the probability landscape itself fluctuates dynamically from one iteration to the next. Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate: * fluctuating concentration regions; * transient attractor-like structures; * noise-driven entropy evolution; * intermittent probability concentration; * metastable probabilistic configurations. === Conceptual interpretation === A qualitative stochastic evolution may be visualized as: <pre> Broad initial distribution ↓ Random localized amplification ↓ Fluctuating concentration regions ↓ Dynamic probabilistic structure </pre> Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit: * partial concentration, * persistent fluctuations, * stochastic stabilization, * or continuously evolving probabilistic structure. These ideas connect PDT to broader areas of: * stochastic processes; * random multiplicative systems; * statistical mechanics; * noise-driven dynamical systems; * probabilistic geometry. At present these behaviors remain exploratory computational possibilities within finite-state toy models. == Qualitative classes of iterative PDT behavior == Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation. The following table summarizes several representative classes explored within finite-state toy models. {| class="wikitable" ! Dilation-field class ! Typical iterative behavior ! Representative qualitative structure |- | Localized fields | Strong entropy reduction and concentration toward a dominant region | Single attractor-like concentration |- | Oscillatory fields | Distributed amplification with slower entropy reduction | Patterned multimodal structure |- | Multi-peak localized fields | Competition between several concentration regions | Hierarchical or metastable probability structure |- | Random and stochastic fields | Fluctuating amplification and noise-driven evolution | Dynamic probabilistic landscapes |} These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field. Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself. At present these qualitative behaviors remain exploratory computational observations within finite-state toy models. == Numerical simulation and iterative models == === Simulation model description === In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points. Two equivalent discrete implementations are common: * '''weighted evaluation''': retain all points and assign weights proportional to <math>D</math>; * '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to <math>D</math>. === Demonstration: reweighting mock galaxy catalogs === A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box. The demonstration pipeline is: # generate a baseline mock catalog; # define a positive dilation field over the configuration space; # perform PDT-style importance resampling; # compute the resulting two-point correlation function <math>\xi(r)</math>; # compare transformed and baseline catalogs. One example dilation field is: <math> D(x)=\exp(\lambda\phi(x)) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\phi(x)\ge0</math> is a nonnegative configuration-space field. An example seed-field construction is: <math> \phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right) </math> where <math>s_k</math> are seed locations and <math>\sigma</math> controls the width of the seed influence. The two-point correlation function may be estimated using the normalized Landy–Szalay estimator: <math> \xi(r) = \frac{DD(r)-2DR(r)+RR(r)}{RR(r)} </math> where <math>DD</math>, <math>DR</math>, and <math>RR</math> are normalized pair counts. {{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}} When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration. === Computational demonstrations === Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages. {{collapse top|Python demonstration placeholder}} <syntaxhighlight lang="python"> # Example implementations may be maintained separately # on GitHub, OSF, or supplementary Wikiversity pages. </syntaxhighlight> {{collapse bottom}} == Scope and Limitations == PDT is a mathematical framework for measure transformations. It does not claim: * a replacement theory for General Relativity or Quantum Mechanics; * empirical confirmation without explicit predictions and tests; * observational validation without independently reproducible analysis. The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations. == Speculative Extensions and Geometric Renormalization == ''This section is speculative and exploratory in nature.'' Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry. Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects. Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation. At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations. == Future directions == * develop canonical families of dilation fields and invariants; * clarify “structure-from-measure” diagnostics; * publish reproducible simulation notebooks and parameter sweeps; * compare multiple dilation families under shared evaluation criteria; * investigate connections between probabilistic geometry and curvature-dependent statistical measures. == Future Directions: Probability Element (PE) == A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics. The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution. This can be expressed in terms of a dimensionless ratio: <math>\eta = \frac{\sigma_P}{\sigma}</math> where: <math>\sigma_P</math> is a hypothesized minimal probability-resolution scale, <math>\sigma</math> is an effective distinguishability scale in probability-state space. === Conceptual motivation === Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry. === Illustrative toy model (not derived physics) === As a heuristic example, one may consider a modification to special relativistic time dilation of the form: <math>d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}</math> where: <math>v</math> is velocity, <math>c</math> is the speed of light, <math>\eta = \sigma_P / \sigma</math> encodes a proposed probability-resolution scale. This expression is constructed such that standard special relativity is recovered exactly in the limit <math>\eta \to 0</math>. === Status === The Probability Element concept is: Not part of standard Fisher information geometry not derived from quantum mechanics or general relativity not currently empirically established. It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale. === Open questions === Key open research directions include: Whether a consistent discrete formulation of probability geometry can be constructed. Whether a fundamental probability-resolution scale <math>\sigma_P</math> can be derived from known physical principles. Whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions. == Convergence behavior == Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations. === Qualitative convergence classes === Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior: * '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy. * '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration. * '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions. * '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior. === Entropy and convergence === In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time. The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations. === Attractor-like behavior === Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense. Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks. == Current limitations == PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation. Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms. Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks. == See also == * [[w:Buffon's needle problem|Buffon's needle problem]] * [[w:Probability measure|Probability measure]] * [[w:Importance sampling|Importance sampling]] * [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]] * [[w:Dynamical system|Dynamical systems]] * [[w:Entropy (information theory)|Entropy]] * [[w:Information theory|Information theory]] * [[w:Measure theory|Measure theory]] * [[w:Geometric probability|Geometric probability]] * [[w:Shannon entropy|Shannon entropy]] * [[w:Stochastic process|Stochastic process]] * [[w:Fixed point (mathematics)|Fixed point]] * [[w:Convergence (mathematics)|Convergence]] == Subpages == The following subpages develop mathematical extensions and specialized topics related to Probability Dilation Theory (PDT). * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows|Fisher Geometry and Dilation Flows]] – studies information geometry, Fisher distance, and geodesic properties of PDT trajectories. * [[Probability Dilation Theory/Logit Representation of PE|Logit Representation of PE]] – develops the log-odds representation of probability elements and exponential PDT flows. * [[Probability Dilation Theory/Convergence and Fixed Points|Convergence and Fixed Points]] – investigates invariant measures, attractors, and stability of iterative PDT transformations. * [[Probability Dilation Theory/Stochastic Dilation Fields|Stochastic Dilation Fields]] – studies random and time-dependent dilation fields, ergodicity, and stochastic measure evolution. * [[Probability Dilation Theory/Entropy Evolution|Entropy Evolution]] – examines Shannon entropy under repeated probability dilation. * [[Probability Dilation Theory/Wasserstein Geometry|Wasserstein Geometry]] – explores distances between probability measures and convergence in measure space. * [[Probability Dilation Theory/Measure-Theoretic Foundations|Measure-Theoretic Foundations]] – develops rigorous measure-theoretic aspects of PDT including normalization and existence conditions. * [[Probability Dilation Theory/Euler Methods and Continuous-Time PDT]] – investigates continuous probability flows and Euler approximations of PDT. * [[Probability Dilation Theory/Worked Example]] – canonical binary example illustrating PDT transformations and geometry. ==Future Research Directions== = Probability Dilation Theory / Decoherence Analogy and Simulation = ''This page is a subpage of [[Probability Dilation Theory]] and develops a speculative but mathematically well‑defined analogy between iterative probability dilation and quantum decoherence. It includes definitions, motivation, and a reproducible Python simulation.'' == 1. Overview == This page explores how Probability Dilation Theory (PDT) can be used to construct toy models that resemble certain statistical aspects of quantum decoherence. This work is: mathematical, not physical exploratory, not authoritative analogical, not a claim about quantum mechanics open for critique and improvement The goal is to provide a clear, reproducible framework for studying how iterative reweighting of probability measures can mimic the probability‑flow behavior seen in decoherence processes. == 2. Background: PDT and Decoherence == === 2.1 Probability Dilation Theory (PDT) === PDT studies how a probability measure :<math>P</math> is transformed by a positive dilation field :<math>D(x) > 0</math> via the operator :<math>P_D(A) = \frac{\int_A D(x)\, dP(x)}{\int_X D(x)\, dP(x)}.</math> Iterating this operator produces a sequence :<math>P_{n+1} = \mathcal{D}(P_n)</math> which may converge to attractors, fixed points, or stable distributions. === 2.2 Decoherence (informal summary) === In quantum mechanics, decoherence describes how a system loses phase coherence due to interaction with an environment. A density matrix :<math>\rho</math> evolves under a completely positive trace‑preserving (CPTP) map :<math>\mathcal{E}(\rho)</math> that typically: leaves diagonal probabilities unchanged or slowly biased exponentially suppresses off‑diagonal terms (coherences) This produces an effectively classical mixture. === 2.3 Why compare them? === Although PDT is purely classical, the diagonal part of many decoherence channels behaves like a dilation step, and the off‑diagonal decay can be modeled as a simple contraction. This makes PDT a useful toy model for exploring: probability concentration pointer‑state attractors emergent classicality iterative reweighting dynamics No physical claims are made. == 3. A Minimal Toy Model == We consider a two‑state system with classical probabilities :<math>P_n = (p_0^{(n)}, p_1^{(n)})</math> and a coherence magnitude :<math>r_n = |c_n|.</math> The model consists of two coupled updates: === 3.1 PDT dilation on probabilities === Given a dilation field :<math>D = (D_0, D_1)</math> we update :<math>p_i^{(n+1)} = \frac{D_i\, p_i^{(n)}}{D_0 p_0^{(n)} + D_1 p_1^{(n)}}.</math> This biases the system toward states with larger <math>D_i</math>. === 3.2 Coherence decay === We model decoherence by :<math>r_{n+1} = \alpha\, r_n</math> with :<math>0 \le \alpha < 1.</math> This is not quantum mechanical — it is a simple exponential decay rule. === 3.3 Combined update === Each iteration applies: PDT dilation on the diagonal Exponential decay on the coherence magnitude This produces a system that: flows toward a classical attractor loses coherence over time resembles decoherence in its statistical behavior == 4. Numerical Simulation (Python) == Below is a complete, runnable Python script that simulates the toy model and produces probability and coherence plots. <syntaxhighlight lang="python"> import numpy as np import matplotlib.pyplot as plt def simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=20 ): p0 = p0_init p1 = 1.0 - p0_init coh = coherence_init p0_hist = [p0] p1_hist = [p1] coh_hist = [coh] for n in range(steps): # PDT dilation step Z = D0 * p0 + D1 * p1 p0 = (D0 * p0) / Z p1 = (D1 * p1) / Z Coherence decay coh = alpha * coh p0_hist.append(p0) p1_hist.append(p1) coh_hist.append(coh) return np.array(p0_hist), np.array(p1_hist), np.array(coh_hist) if name == "main": p0_hist, p1_hist, coh_hist = simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=25 ) t = np.arange(len(p0_hist)) fig, ax1 = plt.subplots() ax1.set_xlabel("Iteration") ax1.set_ylabel("Probability", color="tab:blue") ax1.plot(t, p0_hist, label="p0", color="tab:blue") ax1.plot(t, p1_hist, label="p1", color="tab:cyan", linestyle="--") ax1.tick_params(axis="y", labelcolor="tab:blue") ax1.legend(loc="upper left") ax2 = ax1.twinx() ax2.set_ylabel("Coherence |c|", color="tab:red") ax2.plot(t, coh_hist, label="|c|", color="tab:red") ax2.tick_params(axis="y", labelcolor="tab:red") fig.tight_layout() plt.title("Iterative Dilation + Decoherence Toy Model") plt.show() </syntaxhighlight> == 5. Interpretation == === 5.1 Probability flow === If <math>D_0 > D_1</math>, the system flows toward :<math>p_0 \to 1.</math> This is analogous to a pointer state in decoherence. === 5.2 Coherence decay === The coherence magnitude :<math>r_n</math> decays exponentially, mimicking the suppression of off‑diagonal terms in a density matrix. === 5.3 Combined effect === The system becomes: more classical (probabilities concentrate) less coherent (off‑diagonal terms vanish) This mirrors the qualitative behavior of decoherence. == 6. Limitations == This is not a quantum model. No physical claims are made about dilation fields. The analogy is structural, not ontological. Decoherence involves entanglement; PDT does not. The model is intended for intuition and exploration only. == 7. Open Questions == Can more general decoherence channels be embedded in PDT? What are the fixed points of multi‑state dilation systems? Can continuous‑time dilation flows be defined? Are there PDE analogues of iterative dilation? Can this framework be useful in machine learning or statistical physics? == 8. See Also == [[Probability Dilation Theory]] [[Quantum decoherence]] [[Density matrix]] [[Bayesian updating]] == 9. Invitation for Collaboration == This page is part of an ongoing exploration of PDT and its possible mathematical analogies. Feedback, critique, and contributions from mathematicians, physicists, and computer scientists are welcome. == Numerical Iteration Table == The canonical PDT example may be iterated repeatedly using the dilation field <math> D=(2,1). </math> Starting from <math> P_0=(0.30,0.70), </math> successive PDT iterations produce the following approximate values. {| class="wikitable" ! Iteration ! First-State Probability ! Second-State Probability | ! Shannon Entropy | | ----------------- | | 0 | | 0.300 | | 0.700 | | 0.611 | | - | | 1 | | 0.462 | | 0.538 | | 0.690 | | - | | 2 | | 0.632 | | 0.368 | | 0.658 | | - | | 3 | | 0.774 | | 0.226 | | 0.534 | | - | | 4 | | 0.873 | | 0.127 | | 0.381 | | - | | 5 | | 0.932 | | 0.068 | | 0.249 | | } | The entropy initially increases as the distribution moves closer to uniformity. After passing near the maximum-entropy state, entropy decreases as probability becomes increasingly concentrated in the first state. This behavior illustrates that entropy evolution under PDT need not be monotonic. == Notation == Throughout PDT, the following notation is used: {| class="wikitable" ! Symbol ! Meaning |- | <math>P</math> | Probability measure |- | <math>P_n</math> | nth iterate of PDT |- | <math>T_D</math> | Probability dilation operator |- | <math>D(x)</math> | Dilation field |- | <math>Z(P,D)</math> | Normalization factor |- | <math>H(P)</math> | Shannon entropy |- | <math>d_F</math> | Fisher-Rao distance |- | <math>W_p</math> | Wasserstein distance |- | <math>\ell</math> | Logit coordinate |- | <math>PE</math> | Probability Element |} == Related probabilistic and geometric literature == Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works: * Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014. * Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19 * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997. * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005. * Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007. * Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656. == Copyright and licensing == Text and original figures © Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). Reuse permitted with attribution. rg1z5cg00id1a7ksyhyr476momc3w70 2816259 2816258 2026-06-19T00:14:12Z Howie2024 2995240 /* Numerical Iteration Table */ delete 2816259 wikitext text/x-wiki {{Research project}} {{Original research}} {{To be peer reviewed}} {{subst:proofread}} == Research abstract == '''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values. The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics. At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions. PDT treats a probability measure as the primary mathematical object and investigates: * invariant identities induced by reweighting, * composition and iteration of dilations, * fixed points and near-fixed behavior, * whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations). PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions. == Overview == PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is: * mathematically well-defined (positivity and normalization), * composable under iteration, * analyzable for invariants and fixed points. === Conceptual interpretation === A simplified conceptual flow of the PDT framework is: <pre> Baseline probability measure P ↓ Positive dilation field D(x) ↓ Reweighted probability measure P~ ↓ Observable statistical changes </pre> Repeated dilation may qualitatively behave as: <pre> Broad initial distribution ↓ Localized reweighting ↓ Probability concentration ↓ Emergent multiscale structure </pre> Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field. Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as: <pre> P₀ ↓ D₁ P₁ ↓ D₂ P₂ ↓ D₃ P₃ ↓ ⋯ </pre> where each dilation field reweights the probability structure generated by the previous step. Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. = Mathematical framework = == Definitions and notation == Let <math>(\Omega,\Sigma)</math> be a measurable space. * <math>P</math> denotes a probability measure on <math>(\Omega,\Sigma)</math>. * If <math>P</math> has a density <math>p</math> with respect to a reference measure <math>\mu</math>, then <math>dP=p\,d\mu</math>. * <math>D:\Omega\to(0,\infty)</math> is a measurable '''dilation field''' (a positive weight function). * <math>Z(P,D)</math> is the normalization constant: .<math> Z(P,D)=\int_\Omega D\,dP </math> * For an observable <math>f:\Omega\to\mathbb{R}</math> integrable under the relevant measure, <math> \mathbb{E}_P[f] = \int_\Omega f\,dP </math>. == PDT transformation (probability reweighting) == Given <math>P</math> and <math>D</math> with <math>0<Z(P,D)<\infty</math>, define the '''PDT transform''' <math>\widetilde{P}=\mathrm{PDT}(P;D)</math> by: <math> \widetilde{P}(A) = \frac{ \int_A D\,dP }{ \int_\Omega D\,dP } \quad\text{for all }A\in\Sigma </math> If <math>dP=p\,d\mu</math>, then <math>d\widetilde{P}=\widetilde{p}\,d\mu</math>, where <math> \widetilde{p}(x) = \frac{D(x)\,p(x)}{Z} </math> and <math> Z = \int_\Omega D(x)\,p(x)\,d\mu </math> '''Interpretation:''' the dilation field <math>D</math> shifts probability mass toward regions where <math>D</math> is larger, while renormalization keeps total probability equal to 1. PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures. Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations. A familiar physical example of a strictly positive factor is the Lorentz factor: <math> \gamma(v) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} </math> for <math> |v|<c </math> Lorentz contraction for a rod of rest length <math>L_0</math> moving at speed <math>v</math> is: <math> L(v)=\frac{L_0}{\gamma(v)} </math> To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on <math>\gamma</math>. == Worked finite example == Consider a finite probability space: <math> \Omega=\{a,b,c\} </math> with baseline probabilities: <math> P(a)=0.2,\quad P(b)=0.3,\quad P(c)=0.5 </math> Define a positive dilation field: <math> D(a)=1,\quad D(b)=2,\quad D(c)=4 </math> The normalization constant is: <math> Z=\sum_x D(x)P(x) </math> giving: <math> Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8 </math> The PDT-transformed probabilities become: <math> \widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071 </math> <math> \widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214 </math> <math> \widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714 </math> This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization. == Composition of dilations == An important structural property of sequential PDT transformations is that compose multiplicatively. Suppose two positive dilation fields: <math> D_1(x)>0 </math> and <math> D_2(x)>0 </math> are applied successively to a baseline probability measure <math>P</math>. The first dilation produces: <math> \widetilde{P}_1(A) = \frac{\int_A D_1\,dP} {\int_\Omega D_1\,dP} </math> Applying the second dilation field to <math>\widetilde{P}_1</math> gives: <math> \widetilde{P}_2(A) = \frac{\int_A D_2\,d\widetilde{P}_1} {\int_\Omega D_2\,d\widetilde{P}_1} </math> Substituting the first transformation into the second yields: <math> \widetilde{P}_2(A) = \frac{ \int_A D_2D_1\,dP }{ \int_\Omega D_2D_1\,dP } </math> This shows that sequential PDT transformations compose through multiplication of the dilation fields. This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application. == Fixed points and iterative dynamics == An important question in PDT concerns the long-term behavior of repeated PDT transformations. Given an initial probability measure: <math> P_0 </math> and a sequence of positive dilation fields: <math> D_1,D_2,D_3,\dots </math> successive PDT transformations generate a sequence of measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow P_3 \rightarrow \cdots </math> where each transformed measure is obtained by reweighting the previous one. A measure <math>P</math> is called a fixed point of a dilation field <math>D</math> if: <math> \widetilde{P}=P </math> under the PDT transformation. In the simplest case, this requires the dilation field to be constant almost everywhere with respect to <math>P</math>. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization. More generally, repeated compositions of nontrivial dilation fields may generate: * hierarchical probability structure; * multiscale statistical behavior; * attractor-like distributions; * approximately stable transformed measures. These questions connect PDT to broader areas of: * dynamical systems; * stochastic processes; * iterative renormalization methods; * probabilistic geometry. At present these iterative properties remain largely unexplored within the PDT framework. == Entropy and iterative probability flow == Repeated PDT transformations may alter the entropy structure of a probability measure. For a discrete probability distribution: <math> P=\{p_i\} </math> the Shannon entropy is: <math> H(P) = -\sum_i p_i \log p_i </math> Under iterative PDT transformation, successive transformed measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow \cdots </math> may exhibit changing entropy behavior depending on the structure of the dilation fields. For example: * strongly localized dilation fields may concentrate probability mass and reduce entropy; * broader or smoothing dilation fields may distribute probability more evenly and increase entropy; * iterative compositions may generate approximately stable entropy profiles. These questions connect PDT to: * information theory, * statistical mechanics, * stochastic dynamics, * and renormalization-style iterative systems. At present the entropy behavior of iterative PDT transformations remains an open area for investigation. == Toy experiment: entropy under repeated dilation == A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution. Let the initial probability distribution be: <math> P_0=(0.2,0.2,0.2,0.2,0.2) </math> and define a positive dilation field: <math> D=(1,1,2,4,8) </math> At each step, apply the PDT update: <math> P_{n+1}(i) = \frac{D(i)P_n(i)} {\sum_j D(j)P_n(j)} </math> The Shannon entropy is: <math> H(P_n) = -\sum_i P_n(i)\log P_n(i) </math> In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately: <math> H(P_0)\approx1.6094 </math> to: <math> H(P_{10})\approx0.00775 </math> The final distribution is approximately: <math> P_{10} \approx (0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437) </math> This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior. == Mathematical context == PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature. In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis. The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations. === Example entropy evolution === {| class="wikitable" ! Iteration !! Shannon entropy |- | 0 || 1.6094 |- | 1 || 1.2990 |- | 2 || 0.7790 |- | 3 || 0.4399 |- | 5 || 0.1500 |- | 10 || 0.0078 |} Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states. === Localized dilation fields === A useful class of PDT transformations is generated by localized positive dilation fields. Consider a one-dimensional finite configuration space with states indexed by: <math> x=0,1,2,\dots,N </math> and define a localized dilation field centered at <math>x_0</math>: <math> D(x) = \exp\!\left( \lambda \exp\!\left( -\frac{(x-x_0)^2}{2\sigma^2} \right) \right) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\sigma</math> controls the spatial width of the localized field. Narrow values of <math>\sigma</math> produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space. Under iterative PDT dynamics: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> the probability distribution may progressively concentrate near the center of the dilation field. === Example entropy evolution for localized fields === Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior: {| class="wikitable" ! Field width <math>\sigma</math> ! Final entropy after 10 iterations ! Maximum probability after 10 iterations |- | 1.5 || 0.0352 || 0.9950 |- | 3.0 || 0.8162 || 0.7141 |- | 6.0 || 1.5367 || 0.3595 |} [[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]] [[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]] These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction. == Comparative entropy-flow experiments == The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions. {| class="wikitable" |+ Comparative entropy-flow behavior under PDT field classes ! Field class ! Final entropy ! Entropy decrease ! Final max probability ! Qualitative behavior |- | Localized | 0.3104 | 3.4032 | 0.9275 | Strong probability concentration |- | Oscillatory | 1.5779 | 2.1357 | 0.3418 | Distributed oscillatory structure |- | Multi-peak | 0.2851 | 3.4284 | 0.9425 | Multiple concentration regions |- | Stochastic | 0.7744 | 2.9392 | 0.7413 | Fluctuating concentration behavior |} These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example. In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space. [[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]] The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Oscillatory dilation fields === Another useful class of PDT transformations is generated by oscillatory positive dilation fields. One example is: <math> D(x) = \exp(\lambda\sin(kx)) </math> where: * <math>\lambda>0</math> controls the strength of the oscillatory amplification; * <math>k</math> controls the spatial frequency of the oscillation. Because the exponential is always positive, the dilation field remains strictly positive for all states. Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space. Under repeated PDT transformation: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor. === Example oscillatory-field experiment === A finite-state experiment was performed using: * 41 discrete states; * an initially uniform probability distribution; * a positive oscillatory dilation field with three spatial oscillation cycles; * 10 successive PDT iterations. Representative entropy behavior was: {| class="wikitable" ! Iteration ! Shannon entropy |- | 0 || 3.7136 |- | 2 || 2.8699 |- | 5 || 2.3018 |- | 10 || 1.9335 |} Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space. After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state. This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures. The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Multi-peak localized dilation fields === A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space. One example is: <math> D(x) = \exp\!\left( \sum_k \lambda_k \exp\!\left( -\frac{(x-x_k)^2}{2\sigma_k^2} \right) \right) </math> where: * <math>x_k</math> are the locations of the dilation peaks; * <math>\lambda_k>0</math> control the amplification strength of each peak; * <math>\sigma_k</math> control the spatial width of each localized region. This construction generates a positive multimodal dilation landscape containing several competing amplification regions. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward multiple partially localized concentration regions. Unlike single localized dilation fields, multi-peak fields may generate: * competing attractor-like regions; * hierarchical probability concentration; * partially stabilized multimodal distributions; * multiscale probability structure. Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor: * dominance by a single peak; * coexistence of several concentration regions; * or slowly evolving metastable probability structures. === Conceptual interpretation === A qualitative iterative evolution may be visualized as: <pre> Broad initial distribution ↓ Multiple localized amplifications ↓ Competing concentration regions ↓ Emergent multimodal probability structure </pre> This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone. At present these behaviors remain exploratory computational observations within finite-state toy models. === Random and stochastic dilation fields === Another important class of PDT transformations arises when the dilation field itself varies stochastically. A simple stochastic dilation field may be written schematically as: <math> D_n(x) = \exp\!\left( \sigma \eta_n(x) \right) </math> where: * <math>\eta_n(x)</math> is a random field or stochastic fluctuation at iteration <math>n</math>; * <math>\sigma>0</math> controls the strength of the stochastic variation. Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D_n(x)P_n(x) }{ \sum_y D_n(y)P_n(y) } </math> the probability landscape itself fluctuates dynamically from one iteration to the next. Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate: * fluctuating concentration regions; * transient attractor-like structures; * noise-driven entropy evolution; * intermittent probability concentration; * metastable probabilistic configurations. === Conceptual interpretation === A qualitative stochastic evolution may be visualized as: <pre> Broad initial distribution ↓ Random localized amplification ↓ Fluctuating concentration regions ↓ Dynamic probabilistic structure </pre> Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit: * partial concentration, * persistent fluctuations, * stochastic stabilization, * or continuously evolving probabilistic structure. These ideas connect PDT to broader areas of: * stochastic processes; * random multiplicative systems; * statistical mechanics; * noise-driven dynamical systems; * probabilistic geometry. At present these behaviors remain exploratory computational possibilities within finite-state toy models. == Qualitative classes of iterative PDT behavior == Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation. The following table summarizes several representative classes explored within finite-state toy models. {| class="wikitable" ! Dilation-field class ! Typical iterative behavior ! Representative qualitative structure |- | Localized fields | Strong entropy reduction and concentration toward a dominant region | Single attractor-like concentration |- | Oscillatory fields | Distributed amplification with slower entropy reduction | Patterned multimodal structure |- | Multi-peak localized fields | Competition between several concentration regions | Hierarchical or metastable probability structure |- | Random and stochastic fields | Fluctuating amplification and noise-driven evolution | Dynamic probabilistic landscapes |} These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field. Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself. At present these qualitative behaviors remain exploratory computational observations within finite-state toy models. == Numerical simulation and iterative models == === Simulation model description === In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points. Two equivalent discrete implementations are common: * '''weighted evaluation''': retain all points and assign weights proportional to <math>D</math>; * '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to <math>D</math>. === Demonstration: reweighting mock galaxy catalogs === A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box. The demonstration pipeline is: # generate a baseline mock catalog; # define a positive dilation field over the configuration space; # perform PDT-style importance resampling; # compute the resulting two-point correlation function <math>\xi(r)</math>; # compare transformed and baseline catalogs. One example dilation field is: <math> D(x)=\exp(\lambda\phi(x)) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\phi(x)\ge0</math> is a nonnegative configuration-space field. An example seed-field construction is: <math> \phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right) </math> where <math>s_k</math> are seed locations and <math>\sigma</math> controls the width of the seed influence. The two-point correlation function may be estimated using the normalized Landy–Szalay estimator: <math> \xi(r) = \frac{DD(r)-2DR(r)+RR(r)}{RR(r)} </math> where <math>DD</math>, <math>DR</math>, and <math>RR</math> are normalized pair counts. {{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}} When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration. === Computational demonstrations === Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages. {{collapse top|Python demonstration placeholder}} <syntaxhighlight lang="python"> # Example implementations may be maintained separately # on GitHub, OSF, or supplementary Wikiversity pages. </syntaxhighlight> {{collapse bottom}} == Scope and Limitations == PDT is a mathematical framework for measure transformations. It does not claim: * a replacement theory for General Relativity or Quantum Mechanics; * empirical confirmation without explicit predictions and tests; * observational validation without independently reproducible analysis. The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations. == Speculative Extensions and Geometric Renormalization == ''This section is speculative and exploratory in nature.'' Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry. Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects. Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation. At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations. == Future directions == * develop canonical families of dilation fields and invariants; * clarify “structure-from-measure” diagnostics; * publish reproducible simulation notebooks and parameter sweeps; * compare multiple dilation families under shared evaluation criteria; * investigate connections between probabilistic geometry and curvature-dependent statistical measures. == Future Directions: Probability Element (PE) == A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics. The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution. This can be expressed in terms of a dimensionless ratio: <math>\eta = \frac{\sigma_P}{\sigma}</math> where: <math>\sigma_P</math> is a hypothesized minimal probability-resolution scale, <math>\sigma</math> is an effective distinguishability scale in probability-state space. === Conceptual motivation === Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry. === Illustrative toy model (not derived physics) === As a heuristic example, one may consider a modification to special relativistic time dilation of the form: <math>d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}</math> where: <math>v</math> is velocity, <math>c</math> is the speed of light, <math>\eta = \sigma_P / \sigma</math> encodes a proposed probability-resolution scale. This expression is constructed such that standard special relativity is recovered exactly in the limit <math>\eta \to 0</math>. === Status === The Probability Element concept is: Not part of standard Fisher information geometry not derived from quantum mechanics or general relativity not currently empirically established. It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale. === Open questions === Key open research directions include: Whether a consistent discrete formulation of probability geometry can be constructed. Whether a fundamental probability-resolution scale <math>\sigma_P</math> can be derived from known physical principles. Whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions. == Convergence behavior == Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations. === Qualitative convergence classes === Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior: * '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy. * '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration. * '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions. * '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior. === Entropy and convergence === In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time. The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations. === Attractor-like behavior === Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense. Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks. == Current limitations == PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation. Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms. Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks. == See also == * [[w:Buffon's needle problem|Buffon's needle problem]] * [[w:Probability measure|Probability measure]] * [[w:Importance sampling|Importance sampling]] * [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]] * [[w:Dynamical system|Dynamical systems]] * [[w:Entropy (information theory)|Entropy]] * [[w:Information theory|Information theory]] * [[w:Measure theory|Measure theory]] * [[w:Geometric probability|Geometric probability]] * [[w:Shannon entropy|Shannon entropy]] * [[w:Stochastic process|Stochastic process]] * [[w:Fixed point (mathematics)|Fixed point]] * [[w:Convergence (mathematics)|Convergence]] == Subpages == The following subpages develop mathematical extensions and specialized topics related to Probability Dilation Theory (PDT). * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows|Fisher Geometry and Dilation Flows]] – studies information geometry, Fisher distance, and geodesic properties of PDT trajectories. * [[Probability Dilation Theory/Logit Representation of PE|Logit Representation of PE]] – develops the log-odds representation of probability elements and exponential PDT flows. * [[Probability Dilation Theory/Convergence and Fixed Points|Convergence and Fixed Points]] – investigates invariant measures, attractors, and stability of iterative PDT transformations. * [[Probability Dilation Theory/Stochastic Dilation Fields|Stochastic Dilation Fields]] – studies random and time-dependent dilation fields, ergodicity, and stochastic measure evolution. * [[Probability Dilation Theory/Entropy Evolution|Entropy Evolution]] – examines Shannon entropy under repeated probability dilation. * [[Probability Dilation Theory/Wasserstein Geometry|Wasserstein Geometry]] – explores distances between probability measures and convergence in measure space. * [[Probability Dilation Theory/Measure-Theoretic Foundations|Measure-Theoretic Foundations]] – develops rigorous measure-theoretic aspects of PDT including normalization and existence conditions. * [[Probability Dilation Theory/Euler Methods and Continuous-Time PDT]] – investigates continuous probability flows and Euler approximations of PDT. * [[Probability Dilation Theory/Worked Example]] – canonical binary example illustrating PDT transformations and geometry. ==Future Research Directions== = Probability Dilation Theory / Decoherence Analogy and Simulation = ''This page is a subpage of [[Probability Dilation Theory]] and develops a speculative but mathematically well‑defined analogy between iterative probability dilation and quantum decoherence. It includes definitions, motivation, and a reproducible Python simulation.'' == 1. Overview == This page explores how Probability Dilation Theory (PDT) can be used to construct toy models that resemble certain statistical aspects of quantum decoherence. This work is: mathematical, not physical exploratory, not authoritative analogical, not a claim about quantum mechanics open for critique and improvement The goal is to provide a clear, reproducible framework for studying how iterative reweighting of probability measures can mimic the probability‑flow behavior seen in decoherence processes. == 2. Background: PDT and Decoherence == === 2.1 Probability Dilation Theory (PDT) === PDT studies how a probability measure :<math>P</math> is transformed by a positive dilation field :<math>D(x) > 0</math> via the operator :<math>P_D(A) = \frac{\int_A D(x)\, dP(x)}{\int_X D(x)\, dP(x)}.</math> Iterating this operator produces a sequence :<math>P_{n+1} = \mathcal{D}(P_n)</math> which may converge to attractors, fixed points, or stable distributions. === 2.2 Decoherence (informal summary) === In quantum mechanics, decoherence describes how a system loses phase coherence due to interaction with an environment. A density matrix :<math>\rho</math> evolves under a completely positive trace‑preserving (CPTP) map :<math>\mathcal{E}(\rho)</math> that typically: leaves diagonal probabilities unchanged or slowly biased exponentially suppresses off‑diagonal terms (coherences) This produces an effectively classical mixture. === 2.3 Why compare them? === Although PDT is purely classical, the diagonal part of many decoherence channels behaves like a dilation step, and the off‑diagonal decay can be modeled as a simple contraction. This makes PDT a useful toy model for exploring: probability concentration pointer‑state attractors emergent classicality iterative reweighting dynamics No physical claims are made. == 3. A Minimal Toy Model == We consider a two‑state system with classical probabilities :<math>P_n = (p_0^{(n)}, p_1^{(n)})</math> and a coherence magnitude :<math>r_n = |c_n|.</math> The model consists of two coupled updates: === 3.1 PDT dilation on probabilities === Given a dilation field :<math>D = (D_0, D_1)</math> we update :<math>p_i^{(n+1)} = \frac{D_i\, p_i^{(n)}}{D_0 p_0^{(n)} + D_1 p_1^{(n)}}.</math> This biases the system toward states with larger <math>D_i</math>. === 3.2 Coherence decay === We model decoherence by :<math>r_{n+1} = \alpha\, r_n</math> with :<math>0 \le \alpha < 1.</math> This is not quantum mechanical — it is a simple exponential decay rule. === 3.3 Combined update === Each iteration applies: PDT dilation on the diagonal Exponential decay on the coherence magnitude This produces a system that: flows toward a classical attractor loses coherence over time resembles decoherence in its statistical behavior == 4. Numerical Simulation (Python) == Below is a complete, runnable Python script that simulates the toy model and produces probability and coherence plots. <syntaxhighlight lang="python"> import numpy as np import matplotlib.pyplot as plt def simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=20 ): p0 = p0_init p1 = 1.0 - p0_init coh = coherence_init p0_hist = [p0] p1_hist = [p1] coh_hist = [coh] for n in range(steps): # PDT dilation step Z = D0 * p0 + D1 * p1 p0 = (D0 * p0) / Z p1 = (D1 * p1) / Z Coherence decay coh = alpha * coh p0_hist.append(p0) p1_hist.append(p1) coh_hist.append(coh) return np.array(p0_hist), np.array(p1_hist), np.array(coh_hist) if name == "main": p0_hist, p1_hist, coh_hist = simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=25 ) t = np.arange(len(p0_hist)) fig, ax1 = plt.subplots() ax1.set_xlabel("Iteration") ax1.set_ylabel("Probability", color="tab:blue") ax1.plot(t, p0_hist, label="p0", color="tab:blue") ax1.plot(t, p1_hist, label="p1", color="tab:cyan", linestyle="--") ax1.tick_params(axis="y", labelcolor="tab:blue") ax1.legend(loc="upper left") ax2 = ax1.twinx() ax2.set_ylabel("Coherence |c|", color="tab:red") ax2.plot(t, coh_hist, label="|c|", color="tab:red") ax2.tick_params(axis="y", labelcolor="tab:red") fig.tight_layout() plt.title("Iterative Dilation + Decoherence Toy Model") plt.show() </syntaxhighlight> == 5. Interpretation == === 5.1 Probability flow === If <math>D_0 > D_1</math>, the system flows toward :<math>p_0 \to 1.</math> This is analogous to a pointer state in decoherence. === 5.2 Coherence decay === The coherence magnitude :<math>r_n</math> decays exponentially, mimicking the suppression of off‑diagonal terms in a density matrix. === 5.3 Combined effect === The system becomes: more classical (probabilities concentrate) less coherent (off‑diagonal terms vanish) This mirrors the qualitative behavior of decoherence. == 6. Limitations == This is not a quantum model. No physical claims are made about dilation fields. The analogy is structural, not ontological. Decoherence involves entanglement; PDT does not. The model is intended for intuition and exploration only. == 7. Open Questions == Can more general decoherence channels be embedded in PDT? What are the fixed points of multi‑state dilation systems? Can continuous‑time dilation flows be defined? Are there PDE analogues of iterative dilation? Can this framework be useful in machine learning or statistical physics? == 8. See Also == [[Probability Dilation Theory]] [[Quantum decoherence]] [[Density matrix]] [[Bayesian updating]] == 9. Invitation for Collaboration == This page is part of an ongoing exploration of PDT and its possible mathematical analogies. Feedback, critique, and contributions from mathematicians, physicists, and computer scientists are welcome. == Notation == Throughout PDT, the following notation is used: {| class="wikitable" ! Symbol ! Meaning |- | <math>P</math> | Probability measure |- | <math>P_n</math> | nth iterate of PDT |- | <math>T_D</math> | Probability dilation operator |- | <math>D(x)</math> | Dilation field |- | <math>Z(P,D)</math> | Normalization factor |- | <math>H(P)</math> | Shannon entropy |- | <math>d_F</math> | Fisher-Rao distance |- | <math>W_p</math> | Wasserstein distance |- | <math>\ell</math> | Logit coordinate |- | <math>PE</math> | Probability Element |} == Related probabilistic and geometric literature == Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works: * Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014. * Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19 * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997. * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005. * Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007. * Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656. == Copyright and licensing == Text and original figures © Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). Reuse permitted with attribution. i6zbd5pkuktr4p0krjgon0ck5jgelih 2816260 2816259 2026-06-19T00:19:29Z Howie2024 2995240 Create the subpage link for PDT and Decoherence 2816260 wikitext text/x-wiki {{Research project}} {{Original research}} {{To be peer reviewed}} {{subst:proofread}} == Research abstract == '''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values. The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics. At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions. PDT treats a probability measure as the primary mathematical object and investigates: * invariant identities induced by reweighting, * composition and iteration of dilations, * fixed points and near-fixed behavior, * whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations). PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions. == Overview == PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is: * mathematically well-defined (positivity and normalization), * composable under iteration, * analyzable for invariants and fixed points. === Conceptual interpretation === A simplified conceptual flow of the PDT framework is: <pre> Baseline probability measure P ↓ Positive dilation field D(x) ↓ Reweighted probability measure P~ ↓ Observable statistical changes </pre> Repeated dilation may qualitatively behave as: <pre> Broad initial distribution ↓ Localized reweighting ↓ Probability concentration ↓ Emergent multiscale structure </pre> Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field. Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as: <pre> P₀ ↓ D₁ P₁ ↓ D₂ P₂ ↓ D₃ P₃ ↓ ⋯ </pre> where each dilation field reweights the probability structure generated by the previous step. Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. = Mathematical framework = == Definitions and notation == Let <math>(\Omega,\Sigma)</math> be a measurable space. * <math>P</math> denotes a probability measure on <math>(\Omega,\Sigma)</math>. * If <math>P</math> has a density <math>p</math> with respect to a reference measure <math>\mu</math>, then <math>dP=p\,d\mu</math>. * <math>D:\Omega\to(0,\infty)</math> is a measurable '''dilation field''' (a positive weight function). * <math>Z(P,D)</math> is the normalization constant: .<math> Z(P,D)=\int_\Omega D\,dP </math> * For an observable <math>f:\Omega\to\mathbb{R}</math> integrable under the relevant measure, <math> \mathbb{E}_P[f] = \int_\Omega f\,dP </math>. == PDT transformation (probability reweighting) == Given <math>P</math> and <math>D</math> with <math>0<Z(P,D)<\infty</math>, define the '''PDT transform''' <math>\widetilde{P}=\mathrm{PDT}(P;D)</math> by: <math> \widetilde{P}(A) = \frac{ \int_A D\,dP }{ \int_\Omega D\,dP } \quad\text{for all }A\in\Sigma </math> If <math>dP=p\,d\mu</math>, then <math>d\widetilde{P}=\widetilde{p}\,d\mu</math>, where <math> \widetilde{p}(x) = \frac{D(x)\,p(x)}{Z} </math> and <math> Z = \int_\Omega D(x)\,p(x)\,d\mu </math> '''Interpretation:''' the dilation field <math>D</math> shifts probability mass toward regions where <math>D</math> is larger, while renormalization keeps total probability equal to 1. PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures. Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations. A familiar physical example of a strictly positive factor is the Lorentz factor: <math> \gamma(v) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} </math> for <math> |v|<c </math> Lorentz contraction for a rod of rest length <math>L_0</math> moving at speed <math>v</math> is: <math> L(v)=\frac{L_0}{\gamma(v)} </math> To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on <math>\gamma</math>. == Worked finite example == Consider a finite probability space: <math> \Omega=\{a,b,c\} </math> with baseline probabilities: <math> P(a)=0.2,\quad P(b)=0.3,\quad P(c)=0.5 </math> Define a positive dilation field: <math> D(a)=1,\quad D(b)=2,\quad D(c)=4 </math> The normalization constant is: <math> Z=\sum_x D(x)P(x) </math> giving: <math> Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8 </math> The PDT-transformed probabilities become: <math> \widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071 </math> <math> \widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214 </math> <math> \widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714 </math> This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization. == Composition of dilations == An important structural property of sequential PDT transformations is that compose multiplicatively. Suppose two positive dilation fields: <math> D_1(x)>0 </math> and <math> D_2(x)>0 </math> are applied successively to a baseline probability measure <math>P</math>. The first dilation produces: <math> \widetilde{P}_1(A) = \frac{\int_A D_1\,dP} {\int_\Omega D_1\,dP} </math> Applying the second dilation field to <math>\widetilde{P}_1</math> gives: <math> \widetilde{P}_2(A) = \frac{\int_A D_2\,d\widetilde{P}_1} {\int_\Omega D_2\,d\widetilde{P}_1} </math> Substituting the first transformation into the second yields: <math> \widetilde{P}_2(A) = \frac{ \int_A D_2D_1\,dP }{ \int_\Omega D_2D_1\,dP } </math> This shows that sequential PDT transformations compose through multiplication of the dilation fields. This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application. == Fixed points and iterative dynamics == An important question in PDT concerns the long-term behavior of repeated PDT transformations. Given an initial probability measure: <math> P_0 </math> and a sequence of positive dilation fields: <math> D_1,D_2,D_3,\dots </math> successive PDT transformations generate a sequence of measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow P_3 \rightarrow \cdots </math> where each transformed measure is obtained by reweighting the previous one. A measure <math>P</math> is called a fixed point of a dilation field <math>D</math> if: <math> \widetilde{P}=P </math> under the PDT transformation. In the simplest case, this requires the dilation field to be constant almost everywhere with respect to <math>P</math>. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization. More generally, repeated compositions of nontrivial dilation fields may generate: * hierarchical probability structure; * multiscale statistical behavior; * attractor-like distributions; * approximately stable transformed measures. These questions connect PDT to broader areas of: * dynamical systems; * stochastic processes; * iterative renormalization methods; * probabilistic geometry. At present these iterative properties remain largely unexplored within the PDT framework. == Entropy and iterative probability flow == Repeated PDT transformations may alter the entropy structure of a probability measure. For a discrete probability distribution: <math> P=\{p_i\} </math> the Shannon entropy is: <math> H(P) = -\sum_i p_i \log p_i </math> Under iterative PDT transformation, successive transformed measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow \cdots </math> may exhibit changing entropy behavior depending on the structure of the dilation fields. For example: * strongly localized dilation fields may concentrate probability mass and reduce entropy; * broader or smoothing dilation fields may distribute probability more evenly and increase entropy; * iterative compositions may generate approximately stable entropy profiles. These questions connect PDT to: * information theory, * statistical mechanics, * stochastic dynamics, * and renormalization-style iterative systems. At present the entropy behavior of iterative PDT transformations remains an open area for investigation. == Toy experiment: entropy under repeated dilation == A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution. Let the initial probability distribution be: <math> P_0=(0.2,0.2,0.2,0.2,0.2) </math> and define a positive dilation field: <math> D=(1,1,2,4,8) </math> At each step, apply the PDT update: <math> P_{n+1}(i) = \frac{D(i)P_n(i)} {\sum_j D(j)P_n(j)} </math> The Shannon entropy is: <math> H(P_n) = -\sum_i P_n(i)\log P_n(i) </math> In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately: <math> H(P_0)\approx1.6094 </math> to: <math> H(P_{10})\approx0.00775 </math> The final distribution is approximately: <math> P_{10} \approx (0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437) </math> This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior. == Mathematical context == PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature. In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis. The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations. === Example entropy evolution === {| class="wikitable" ! Iteration !! Shannon entropy |- | 0 || 1.6094 |- | 1 || 1.2990 |- | 2 || 0.7790 |- | 3 || 0.4399 |- | 5 || 0.1500 |- | 10 || 0.0078 |} Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states. === Localized dilation fields === A useful class of PDT transformations is generated by localized positive dilation fields. Consider a one-dimensional finite configuration space with states indexed by: <math> x=0,1,2,\dots,N </math> and define a localized dilation field centered at <math>x_0</math>: <math> D(x) = \exp\!\left( \lambda \exp\!\left( -\frac{(x-x_0)^2}{2\sigma^2} \right) \right) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\sigma</math> controls the spatial width of the localized field. Narrow values of <math>\sigma</math> produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space. Under iterative PDT dynamics: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> the probability distribution may progressively concentrate near the center of the dilation field. === Example entropy evolution for localized fields === Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior: {| class="wikitable" ! Field width <math>\sigma</math> ! Final entropy after 10 iterations ! Maximum probability after 10 iterations |- | 1.5 || 0.0352 || 0.9950 |- | 3.0 || 0.8162 || 0.7141 |- | 6.0 || 1.5367 || 0.3595 |} [[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]] [[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]] These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction. == Comparative entropy-flow experiments == The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions. {| class="wikitable" |+ Comparative entropy-flow behavior under PDT field classes ! Field class ! Final entropy ! Entropy decrease ! Final max probability ! Qualitative behavior |- | Localized | 0.3104 | 3.4032 | 0.9275 | Strong probability concentration |- | Oscillatory | 1.5779 | 2.1357 | 0.3418 | Distributed oscillatory structure |- | Multi-peak | 0.2851 | 3.4284 | 0.9425 | Multiple concentration regions |- | Stochastic | 0.7744 | 2.9392 | 0.7413 | Fluctuating concentration behavior |} These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example. In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space. [[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]] The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Oscillatory dilation fields === Another useful class of PDT transformations is generated by oscillatory positive dilation fields. One example is: <math> D(x) = \exp(\lambda\sin(kx)) </math> where: * <math>\lambda>0</math> controls the strength of the oscillatory amplification; * <math>k</math> controls the spatial frequency of the oscillation. Because the exponential is always positive, the dilation field remains strictly positive for all states. Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space. Under repeated PDT transformation: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor. === Example oscillatory-field experiment === A finite-state experiment was performed using: * 41 discrete states; * an initially uniform probability distribution; * a positive oscillatory dilation field with three spatial oscillation cycles; * 10 successive PDT iterations. Representative entropy behavior was: {| class="wikitable" ! Iteration ! Shannon entropy |- | 0 || 3.7136 |- | 2 || 2.8699 |- | 5 || 2.3018 |- | 10 || 1.9335 |} Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space. After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state. This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures. The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Multi-peak localized dilation fields === A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space. One example is: <math> D(x) = \exp\!\left( \sum_k \lambda_k \exp\!\left( -\frac{(x-x_k)^2}{2\sigma_k^2} \right) \right) </math> where: * <math>x_k</math> are the locations of the dilation peaks; * <math>\lambda_k>0</math> control the amplification strength of each peak; * <math>\sigma_k</math> control the spatial width of each localized region. This construction generates a positive multimodal dilation landscape containing several competing amplification regions. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward multiple partially localized concentration regions. Unlike single localized dilation fields, multi-peak fields may generate: * competing attractor-like regions; * hierarchical probability concentration; * partially stabilized multimodal distributions; * multiscale probability structure. Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor: * dominance by a single peak; * coexistence of several concentration regions; * or slowly evolving metastable probability structures. === Conceptual interpretation === A qualitative iterative evolution may be visualized as: <pre> Broad initial distribution ↓ Multiple localized amplifications ↓ Competing concentration regions ↓ Emergent multimodal probability structure </pre> This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone. At present these behaviors remain exploratory computational observations within finite-state toy models. === Random and stochastic dilation fields === Another important class of PDT transformations arises when the dilation field itself varies stochastically. A simple stochastic dilation field may be written schematically as: <math> D_n(x) = \exp\!\left( \sigma \eta_n(x) \right) </math> where: * <math>\eta_n(x)</math> is a random field or stochastic fluctuation at iteration <math>n</math>; * <math>\sigma>0</math> controls the strength of the stochastic variation. Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D_n(x)P_n(x) }{ \sum_y D_n(y)P_n(y) } </math> the probability landscape itself fluctuates dynamically from one iteration to the next. Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate: * fluctuating concentration regions; * transient attractor-like structures; * noise-driven entropy evolution; * intermittent probability concentration; * metastable probabilistic configurations. === Conceptual interpretation === A qualitative stochastic evolution may be visualized as: <pre> Broad initial distribution ↓ Random localized amplification ↓ Fluctuating concentration regions ↓ Dynamic probabilistic structure </pre> Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit: * partial concentration, * persistent fluctuations, * stochastic stabilization, * or continuously evolving probabilistic structure. These ideas connect PDT to broader areas of: * stochastic processes; * random multiplicative systems; * statistical mechanics; * noise-driven dynamical systems; * probabilistic geometry. At present these behaviors remain exploratory computational possibilities within finite-state toy models. == Qualitative classes of iterative PDT behavior == Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation. The following table summarizes several representative classes explored within finite-state toy models. {| class="wikitable" ! Dilation-field class ! Typical iterative behavior ! Representative qualitative structure |- | Localized fields | Strong entropy reduction and concentration toward a dominant region | Single attractor-like concentration |- | Oscillatory fields | Distributed amplification with slower entropy reduction | Patterned multimodal structure |- | Multi-peak localized fields | Competition between several concentration regions | Hierarchical or metastable probability structure |- | Random and stochastic fields | Fluctuating amplification and noise-driven evolution | Dynamic probabilistic landscapes |} These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field. Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself. At present these qualitative behaviors remain exploratory computational observations within finite-state toy models. == Numerical simulation and iterative models == === Simulation model description === In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points. Two equivalent discrete implementations are common: * '''weighted evaluation''': retain all points and assign weights proportional to <math>D</math>; * '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to <math>D</math>. === Demonstration: reweighting mock galaxy catalogs === A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box. The demonstration pipeline is: # generate a baseline mock catalog; # define a positive dilation field over the configuration space; # perform PDT-style importance resampling; # compute the resulting two-point correlation function <math>\xi(r)</math>; # compare transformed and baseline catalogs. One example dilation field is: <math> D(x)=\exp(\lambda\phi(x)) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\phi(x)\ge0</math> is a nonnegative configuration-space field. An example seed-field construction is: <math> \phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right) </math> where <math>s_k</math> are seed locations and <math>\sigma</math> controls the width of the seed influence. The two-point correlation function may be estimated using the normalized Landy–Szalay estimator: <math> \xi(r) = \frac{DD(r)-2DR(r)+RR(r)}{RR(r)} </math> where <math>DD</math>, <math>DR</math>, and <math>RR</math> are normalized pair counts. {{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}} When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration. === Computational demonstrations === Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages. {{collapse top|Python demonstration placeholder}} <syntaxhighlight lang="python"> # Example implementations may be maintained separately # on GitHub, OSF, or supplementary Wikiversity pages. </syntaxhighlight> {{collapse bottom}} == Scope and Limitations == PDT is a mathematical framework for measure transformations. It does not claim: * a replacement theory for General Relativity or Quantum Mechanics; * empirical confirmation without explicit predictions and tests; * observational validation without independently reproducible analysis. The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations. == Speculative Extensions and Geometric Renormalization == ''This section is speculative and exploratory in nature.'' Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry. Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects. Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation. At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations. == Future directions == * develop canonical families of dilation fields and invariants; * clarify “structure-from-measure” diagnostics; * publish reproducible simulation notebooks and parameter sweeps; * compare multiple dilation families under shared evaluation criteria; * investigate connections between probabilistic geometry and curvature-dependent statistical measures. == Future Directions: Probability Element (PE) == A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics. The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution. This can be expressed in terms of a dimensionless ratio: <math>\eta = \frac{\sigma_P}{\sigma}</math> where: <math>\sigma_P</math> is a hypothesized minimal probability-resolution scale, <math>\sigma</math> is an effective distinguishability scale in probability-state space. === Conceptual motivation === Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry. === Illustrative toy model (not derived physics) === As a heuristic example, one may consider a modification to special relativistic time dilation of the form: <math>d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}</math> where: <math>v</math> is velocity, <math>c</math> is the speed of light, <math>\eta = \sigma_P / \sigma</math> encodes a proposed probability-resolution scale. This expression is constructed such that standard special relativity is recovered exactly in the limit <math>\eta \to 0</math>. === Status === The Probability Element concept is: Not part of standard Fisher information geometry not derived from quantum mechanics or general relativity not currently empirically established. It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale. === Open questions === Key open research directions include: Whether a consistent discrete formulation of probability geometry can be constructed. Whether a fundamental probability-resolution scale <math>\sigma_P</math> can be derived from known physical principles. Whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions. == Convergence behavior == Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations. === Qualitative convergence classes === Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior: * '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy. * '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration. * '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions. * '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior. === Entropy and convergence === In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time. The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations. === Attractor-like behavior === Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense. Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks. == Current limitations == PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation. Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms. Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks. == See also == * [[w:Buffon's needle problem|Buffon's needle problem]] * [[w:Probability measure|Probability measure]] * [[w:Importance sampling|Importance sampling]] * [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]] * [[w:Dynamical system|Dynamical systems]] * [[w:Entropy (information theory)|Entropy]] * [[w:Information theory|Information theory]] * [[w:Measure theory|Measure theory]] * [[w:Geometric probability|Geometric probability]] * [[w:Shannon entropy|Shannon entropy]] * [[w:Stochastic process|Stochastic process]] * [[w:Fixed point (mathematics)|Fixed point]] * [[w:Convergence (mathematics)|Convergence]] == Subpages == The following subpages develop mathematical extensions and specialized topics related to Probability Dilation Theory (PDT). * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows|Fisher Geometry and Dilation Flows]] – studies information geometry, Fisher distance, and geodesic properties of PDT trajectories. * [[Probability Dilation Theory/Logit Representation of PE|Logit Representation of PE]] – develops the log-odds representation of probability elements and exponential PDT flows. * [[Probability Dilation Theory/Convergence and Fixed Points|Convergence and Fixed Points]] – investigates invariant measures, attractors, and stability of iterative PDT transformations. * [[Probability Dilation Theory/Stochastic Dilation Fields|Stochastic Dilation Fields]] – studies random and time-dependent dilation fields, ergodicity, and stochastic measure evolution. * [[Probability Dilation Theory/Entropy Evolution|Entropy Evolution]] – examines Shannon entropy under repeated probability dilation. * [[Probability Dilation Theory/Wasserstein Geometry|Wasserstein Geometry]] – explores distances between probability measures and convergence in measure space. * [[Probability Dilation Theory/Measure-Theoretic Foundations|Measure-Theoretic Foundations]] – develops rigorous measure-theoretic aspects of PDT including normalization and existence conditions. * [[Probability Dilation Theory/Euler Methods and Continuous-Time PDT]] – investigates continuous probability flows and Euler approximations of PDT. * [[Probability Dilation Theory/Worked Example]] – canonical binary example illustrating PDT transformations and geometry. [[Probability Dilation Theory/Decoherence Analogy and Simulation]] = Probability Dilation Theory / Decoherence Analogy and Simulation = ''This page is a subpage of [[Probability Dilation Theory]] and develops a speculative but mathematically well‑defined analogy between iterative probability dilation and quantum decoherence. It includes definitions, motivation, and a reproducible Python simulation.'' == 1. Overview == This page explores how Probability Dilation Theory (PDT) can be used to construct toy models that resemble certain statistical aspects of quantum decoherence. This work is: mathematical, not physical exploratory, not authoritative analogical, not a claim about quantum mechanics open for critique and improvement The goal is to provide a clear, reproducible framework for studying how iterative reweighting of probability measures can mimic the probability‑flow behavior seen in decoherence processes. == 2. Background: PDT and Decoherence == === 2.1 Probability Dilation Theory (PDT) === PDT studies how a probability measure :<math>P</math> is transformed by a positive dilation field :<math>D(x) > 0</math> via the operator :<math>P_D(A) = \frac{\int_A D(x)\, dP(x)}{\int_X D(x)\, dP(x)}.</math> Iterating this operator produces a sequence :<math>P_{n+1} = \mathcal{D}(P_n)</math> which may converge to attractors, fixed points, or stable distributions. === 2.2 Decoherence (informal summary) === In quantum mechanics, decoherence describes how a system loses phase coherence due to interaction with an environment. A density matrix :<math>\rho</math> evolves under a completely positive trace‑preserving (CPTP) map :<math>\mathcal{E}(\rho)</math> that typically: leaves diagonal probabilities unchanged or slowly biased exponentially suppresses off‑diagonal terms (coherences) This produces an effectively classical mixture. === 2.3 Why compare them? === Although PDT is purely classical, the diagonal part of many decoherence channels behaves like a dilation step, and the off‑diagonal decay can be modeled as a simple contraction. This makes PDT a useful toy model for exploring: probability concentration pointer‑state attractors emergent classicality iterative reweighting dynamics No physical claims are made. == 3. A Minimal Toy Model == We consider a two‑state system with classical probabilities :<math>P_n = (p_0^{(n)}, p_1^{(n)})</math> and a coherence magnitude :<math>r_n = |c_n|.</math> The model consists of two coupled updates: === 3.1 PDT dilation on probabilities === Given a dilation field :<math>D = (D_0, D_1)</math> we update :<math>p_i^{(n+1)} = \frac{D_i\, p_i^{(n)}}{D_0 p_0^{(n)} + D_1 p_1^{(n)}}.</math> This biases the system toward states with larger <math>D_i</math>. === 3.2 Coherence decay === We model decoherence by :<math>r_{n+1} = \alpha\, r_n</math> with :<math>0 \le \alpha < 1.</math> This is not quantum mechanical — it is a simple exponential decay rule. === 3.3 Combined update === Each iteration applies: PDT dilation on the diagonal Exponential decay on the coherence magnitude This produces a system that: flows toward a classical attractor loses coherence over time resembles decoherence in its statistical behavior == 4. Numerical Simulation (Python) == Below is a complete, runnable Python script that simulates the toy model and produces probability and coherence plots. <syntaxhighlight lang="python"> import numpy as np import matplotlib.pyplot as plt def simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=20 ): p0 = p0_init p1 = 1.0 - p0_init coh = coherence_init p0_hist = [p0] p1_hist = [p1] coh_hist = [coh] for n in range(steps): # PDT dilation step Z = D0 * p0 + D1 * p1 p0 = (D0 * p0) / Z p1 = (D1 * p1) / Z Coherence decay coh = alpha * coh p0_hist.append(p0) p1_hist.append(p1) coh_hist.append(coh) return np.array(p0_hist), np.array(p1_hist), np.array(coh_hist) if name == "main": p0_hist, p1_hist, coh_hist = simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=25 ) t = np.arange(len(p0_hist)) fig, ax1 = plt.subplots() ax1.set_xlabel("Iteration") ax1.set_ylabel("Probability", color="tab:blue") ax1.plot(t, p0_hist, label="p0", color="tab:blue") ax1.plot(t, p1_hist, label="p1", color="tab:cyan", linestyle="--") ax1.tick_params(axis="y", labelcolor="tab:blue") ax1.legend(loc="upper left") ax2 = ax1.twinx() ax2.set_ylabel("Coherence |c|", color="tab:red") ax2.plot(t, coh_hist, label="|c|", color="tab:red") ax2.tick_params(axis="y", labelcolor="tab:red") fig.tight_layout() plt.title("Iterative Dilation + Decoherence Toy Model") plt.show() </syntaxhighlight> == 5. Interpretation == === 5.1 Probability flow === If <math>D_0 > D_1</math>, the system flows toward :<math>p_0 \to 1.</math> This is analogous to a pointer state in decoherence. === 5.2 Coherence decay === The coherence magnitude :<math>r_n</math> decays exponentially, mimicking the suppression of off‑diagonal terms in a density matrix. === 5.3 Combined effect === The system becomes: more classical (probabilities concentrate) less coherent (off‑diagonal terms vanish) This mirrors the qualitative behavior of decoherence. == 6. Limitations == This is not a quantum model. No physical claims are made about dilation fields. The analogy is structural, not ontological. Decoherence involves entanglement; PDT does not. The model is intended for intuition and exploration only. == 7. Open Questions == Can more general decoherence channels be embedded in PDT? What are the fixed points of multi‑state dilation systems? Can continuous‑time dilation flows be defined? Are there PDE analogues of iterative dilation? Can this framework be useful in machine learning or statistical physics? == 8. See Also == [[Probability Dilation Theory]] [[Quantum decoherence]] [[Density matrix]] [[Bayesian updating]] == 9. Invitation for Collaboration == This page is part of an ongoing exploration of PDT and its possible mathematical analogies. Feedback, critique, and contributions from mathematicians, physicists, and computer scientists are welcome. == Notation == Throughout PDT, the following notation is used: {| class="wikitable" ! Symbol ! Meaning |- | <math>P</math> | Probability measure |- | <math>P_n</math> | nth iterate of PDT |- | <math>T_D</math> | Probability dilation operator |- | <math>D(x)</math> | Dilation field |- | <math>Z(P,D)</math> | Normalization factor |- | <math>H(P)</math> | Shannon entropy |- | <math>d_F</math> | Fisher-Rao distance |- | <math>W_p</math> | Wasserstein distance |- | <math>\ell</math> | Logit coordinate |- | <math>PE</math> | Probability Element |} == Related probabilistic and geometric literature == Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works: * Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014. * Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19 * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997. * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005. * Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007. * Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656. == Copyright and licensing == Text and original figures © Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). Reuse permitted with attribution. 378uuatkqqspueqsx1qthh2zxno0zwc 2816262 2816260 2026-06-19T00:27:04Z Howie2024 2995240 /* Probability Dilation Theory / Decoherence Analogy and Simulation */ 2816262 wikitext text/x-wiki {{Research project}} {{Original research}} {{To be peer reviewed}} {{subst:proofread}} == Research abstract == '''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values. The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics. At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions. PDT treats a probability measure as the primary mathematical object and investigates: * invariant identities induced by reweighting, * composition and iteration of dilations, * fixed points and near-fixed behavior, * whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations). PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions. == Overview == PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is: * mathematically well-defined (positivity and normalization), * composable under iteration, * analyzable for invariants and fixed points. === Conceptual interpretation === A simplified conceptual flow of the PDT framework is: <pre> Baseline probability measure P ↓ Positive dilation field D(x) ↓ Reweighted probability measure P~ ↓ Observable statistical changes </pre> Repeated dilation may qualitatively behave as: <pre> Broad initial distribution ↓ Localized reweighting ↓ Probability concentration ↓ Emergent multiscale structure </pre> Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field. Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as: <pre> P₀ ↓ D₁ P₁ ↓ D₂ P₂ ↓ D₃ P₃ ↓ ⋯ </pre> where each dilation field reweights the probability structure generated by the previous step. Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. = Mathematical framework = == Definitions and notation == Let <math>(\Omega,\Sigma)</math> be a measurable space. * <math>P</math> denotes a probability measure on <math>(\Omega,\Sigma)</math>. * If <math>P</math> has a density <math>p</math> with respect to a reference measure <math>\mu</math>, then <math>dP=p\,d\mu</math>. * <math>D:\Omega\to(0,\infty)</math> is a measurable '''dilation field''' (a positive weight function). * <math>Z(P,D)</math> is the normalization constant: .<math> Z(P,D)=\int_\Omega D\,dP </math> * For an observable <math>f:\Omega\to\mathbb{R}</math> integrable under the relevant measure, <math> \mathbb{E}_P[f] = \int_\Omega f\,dP </math>. == PDT transformation (probability reweighting) == Given <math>P</math> and <math>D</math> with <math>0<Z(P,D)<\infty</math>, define the '''PDT transform''' <math>\widetilde{P}=\mathrm{PDT}(P;D)</math> by: <math> \widetilde{P}(A) = \frac{ \int_A D\,dP }{ \int_\Omega D\,dP } \quad\text{for all }A\in\Sigma </math> If <math>dP=p\,d\mu</math>, then <math>d\widetilde{P}=\widetilde{p}\,d\mu</math>, where <math> \widetilde{p}(x) = \frac{D(x)\,p(x)}{Z} </math> and <math> Z = \int_\Omega D(x)\,p(x)\,d\mu </math> '''Interpretation:''' the dilation field <math>D</math> shifts probability mass toward regions where <math>D</math> is larger, while renormalization keeps total probability equal to 1. PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures. Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations. A familiar physical example of a strictly positive factor is the Lorentz factor: <math> \gamma(v) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} </math> for <math> |v|<c </math> Lorentz contraction for a rod of rest length <math>L_0</math> moving at speed <math>v</math> is: <math> L(v)=\frac{L_0}{\gamma(v)} </math> To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on <math>\gamma</math>. == Worked finite example == Consider a finite probability space: <math> \Omega=\{a,b,c\} </math> with baseline probabilities: <math> P(a)=0.2,\quad P(b)=0.3,\quad P(c)=0.5 </math> Define a positive dilation field: <math> D(a)=1,\quad D(b)=2,\quad D(c)=4 </math> The normalization constant is: <math> Z=\sum_x D(x)P(x) </math> giving: <math> Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8 </math> The PDT-transformed probabilities become: <math> \widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071 </math> <math> \widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214 </math> <math> \widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714 </math> This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization. == Composition of dilations == An important structural property of sequential PDT transformations is that compose multiplicatively. Suppose two positive dilation fields: <math> D_1(x)>0 </math> and <math> D_2(x)>0 </math> are applied successively to a baseline probability measure <math>P</math>. The first dilation produces: <math> \widetilde{P}_1(A) = \frac{\int_A D_1\,dP} {\int_\Omega D_1\,dP} </math> Applying the second dilation field to <math>\widetilde{P}_1</math> gives: <math> \widetilde{P}_2(A) = \frac{\int_A D_2\,d\widetilde{P}_1} {\int_\Omega D_2\,d\widetilde{P}_1} </math> Substituting the first transformation into the second yields: <math> \widetilde{P}_2(A) = \frac{ \int_A D_2D_1\,dP }{ \int_\Omega D_2D_1\,dP } </math> This shows that sequential PDT transformations compose through multiplication of the dilation fields. This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application. == Fixed points and iterative dynamics == An important question in PDT concerns the long-term behavior of repeated PDT transformations. Given an initial probability measure: <math> P_0 </math> and a sequence of positive dilation fields: <math> D_1,D_2,D_3,\dots </math> successive PDT transformations generate a sequence of measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow P_3 \rightarrow \cdots </math> where each transformed measure is obtained by reweighting the previous one. A measure <math>P</math> is called a fixed point of a dilation field <math>D</math> if: <math> \widetilde{P}=P </math> under the PDT transformation. In the simplest case, this requires the dilation field to be constant almost everywhere with respect to <math>P</math>. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization. More generally, repeated compositions of nontrivial dilation fields may generate: * hierarchical probability structure; * multiscale statistical behavior; * attractor-like distributions; * approximately stable transformed measures. These questions connect PDT to broader areas of: * dynamical systems; * stochastic processes; * iterative renormalization methods; * probabilistic geometry. At present these iterative properties remain largely unexplored within the PDT framework. == Entropy and iterative probability flow == Repeated PDT transformations may alter the entropy structure of a probability measure. For a discrete probability distribution: <math> P=\{p_i\} </math> the Shannon entropy is: <math> H(P) = -\sum_i p_i \log p_i </math> Under iterative PDT transformation, successive transformed measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow \cdots </math> may exhibit changing entropy behavior depending on the structure of the dilation fields. For example: * strongly localized dilation fields may concentrate probability mass and reduce entropy; * broader or smoothing dilation fields may distribute probability more evenly and increase entropy; * iterative compositions may generate approximately stable entropy profiles. These questions connect PDT to: * information theory, * statistical mechanics, * stochastic dynamics, * and renormalization-style iterative systems. At present the entropy behavior of iterative PDT transformations remains an open area for investigation. == Toy experiment: entropy under repeated dilation == A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution. Let the initial probability distribution be: <math> P_0=(0.2,0.2,0.2,0.2,0.2) </math> and define a positive dilation field: <math> D=(1,1,2,4,8) </math> At each step, apply the PDT update: <math> P_{n+1}(i) = \frac{D(i)P_n(i)} {\sum_j D(j)P_n(j)} </math> The Shannon entropy is: <math> H(P_n) = -\sum_i P_n(i)\log P_n(i) </math> In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately: <math> H(P_0)\approx1.6094 </math> to: <math> H(P_{10})\approx0.00775 </math> The final distribution is approximately: <math> P_{10} \approx (0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437) </math> This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior. == Mathematical context == PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature. In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis. The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations. === Example entropy evolution === {| class="wikitable" ! Iteration !! Shannon entropy |- | 0 || 1.6094 |- | 1 || 1.2990 |- | 2 || 0.7790 |- | 3 || 0.4399 |- | 5 || 0.1500 |- | 10 || 0.0078 |} Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states. === Localized dilation fields === A useful class of PDT transformations is generated by localized positive dilation fields. Consider a one-dimensional finite configuration space with states indexed by: <math> x=0,1,2,\dots,N </math> and define a localized dilation field centered at <math>x_0</math>: <math> D(x) = \exp\!\left( \lambda \exp\!\left( -\frac{(x-x_0)^2}{2\sigma^2} \right) \right) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\sigma</math> controls the spatial width of the localized field. Narrow values of <math>\sigma</math> produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space. Under iterative PDT dynamics: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> the probability distribution may progressively concentrate near the center of the dilation field. === Example entropy evolution for localized fields === Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior: {| class="wikitable" ! Field width <math>\sigma</math> ! Final entropy after 10 iterations ! Maximum probability after 10 iterations |- | 1.5 || 0.0352 || 0.9950 |- | 3.0 || 0.8162 || 0.7141 |- | 6.0 || 1.5367 || 0.3595 |} [[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]] [[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]] These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction. == Comparative entropy-flow experiments == The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions. {| class="wikitable" |+ Comparative entropy-flow behavior under PDT field classes ! Field class ! Final entropy ! Entropy decrease ! Final max probability ! Qualitative behavior |- | Localized | 0.3104 | 3.4032 | 0.9275 | Strong probability concentration |- | Oscillatory | 1.5779 | 2.1357 | 0.3418 | Distributed oscillatory structure |- | Multi-peak | 0.2851 | 3.4284 | 0.9425 | Multiple concentration regions |- | Stochastic | 0.7744 | 2.9392 | 0.7413 | Fluctuating concentration behavior |} These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example. In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space. [[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]] The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Oscillatory dilation fields === Another useful class of PDT transformations is generated by oscillatory positive dilation fields. One example is: <math> D(x) = \exp(\lambda\sin(kx)) </math> where: * <math>\lambda>0</math> controls the strength of the oscillatory amplification; * <math>k</math> controls the spatial frequency of the oscillation. Because the exponential is always positive, the dilation field remains strictly positive for all states. Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space. Under repeated PDT transformation: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor. === Example oscillatory-field experiment === A finite-state experiment was performed using: * 41 discrete states; * an initially uniform probability distribution; * a positive oscillatory dilation field with three spatial oscillation cycles; * 10 successive PDT iterations. Representative entropy behavior was: {| class="wikitable" ! Iteration ! Shannon entropy |- | 0 || 3.7136 |- | 2 || 2.8699 |- | 5 || 2.3018 |- | 10 || 1.9335 |} Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space. After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state. This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures. The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Multi-peak localized dilation fields === A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space. One example is: <math> D(x) = \exp\!\left( \sum_k \lambda_k \exp\!\left( -\frac{(x-x_k)^2}{2\sigma_k^2} \right) \right) </math> where: * <math>x_k</math> are the locations of the dilation peaks; * <math>\lambda_k>0</math> control the amplification strength of each peak; * <math>\sigma_k</math> control the spatial width of each localized region. This construction generates a positive multimodal dilation landscape containing several competing amplification regions. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward multiple partially localized concentration regions. Unlike single localized dilation fields, multi-peak fields may generate: * competing attractor-like regions; * hierarchical probability concentration; * partially stabilized multimodal distributions; * multiscale probability structure. Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor: * dominance by a single peak; * coexistence of several concentration regions; * or slowly evolving metastable probability structures. === Conceptual interpretation === A qualitative iterative evolution may be visualized as: <pre> Broad initial distribution ↓ Multiple localized amplifications ↓ Competing concentration regions ↓ Emergent multimodal probability structure </pre> This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone. At present these behaviors remain exploratory computational observations within finite-state toy models. === Random and stochastic dilation fields === Another important class of PDT transformations arises when the dilation field itself varies stochastically. A simple stochastic dilation field may be written schematically as: <math> D_n(x) = \exp\!\left( \sigma \eta_n(x) \right) </math> where: * <math>\eta_n(x)</math> is a random field or stochastic fluctuation at iteration <math>n</math>; * <math>\sigma>0</math> controls the strength of the stochastic variation. Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D_n(x)P_n(x) }{ \sum_y D_n(y)P_n(y) } </math> the probability landscape itself fluctuates dynamically from one iteration to the next. Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate: * fluctuating concentration regions; * transient attractor-like structures; * noise-driven entropy evolution; * intermittent probability concentration; * metastable probabilistic configurations. === Conceptual interpretation === A qualitative stochastic evolution may be visualized as: <pre> Broad initial distribution ↓ Random localized amplification ↓ Fluctuating concentration regions ↓ Dynamic probabilistic structure </pre> Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit: * partial concentration, * persistent fluctuations, * stochastic stabilization, * or continuously evolving probabilistic structure. These ideas connect PDT to broader areas of: * stochastic processes; * random multiplicative systems; * statistical mechanics; * noise-driven dynamical systems; * probabilistic geometry. At present these behaviors remain exploratory computational possibilities within finite-state toy models. == Qualitative classes of iterative PDT behavior == Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation. The following table summarizes several representative classes explored within finite-state toy models. {| class="wikitable" ! Dilation-field class ! Typical iterative behavior ! Representative qualitative structure |- | Localized fields | Strong entropy reduction and concentration toward a dominant region | Single attractor-like concentration |- | Oscillatory fields | Distributed amplification with slower entropy reduction | Patterned multimodal structure |- | Multi-peak localized fields | Competition between several concentration regions | Hierarchical or metastable probability structure |- | Random and stochastic fields | Fluctuating amplification and noise-driven evolution | Dynamic probabilistic landscapes |} These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field. Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself. At present these qualitative behaviors remain exploratory computational observations within finite-state toy models. == Numerical simulation and iterative models == === Simulation model description === In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points. Two equivalent discrete implementations are common: * '''weighted evaluation''': retain all points and assign weights proportional to <math>D</math>; * '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to <math>D</math>. === Demonstration: reweighting mock galaxy catalogs === A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box. The demonstration pipeline is: # generate a baseline mock catalog; # define a positive dilation field over the configuration space; # perform PDT-style importance resampling; # compute the resulting two-point correlation function <math>\xi(r)</math>; # compare transformed and baseline catalogs. One example dilation field is: <math> D(x)=\exp(\lambda\phi(x)) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\phi(x)\ge0</math> is a nonnegative configuration-space field. An example seed-field construction is: <math> \phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right) </math> where <math>s_k</math> are seed locations and <math>\sigma</math> controls the width of the seed influence. The two-point correlation function may be estimated using the normalized Landy–Szalay estimator: <math> \xi(r) = \frac{DD(r)-2DR(r)+RR(r)}{RR(r)} </math> where <math>DD</math>, <math>DR</math>, and <math>RR</math> are normalized pair counts. {{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}} When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration. === Computational demonstrations === Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages. {{collapse top|Python demonstration placeholder}} <syntaxhighlight lang="python"> # Example implementations may be maintained separately # on GitHub, OSF, or supplementary Wikiversity pages. </syntaxhighlight> {{collapse bottom}} == Scope and Limitations == PDT is a mathematical framework for measure transformations. It does not claim: * a replacement theory for General Relativity or Quantum Mechanics; * empirical confirmation without explicit predictions and tests; * observational validation without independently reproducible analysis. The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations. == Speculative Extensions and Geometric Renormalization == ''This section is speculative and exploratory in nature.'' Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry. Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects. Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation. At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations. == Future directions == * develop canonical families of dilation fields and invariants; * clarify “structure-from-measure” diagnostics; * publish reproducible simulation notebooks and parameter sweeps; * compare multiple dilation families under shared evaluation criteria; * investigate connections between probabilistic geometry and curvature-dependent statistical measures. == Future Directions: Probability Element (PE) == A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics. The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution. This can be expressed in terms of a dimensionless ratio: <math>\eta = \frac{\sigma_P}{\sigma}</math> where: <math>\sigma_P</math> is a hypothesized minimal probability-resolution scale, <math>\sigma</math> is an effective distinguishability scale in probability-state space. === Conceptual motivation === Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry. === Illustrative toy model (not derived physics) === As a heuristic example, one may consider a modification to special relativistic time dilation of the form: <math>d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}</math> where: <math>v</math> is velocity, <math>c</math> is the speed of light, <math>\eta = \sigma_P / \sigma</math> encodes a proposed probability-resolution scale. This expression is constructed such that standard special relativity is recovered exactly in the limit <math>\eta \to 0</math>. === Status === The Probability Element concept is: Not part of standard Fisher information geometry not derived from quantum mechanics or general relativity not currently empirically established. It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale. === Open questions === Key open research directions include: Whether a consistent discrete formulation of probability geometry can be constructed. Whether a fundamental probability-resolution scale <math>\sigma_P</math> can be derived from known physical principles. Whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions. == Convergence behavior == Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations. === Qualitative convergence classes === Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior: * '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy. * '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration. * '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions. * '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior. === Entropy and convergence === In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time. The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations. === Attractor-like behavior === Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense. Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks. == Current limitations == PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation. Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms. Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks. == See also == * [[w:Buffon's needle problem|Buffon's needle problem]] * [[w:Probability measure|Probability measure]] * [[w:Importance sampling|Importance sampling]] * [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]] * [[w:Dynamical system|Dynamical systems]] * [[w:Entropy (information theory)|Entropy]] * [[w:Information theory|Information theory]] * [[w:Measure theory|Measure theory]] * [[w:Geometric probability|Geometric probability]] * [[w:Shannon entropy|Shannon entropy]] * [[w:Stochastic process|Stochastic process]] * [[w:Fixed point (mathematics)|Fixed point]] * [[w:Convergence (mathematics)|Convergence]] == Subpages == The following subpages develop mathematical extensions and specialized topics related to Probability Dilation Theory (PDT). * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows|Fisher Geometry and Dilation Flows]] – studies information geometry, Fisher distance, and geodesic properties of PDT trajectories. * [[Probability Dilation Theory/Logit Representation of PE|Logit Representation of PE]] – develops the log-odds representation of probability elements and exponential PDT flows. * [[Probability Dilation Theory/Convergence and Fixed Points|Convergence and Fixed Points]] – investigates invariant measures, attractors, and stability of iterative PDT transformations. * [[Probability Dilation Theory/Stochastic Dilation Fields|Stochastic Dilation Fields]] – studies random and time-dependent dilation fields, ergodicity, and stochastic measure evolution. * [[Probability Dilation Theory/Entropy Evolution|Entropy Evolution]] – examines Shannon entropy under repeated probability dilation. * [[Probability Dilation Theory/Wasserstein Geometry|Wasserstein Geometry]] – explores distances between probability measures and convergence in measure space. * [[Probability Dilation Theory/Measure-Theoretic Foundations|Measure-Theoretic Foundations]] – develops rigorous measure-theoretic aspects of PDT including normalization and existence conditions. * [[Probability Dilation Theory/Euler Methods and Continuous-Time PDT]] – investigates continuous probability flows and Euler approximations of PDT. * [[Probability Dilation Theory/Worked Example]] – canonical binary example illustrating PDT transformations and geometry. [[Probability Dilation Theory/Decoherence Analogy and Simulation]] == Notation == Throughout PDT, the following notation is used: {| class="wikitable" ! Symbol ! Meaning |- | <math>P</math> | Probability measure |- | <math>P_n</math> | nth iterate of PDT |- | <math>T_D</math> | Probability dilation operator |- | <math>D(x)</math> | Dilation field |- | <math>Z(P,D)</math> | Normalization factor |- | <math>H(P)</math> | Shannon entropy |- | <math>d_F</math> | Fisher-Rao distance |- | <math>W_p</math> | Wasserstein distance |- | <math>\ell</math> | Logit coordinate |- | <math>PE</math> | Probability Element |} == Related probabilistic and geometric literature == Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works: * Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014. * Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19 * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997. * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005. * Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007. * Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656. == Copyright and licensing == Text and original figures © Howard Richardson. Licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). Reuse permitted with attribution. oecubyynfqqlnadgxdh7tid4f4plokr 2816275 2816262 2026-06-19T01:54:55Z Howie2024 2995240 /* Copyright and licensing */ 2816275 wikitext text/x-wiki {{Research project}} {{Original research}} {{To be peer reviewed}} {{subst:proofread}} == Research abstract == '''Probability Dilation Theory (PDT)''' is a measure-theoretic research framework for studying how probability measures transform under '''positive reweighting (dilation)''' while preserving normalization and producing controlled changes in expectation values. The theory is an exploratory framework for iterative probability-measure evolution under positive dilation fields. The framework studies how repeated probabilistic reweighting transformations may generate emergent statistical structure, entropy flow, and multiscale probability dynamics. At its core, PDT studies how repeated positive probability reweighting transformations alter the long-term structure of probability distributions. PDT treats a probability measure as the primary mathematical object and investigates: * invariant identities induced by reweighting, * composition and iteration of dilations, * fixed points and near-fixed behavior, * whether iterative measure updates can generate testable multiscale statistical structure (to be evaluated via explicit models and simulations). PDT is presented as a mathematical framework. Any proposed application to physics or cosmology must be expressed as a concrete model (space, baseline measure, dilation field) and tested against falsifiable predictions. == Overview == PDT is motivated by the observation that some structural information can be recovered from sampling statistics (e.g., [[w:Buffon's needle problem|Buffon’s needle]]). PDT abstracts this idea by focusing on measure transformation itself: a dilation field modifies a baseline probability measure in a way that is: * mathematically well-defined (positivity and normalization), * composable under iteration, * analyzable for invariants and fixed points. === Conceptual interpretation === A simplified conceptual flow of the PDT framework is: <pre> Baseline probability measure P ↓ Positive dilation field D(x) ↓ Reweighted probability measure P~ ↓ Observable statistical changes </pre> Repeated dilation may qualitatively behave as: <pre> Broad initial distribution ↓ Localized reweighting ↓ Probability concentration ↓ Emergent multiscale structure </pre> Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. In this interpretation, PDT does not alter the underlying sample space directly. Instead, it modifies how probability mass is distributed across that space through a positive reweighting field. Regions with larger values of the dilation field contribute more strongly to the transformed measure, while normalization preserves total probability. Earlier exploratory formulations of Probability Dilation Theory (PDT) were informally referred to as the Einstein Buffon Process (EBP), reflecting initial probabilistic-geometric interpretations inspired by Buffon-type constructions and Einstein-style scaling analogies. The framework has since evolved toward a broader iterative theory of probability-measure dynamics under positive dilation fields. A simple iterative interpretation may also be visualized as: <pre> P₀ ↓ D₁ P₁ ↓ D₂ P₂ ↓ D₃ P₃ ↓ ⋯ </pre> where each dilation field reweights the probability structure generated by the previous step. Different classes of dilation fields may therefore generate qualitatively different long-term probability dynamics. = Mathematical framework = == Definitions and notation == Let <math>(\Omega,\Sigma)</math> be a measurable space. * <math>P</math> denotes a probability measure on <math>(\Omega,\Sigma)</math>. * If <math>P</math> has a density <math>p</math> with respect to a reference measure <math>\mu</math>, then <math>dP=p\,d\mu</math>. * <math>D:\Omega\to(0,\infty)</math> is a measurable '''dilation field''' (a positive weight function). * <math>Z(P,D)</math> is the normalization constant: .<math> Z(P,D)=\int_\Omega D\,dP </math> * For an observable <math>f:\Omega\to\mathbb{R}</math> integrable under the relevant measure, <math> \mathbb{E}_P[f] = \int_\Omega f\,dP </math>. == PDT transformation (probability reweighting) == Given <math>P</math> and <math>D</math> with <math>0<Z(P,D)<\infty</math>, define the '''PDT transform''' <math>\widetilde{P}=\mathrm{PDT}(P;D)</math> by: <math> \widetilde{P}(A) = \frac{ \int_A D\,dP }{ \int_\Omega D\,dP } \quad\text{for all }A\in\Sigma </math> If <math>dP=p\,d\mu</math>, then <math>d\widetilde{P}=\widetilde{p}\,d\mu</math>, where <math> \widetilde{p}(x) = \frac{D(x)\,p(x)}{Z} </math> and <math> Z = \int_\Omega D(x)\,p(x)\,d\mu </math> '''Interpretation:''' the dilation field <math>D</math> shifts probability mass toward regions where <math>D</math> is larger, while renormalization keeps total probability equal to 1. PDT is mathematically related to importance sampling, Gibbs-style reweighting, and Radon–Nikodym measure transformations, although the framework emphasizes compositional and geometric interpretations of probability reweighting rather than only numerical estimation procedures. Unlike conventional importance sampling, however, PDT emphasizes the compositional and potentially dynamical behavior of repeated probability reweighting transformations. A familiar physical example of a strictly positive factor is the Lorentz factor: <math> \gamma(v) = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} </math> for <math> |v|<c </math> Lorentz contraction for a rod of rest length <math>L_0</math> moving at speed <math>v</math> is: <math> L(v)=\frac{L_0}{\gamma(v)} </math> To connect this idea to PDT (as an illustration only), one may define a positive dilation field based on <math>\gamma</math>. == Worked finite example == Consider a finite probability space: <math> \Omega=\{a,b,c\} </math> with baseline probabilities: <math> P(a)=0.2,\quad P(b)=0.3,\quad P(c)=0.5 </math> Define a positive dilation field: <math> D(a)=1,\quad D(b)=2,\quad D(c)=4 </math> The normalization constant is: <math> Z=\sum_x D(x)P(x) </math> giving: <math> Z=(1)(0.2)+(2)(0.3)+(4)(0.5)=2.8 </math> The PDT-transformed probabilities become: <math> \widetilde{P}(a)=\frac{0.2}{2.8}\approx0.071 </math> <math> \widetilde{P}(b)=\frac{0.6}{2.8}\approx0.214 </math> <math> \widetilde{P}(c)=\frac{2.0}{2.8}\approx0.714 </math> This illustrates how PDT shifts probability mass toward regions with larger dilation weights while preserving normalization. == Composition of dilations == An important structural property of sequential PDT transformations is that compose multiplicatively. Suppose two positive dilation fields: <math> D_1(x)>0 </math> and <math> D_2(x)>0 </math> are applied successively to a baseline probability measure <math>P</math>. The first dilation produces: <math> \widetilde{P}_1(A) = \frac{\int_A D_1\,dP} {\int_\Omega D_1\,dP} </math> Applying the second dilation field to <math>\widetilde{P}_1</math> gives: <math> \widetilde{P}_2(A) = \frac{\int_A D_2\,d\widetilde{P}_1} {\int_\Omega D_2\,d\widetilde{P}_1} </math> Substituting the first transformation into the second yields: <math> \widetilde{P}_2(A) = \frac{ \int_A D_2D_1\,dP }{ \int_\Omega D_2D_1\,dP } </math> This shows that sequential PDT transformations compose through multiplication of the dilation fields. This compositional structure allows iterative probability reweighting to be studied using products of positive fields, potentially generating multiscale or hierarchical probability structures under repeated application. == Fixed points and iterative dynamics == An important question in PDT concerns the long-term behavior of repeated PDT transformations. Given an initial probability measure: <math> P_0 </math> and a sequence of positive dilation fields: <math> D_1,D_2,D_3,\dots </math> successive PDT transformations generate a sequence of measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow P_3 \rightarrow \cdots </math> where each transformed measure is obtained by reweighting the previous one. A measure <math>P</math> is called a fixed point of a dilation field <math>D</math> if: <math> \widetilde{P}=P </math> under the PDT transformation. In the simplest case, this requires the dilation field to be constant almost everywhere with respect to <math>P</math>. More general fixed-point behavior may arise when iterative compositions balance probability amplification against normalization. More generally, repeated compositions of nontrivial dilation fields may generate: * hierarchical probability structure; * multiscale statistical behavior; * attractor-like distributions; * approximately stable transformed measures. These questions connect PDT to broader areas of: * dynamical systems; * stochastic processes; * iterative renormalization methods; * probabilistic geometry. At present these iterative properties remain largely unexplored within the PDT framework. == Entropy and iterative probability flow == Repeated PDT transformations may alter the entropy structure of a probability measure. For a discrete probability distribution: <math> P=\{p_i\} </math> the Shannon entropy is: <math> H(P) = -\sum_i p_i \log p_i </math> Under iterative PDT transformation, successive transformed measures: <math> P_0 \rightarrow P_1 \rightarrow P_2 \rightarrow \cdots </math> may exhibit changing entropy behavior depending on the structure of the dilation fields. For example: * strongly localized dilation fields may concentrate probability mass and reduce entropy; * broader or smoothing dilation fields may distribute probability more evenly and increase entropy; * iterative compositions may generate approximately stable entropy profiles. These questions connect PDT to: * information theory, * statistical mechanics, * stochastic dynamics, * and renormalization-style iterative systems. At present the entropy behavior of iterative PDT transformations remains an open area for investigation. == Toy experiment: entropy under repeated dilation == A simple finite-state experiment illustrates how repeated PDT transformations can change the entropy of a probability distribution. Let the initial probability distribution be: <math> P_0=(0.2,0.2,0.2,0.2,0.2) </math> and define a positive dilation field: <math> D=(1,1,2,4,8) </math> At each step, apply the PDT update: <math> P_{n+1}(i) = \frac{D(i)P_n(i)} {\sum_j D(j)P_n(j)} </math> The Shannon entropy is: <math> H(P_n) = -\sum_i P_n(i)\log P_n(i) </math> In this toy model, repeated dilation shifts probability mass toward the highest-weight state. Over ten iterations, the entropy decreases from approximately: <math> H(P_0)\approx1.6094 </math> to: <math> H(P_{10})\approx0.00775 </math> The final distribution is approximately: <math> P_{10} \approx (0.000000001,\;0.000000001,\;0.000000953,\;0.000975609,\;0.999023437) </math> This example demonstrates probability concentration under repeated positive dilation. It is a finite-state toy model and should not be interpreted as physical evidence; its purpose is to illustrate iterative PDT behavior. == Mathematical context == PDT transformations may be viewed as exploratory probability-measure reweighting procedures related conceptually to conditioning behavior, stochastic transformations, entropy evolution, and probabilistic dilation phenomena studied in imprecise probability theory and dynamical systems literature. In PDT, the term ''dilation'' refers to probabilistic reweighting and transformation behavior under localized weighting fields rather than the formal operator-theoretic notion of dilation used in functional analysis. The iterative entropy-flow experiments explored in PDT resemble finite-state dynamical systems in which repeated transformations generate convergence, concentration, and emergent probabilistic structure over successive iterations. === Example entropy evolution === {| class="wikitable" ! Iteration !! Shannon entropy |- | 0 || 1.6094 |- | 1 || 1.2990 |- | 2 || 0.7790 |- | 3 || 0.4399 |- | 5 || 0.1500 |- | 10 || 0.0078 |} Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting. Programmatically generated using Python in a ChatGPT-assisted workflow. The entropy decreases under repeated application of the dilation field as probability mass becomes increasingly concentrated in the highest-weight states. === Localized dilation fields === A useful class of PDT transformations is generated by localized positive dilation fields. Consider a one-dimensional finite configuration space with states indexed by: <math> x=0,1,2,\dots,N </math> and define a localized dilation field centered at <math>x_0</math>: <math> D(x) = \exp\!\left( \lambda \exp\!\left( -\frac{(x-x_0)^2}{2\sigma^2} \right) \right) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\sigma</math> controls the spatial width of the localized field. Narrow values of <math>\sigma</math> produce sharply localized amplification, while broader values produce smoother probability reweighting across the configuration space. Under iterative PDT dynamics: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> the probability distribution may progressively concentrate near the center of the dilation field. === Example entropy evolution for localized fields === Using an initially uniform distribution over 21 states and iterating the PDT transformation 10 times produces the following representative entropy behavior: {| class="wikitable" ! Field width <math>\sigma</math> ! Final entropy after 10 iterations ! Maximum probability after 10 iterations |- | 1.5 || 0.0352 || 0.9950 |- | 3.0 || 0.8162 || 0.7141 |- | 6.0 || 1.5367 || 0.3595 |} [[File:PDT entropy evolution localized field.png|thumb|center|600px|Entropy evolution under repeated localized PDT transformation showing entropy reduction and probability concentration under iterative probabilistic reweighting.]] [[File:Epd_entropy_evolution.png|thumb|center|600px|Entropy evolution under repeated localized PDT dilation. Narrow localized dilation fields produce rapid entropy reduction and probability concentration under iterative reweighting.]] These results indicate that narrower localized dilation fields generate stronger probability concentration and more rapid entropy reduction. == Comparative entropy-flow experiments == The following finite-state computational experiments illustrate comparative entropy evolution under several classes of PDT dilation fields. Each experiment begins with the same initially uniform probability distribution and applies repeated PDT transformations under different field structures. The experiments are exploratory and intended to illustrate qualitative differences in iterative probabilistic behavior rather than empirical physical predictions. {| class="wikitable" |+ Comparative entropy-flow behavior under PDT field classes ! Field class ! Final entropy ! Entropy decrease ! Final max probability ! Qualitative behavior |- | Localized | 0.3104 | 3.4032 | 0.9275 | Strong probability concentration |- | Oscillatory | 1.5779 | 2.1357 | 0.3418 | Distributed oscillatory structure |- | Multi-peak | 0.2851 | 3.4284 | 0.9425 | Multiple concentration regions |- | Stochastic | 0.7744 | 2.9392 | 0.7413 | Fluctuating concentration behavior |} These experiments suggest that different classes of dilation fields may generate qualitatively distinct entropy-flow and concentration behavior under iterative PDT dynamics. Localized and multi-peak fields produce strong entropy reduction and probability concentration, while oscillatory fields preserve more distributed probabilistic structure. Stochastic fields exhibit fluctuating but still partially concentrating behavior in this finite-state example. In this toy model, repeated localized dilation behaves qualitatively like an attractor centered on the highest-weight region of the configuration space. [[File:Pdt comparative entropy flow.png|thumb|Comparative entropy evolution under localized, oscillatory, multi-peak, and stochastic PDT dilation fields.]] The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Oscillatory dilation fields === Another useful class of PDT transformations is generated by oscillatory positive dilation fields. One example is: <math> D(x) = \exp(\lambda\sin(kx)) </math> where: * <math>\lambda>0</math> controls the strength of the oscillatory amplification; * <math>k</math> controls the spatial frequency of the oscillation. Because the exponential is always positive, the dilation field remains strictly positive for all states. Unlike localized dilation fields, oscillatory fields may generate multiple competing high-weight regions across the configuration space. Under repeated PDT transformation: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward several distributed concentration regions rather than a single dominant attractor. === Example oscillatory-field experiment === A finite-state experiment was performed using: * 41 discrete states; * an initially uniform probability distribution; * a positive oscillatory dilation field with three spatial oscillation cycles; * 10 successive PDT iterations. Representative entropy behavior was: {| class="wikitable" ! Iteration ! Shannon entropy |- | 0 || 3.7136 |- | 2 || 2.8699 |- | 5 || 2.3018 |- | 10 || 1.9335 |} Unlike sharply localized dilation fields, the oscillatory field produced slower entropy reduction and multiple probability concentration peaks distributed across the configuration space. After 10 iterations, the largest probability concentration remained distributed rather than collapsing into a single dominant state. This suggests that different classes of positive dilation fields may generate qualitatively different long-term iterative probability structures. The experiment is intended only as a finite-state demonstration of iterative PDT dynamics and should not be interpreted as physical evidence. === Multi-peak localized dilation fields === A broader class of PDT transformations may be generated using multiple localized dilation peaks distributed across the configuration space. One example is: <math> D(x) = \exp\!\left( \sum_k \lambda_k \exp\!\left( -\frac{(x-x_k)^2}{2\sigma_k^2} \right) \right) </math> where: * <math>x_k</math> are the locations of the dilation peaks; * <math>\lambda_k>0</math> control the amplification strength of each peak; * <math>\sigma_k</math> control the spatial width of each localized region. This construction generates a positive multimodal dilation landscape containing several competing amplification regions. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D(x)P_n(x) }{ \sum_y D(y)P_n(y) } </math> probability mass may evolve toward multiple partially localized concentration regions. Unlike single localized dilation fields, multi-peak fields may generate: * competing attractor-like regions; * hierarchical probability concentration; * partially stabilized multimodal distributions; * multiscale probability structure. Depending on the relative strengths and widths of the peaks, the iterative dynamics may favor: * dominance by a single peak; * coexistence of several concentration regions; * or slowly evolving metastable probability structures. === Conceptual interpretation === A qualitative iterative evolution may be visualized as: <pre> Broad initial distribution ↓ Multiple localized amplifications ↓ Competing concentration regions ↓ Emergent multimodal probability structure </pre> This class of dilation fields suggests that iterative PDT dynamics may generate richer probability organization than either single localized attractors or simple oscillatory fields alone. At present these behaviors remain exploratory computational observations within finite-state toy models. === Random and stochastic dilation fields === Another important class of PDT transformations arises when the dilation field itself varies stochastically. A simple stochastic dilation field may be written schematically as: <math> D_n(x) = \exp\!\left( \sigma \eta_n(x) \right) </math> where: * <math>\eta_n(x)</math> is a random field or stochastic fluctuation at iteration <math>n</math>; * <math>\sigma>0</math> controls the strength of the stochastic variation. Because the exponential is strictly positive, the dilation field remains positive for all realizations of the random process. Under repeated PDT iteration: <math> P_{n+1}(x) = \frac{ D_n(x)P_n(x) }{ \sum_y D_n(y)P_n(y) } </math> the probability landscape itself fluctuates dynamically from one iteration to the next. Unlike deterministic localized or oscillatory dilation fields, stochastic dilation fields may generate: * fluctuating concentration regions; * transient attractor-like structures; * noise-driven entropy evolution; * intermittent probability concentration; * metastable probabilistic configurations. === Conceptual interpretation === A qualitative stochastic evolution may be visualized as: <pre> Broad initial distribution ↓ Random localized amplification ↓ Fluctuating concentration regions ↓ Dynamic probabilistic structure </pre> Depending on the stochastic process used to generate the dilation fields, the long-term dynamics may exhibit: * partial concentration, * persistent fluctuations, * stochastic stabilization, * or continuously evolving probabilistic structure. These ideas connect PDT to broader areas of: * stochastic processes; * random multiplicative systems; * statistical mechanics; * noise-driven dynamical systems; * probabilistic geometry. At present these behaviors remain exploratory computational possibilities within finite-state toy models. == Qualitative classes of iterative PDT behavior == Different classes of positive dilation fields may generate qualitatively different long-term probability dynamics under repeated PDT transformation. The following table summarizes several representative classes explored within finite-state toy models. {| class="wikitable" ! Dilation-field class ! Typical iterative behavior ! Representative qualitative structure |- | Localized fields | Strong entropy reduction and concentration toward a dominant region | Single attractor-like concentration |- | Oscillatory fields | Distributed amplification with slower entropy reduction | Patterned multimodal structure |- | Multi-peak localized fields | Competition between several concentration regions | Hierarchical or metastable probability structure |- | Random and stochastic fields | Fluctuating amplification and noise-driven evolution | Dynamic probabilistic landscapes |} These examples suggest that iterative PDT reweighting may generate a broad spectrum of emergent statistical structures depending on the geometry and dynamics of the dilation field. Within the PDT framework, the iterative behavior of probability measures may therefore depend as strongly on the structure of the dilation field as on the initial probability distribution itself. At present these qualitative behaviors remain exploratory computational observations within finite-state toy models. == Numerical simulation and iterative models == === Simulation model description === In discrete demonstrations, the “state space” may be represented by a finite set such as bins, configurations, or catalog points. Two equivalent discrete implementations are common: * '''weighted evaluation''': retain all points and assign weights proportional to <math>D</math>; * '''importance resampling''': generate a new empirical catalog with sampling probabilities proportional to <math>D</math>. === Demonstration: reweighting mock galaxy catalogs === A simple computational demonstration of PDT may be constructed using synthetic galaxy catalogs in a periodic simulation box. The demonstration pipeline is: # generate a baseline mock catalog; # define a positive dilation field over the configuration space; # perform PDT-style importance resampling; # compute the resulting two-point correlation function <math>\xi(r)</math>; # compare transformed and baseline catalogs. One example dilation field is: <math> D(x)=\exp(\lambda\phi(x)) </math> where: * <math>\lambda>0</math> controls the strength of the dilation; * <math>\phi(x)\ge0</math> is a nonnegative configuration-space field. An example seed-field construction is: <math> \phi(x)=\sum_k \exp\!\left(-\frac{\|x-s_k\|^2}{2\sigma^2}\right) </math> where <math>s_k</math> are seed locations and <math>\sigma</math> controls the width of the seed influence. The two-point correlation function may be estimated using the normalized Landy–Szalay estimator: <math> \xi(r) = \frac{DD(r)-2DR(r)+RR(r)}{RR(r)} </math> where <math>DD</math>, <math>DR</math>, and <math>RR</math> are normalized pair counts. {{Note|Unless observational datasets are explicitly supplied, demonstrations may use synthetic target correlation curves for methodological illustration only. Synthetic demonstrations should not be interpreted as empirical cosmological evidence.}} When run using synthetic target curves, PDT-resampled catalogs may exhibit enhanced small-scale clustering relative to the baseline configuration. === Computational demonstrations === Reference implementations and supplementary simulation notebooks may be maintained on external repositories or supplementary Wikiversity pages. {{collapse top|Python demonstration placeholder}} <syntaxhighlight lang="python"> # Example implementations may be maintained separately # on GitHub, OSF, or supplementary Wikiversity pages. </syntaxhighlight> {{collapse bottom}} == Scope and Limitations == PDT is a mathematical framework for measure transformations. It does not claim: * a replacement theory for General Relativity or Quantum Mechanics; * empirical confirmation without explicit predictions and tests; * observational validation without independently reproducible analysis. The following discussion extends beyond the primary mathematical framework developed earlier in the article and explores possible conceptual implications and speculative generalizations. == Speculative Extensions and Geometric Renormalization == ''This section is speculative and exploratory in nature.'' Recent mathematical work published in the ''Journal of Applied Probability'' by Baryshnikov, Cao, Kahle, and Liu suggests a possible connection between probability distributions and intrinsic geometry. Studies of “Buffon deficits” on curved manifolds indicate that deviations from classical flat-space Buffon probabilities may encode curvature-dependent geometric information. Within the PDT framework, these observations motivate the broader possibility that geometric structure may influence iterative probabilistic dynamics through curvature-dependent statistical weighting effects. Within PDT, these results are conceptually relevant because they suggest that probabilistic weighting structures may encode nontrivial geometric information. In particular, the Cambridge analysis demonstrates that generalized Buffon-type probabilistic constructions can reflect Gaussian curvature in different geometries. PDT extends this probabilistic perspective by exploring how iterative probability-measure transformations under positive dilation fields may generate evolving statistical structure, entropy flow, and geometry-dependent probabilistic behavior under repeated transformation. At present these ideas remain exploratory and heuristic. No direct physical interpretation is presently established within the PDT framework. Within the PDT framework, this motivates the speculative possibility that curvature could act as a statistical weighting mechanism on classes of admissible paths or configurations. == Future directions == * develop canonical families of dilation fields and invariants; * clarify “structure-from-measure” diagnostics; * publish reproducible simulation notebooks and parameter sweeps; * compare multiple dilation families under shared evaluation criteria; * investigate connections between probabilistic geometry and curvature-dependent statistical measures. == Future Directions: Probability Element (PE) == A speculative extension of Probability Dilation Theory (PDT) is the introduction of a minimal invariant scale in probability-state space, referred to as a '''Probability Element (PE)'''. This concept lies outside standard Fisher information geometry and is not part of established physics. The PE hypothesis proposes that probability-state space may not be fully continuous, but may instead admit a smallest distinguishable scale of structure in terms of information-theoretic resolution. This can be expressed in terms of a dimensionless ratio: <math>\eta = \frac{\sigma_P}{\sigma}</math> where: <math>\sigma_P</math> is a hypothesized minimal probability-resolution scale, <math>\sigma</math> is an effective distinguishability scale in probability-state space. === Conceptual motivation === Standard Fisher information geometry treats probability distributions as points on a smooth manifold with arbitrarily fine distinguishability. The PE hypothesis explores the possibility that this distinguishability may have a lower bound, introducing a form of discreteness in probability-state geometry. === Illustrative toy model (not derived physics) === As a heuristic example, one may consider a modification to special relativistic time dilation of the form: <math>d\tau = dt\sqrt{1 - \frac{v^2}{c^2}}\sqrt{1 - \eta^2}</math> where: <math>v</math> is velocity, <math>c</math> is the speed of light, <math>\eta = \sigma_P / \sigma</math> encodes a proposed probability-resolution scale. This expression is constructed such that standard special relativity is recovered exactly in the limit <math>\eta \to 0</math>. === Status === The Probability Element concept is: Not part of standard Fisher information geometry not derived from quantum mechanics or general relativity not currently empirically established. It is included only as a speculative direction for exploring whether probability-state space admits a minimal geometric resolution scale. === Open questions === Key open research directions include: Whether a consistent discrete formulation of probability geometry can be constructed. Whether a fundamental probability-resolution scale <math>\sigma_P</math> can be derived from known physical principles. Whether such a structure could lead to measurable deviations from standard statistical or relativistic predictions. == Convergence behavior == Iterative PDT transformations may exhibit qualitatively different convergence behavior depending on the structure of the applied dilation field. Repeated probabilistic reweighting can produce entropy reduction, probability concentration, oscillatory behavior, or fluctuating stochastic dynamics over successive iterations. === Qualitative convergence classes === Exploratory finite-state PDT experiments suggest several broad classes of iterative behavior: * '''Concentrating regimes''' — repeated transformations progressively concentrate probability mass into localized regions, often accompanied by decreasing Shannon entropy. * '''Oscillatory regimes''' — probability structure evolves through recurring redistribution patterns without strong long-term concentration. * '''Multi-peak regimes''' — multiple semi-stable concentration regions emerge simultaneously, producing persistent structured probability distributions. * '''Stochastic regimes''' — fluctuating probabilistic structure evolves under partially random or time-dependent weighting behavior. === Entropy and convergence === In many exploratory PDT experiments, entropy reduction correlates with increasing probability concentration under repeated transformation. However, some oscillatory and stochastic field classes may preserve higher entropy distributions or exhibit fluctuating convergence behavior over time. The relationship between entropy evolution and convergence remains an open area of investigation. Future work may examine entropy rates, stability properties, and long-term probabilistic structure under repeated PDT transformations. === Attractor-like behavior === Some iterative PDT systems may exhibit transient attractor-like probabilistic structure in finite-state computational experiments. These behaviors are presently exploratory and are not established mathematical attractors in the formal dynamical-systems sense. Future investigation of PDT convergence behavior may include stability analysis, fixed-point structure, stochastic convergence properties, and comparison with established dynamical systems and probabilistic evolution frameworks. == Current limitations == PDT presently operates as an exploratory probabilistic and computational framework. The theory does not presently derive known physical laws from first principles, nor does it replace established formulations of quantum mechanics or general relativity. Current PDT investigations primarily focus on iterative probability transformations, entropy evolution, probabilistic weighting behavior, and computationally modeled structure formation. Many proposed physical interpretations associated with PDT remain speculative and exploratory. Existing computational experiments are finite-state toy models intended to illustrate qualitative probabilistic behavior rather than experimentally verified physical mechanisms. Future development of PDT would likely require additional mathematical formalization, convergence analysis, stochastic modeling, and comparison with established probabilistic and dynamical systems frameworks. == See also == * [[w:Buffon's needle problem|Buffon's needle problem]] * [[w:Probability measure|Probability measure]] * [[w:Importance sampling|Importance sampling]] * [[w:Radon–Nikodym theorem|Radon–Nikodym theorem]] * [[w:Dynamical system|Dynamical systems]] * [[w:Entropy (information theory)|Entropy]] * [[w:Information theory|Information theory]] * [[w:Measure theory|Measure theory]] * [[w:Geometric probability|Geometric probability]] * [[w:Shannon entropy|Shannon entropy]] * [[w:Stochastic process|Stochastic process]] * [[w:Fixed point (mathematics)|Fixed point]] * [[w:Convergence (mathematics)|Convergence]] == Subpages == The following subpages develop mathematical extensions and specialized topics related to Probability Dilation Theory (PDT). * [[Probability Dilation Theory/Fisher Geometry and Dilation Flows|Fisher Geometry and Dilation Flows]] – studies information geometry, Fisher distance, and geodesic properties of PDT trajectories. * [[Probability Dilation Theory/Logit Representation of PE|Logit Representation of PE]] – develops the log-odds representation of probability elements and exponential PDT flows. * [[Probability Dilation Theory/Convergence and Fixed Points|Convergence and Fixed Points]] – investigates invariant measures, attractors, and stability of iterative PDT transformations. * [[Probability Dilation Theory/Stochastic Dilation Fields|Stochastic Dilation Fields]] – studies random and time-dependent dilation fields, ergodicity, and stochastic measure evolution. * [[Probability Dilation Theory/Entropy Evolution|Entropy Evolution]] – examines Shannon entropy under repeated probability dilation. * [[Probability Dilation Theory/Wasserstein Geometry|Wasserstein Geometry]] – explores distances between probability measures and convergence in measure space. * [[Probability Dilation Theory/Measure-Theoretic Foundations|Measure-Theoretic Foundations]] – develops rigorous measure-theoretic aspects of PDT including normalization and existence conditions. * [[Probability Dilation Theory/Euler Methods and Continuous-Time PDT]] – investigates continuous probability flows and Euler approximations of PDT. * [[Probability Dilation Theory/Worked Example]] – canonical binary example illustrating PDT transformations and geometry. [[Probability Dilation Theory/Decoherence Analogy and Simulation]] == Notation == Throughout PDT, the following notation is used: {| class="wikitable" ! Symbol ! Meaning |- | <math>P</math> | Probability measure |- | <math>P_n</math> | nth iterate of PDT |- | <math>T_D</math> | Probability dilation operator |- | <math>D(x)</math> | Dilation field |- | <math>Z(P,D)</math> | Normalization factor |- | <math>H(P)</math> | Shannon entropy |- | <math>d_F</math> | Fisher-Rao distance |- | <math>W_p</math> | Wasserstein distance |- | <math>\ell</math> | Logit coordinate |- | <math>PE</math> | Probability Element |} == Related probabilistic and geometric literature == Related literature on probabilistic dilation, conditioning behavior, geometric probability, and curvature-dependent probabilistic structure includes the following works: * Augustin, T.; Coolen, F. P. A.; de Cooman, G.; Troffaes, M. C. M. ''Introduction to Imprecise Probabilities''. Wiley, 2014. * Baryshnikov, Y.; Cao, Y.; Kahle, M.; Liu, J. (2024). ''Buffon’s problem on curved surfaces and Gaussian curvature''. ''Journal of Applied Probability''. Cambridge University Press. doi:10.1017/jpr.2024.19 * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Divisive Conditioning: Further Results on Dilation''. Philosophy of Science, Vol. 64, No. 3, 1997. * Herron, T.; Seidenfeld, T.; Wasserman, L. ''Distention for Sets of Probabilities''. Annals of Mathematics and Artificial Intelligence, Vol. 45, 2005. * Moral, S.; Wilson, N. ''Dilation Properties of Coherent Nearly-Linear Models''. International Journal of Approximate Reasoning, Vol. 45, 2007. * Shannon, C. E. (1948). ''A Mathematical Theory of Communication''. ''Bell System Technical Journal'', 27(3), 379–423; 27(4), 623–656. Text and original figures © Howard Richardson. Original contributions by the author. Content is available under the Creative Commons Attribution-ShareAlike license (CC BY-SA) as required by Wikiversity. Reuse permitted with attribution. q07el8jvc84iwvugu065zusixp31y66 0 326060 2816229 2815490 2026-06-18T16:26:04Z Jeanne Noiraud 1366702 /* Database search */ 2816229 wikitext text/x-wiki == Contributors == {| class="wikitable" |+ !Name !Affiliation !ORCID !Contribution |- |Adélie Ranville |IAE de Grenoble, CERAG lab (https://ror.org/0509qp208) |https://orcid.org/0000-0002-3993-6135 |Research design, database search, article screening, knowledge modelling |- |Amélie Pereira | | |Meta-data enrichement |- | | | | |} Contribution statistics are visible here : https://xtools.wmcloud.org/pageinfo/en.wikiversity.org/Just_sustainability_transitions:_a_living_review == Introduction == === Definition of living review === The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked<ref name="Why1">{{Cite Q |Q40040379 }}</ref>. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject<ref name="Why1" /><ref name="Why4">{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}</ref><ref name=":6">{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}</ref>. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication<ref name="Why1"/><ref name="Why4" /><ref name=":6" />. Living systematic reviews exists to address this common problem<ref name="Why1" /><ref name="Why4" /><ref name=":6" /><ref name=":2">https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/</ref>. It is therefore particularly useful in rapidly evolving fields of research<ref name="Why1" /><ref name=":6" />, such as just transition. [[wikidata:Q33002955|Knowledge graphs]], a structured representation of knowledge in the form of a graph, linked together by relationships that encode explicit meanings between these entities, are very suitable for conducting living systematic reviews<ref name=":2" /><ref name="Fotopoulou">{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.</ref>. Advances in AI could render certain older methodological types of living systematic reviews obsoletes<ref>{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}</ref>, as IA are useful to extract, filter and classify datas<ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref><ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref>. [[Large language models]] (LLM) are "on the rise" (2025), but "not yet ready for use"<ref>{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}</ref>. The living review method relevant for just transition because it includes topic such as energy democracy which necessitate transdisciplinarity and consolidation of fragmented literature<ref>{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research & Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}</ref>. === Definitions of just transition : === * «a fair and equitable process of moving towards a post-carbon society’. »<ref name=":0">{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}</ref>. The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions<ref name=":0" />. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions<ref name=":0" />. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice<ref name=":3">{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research & Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}</ref><ref name=":1">{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}</ref> and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice<ref name=":0" /><ref name=":3" /><ref name=":1" /><ref name=":4">{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. === Definition of Procedural justice === Procedural justice is about the fairness of decision-making processes related to transitions<ref name=":4" /> such as the inclusion of those impacted by these decisions<ref name=":5">{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}</ref>. Procedural justice can include issues of community and citizen participation in decision making, their political representation their consultation or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities<ref>{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. For example, the participation of affected communities in decisions related to the construction of new infrastructures<ref name=":0" />. == Methodology == === Wikidata and the semantic web ===<!-- Add introduction to what wikidata is and how the triplet works in a pedagogical manner Example of good description here : https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k#fig1 --> "A knowledge graph is a structured representation of knowledge that captures information in a machine-readable format.<ref name=":9">{{Cite journal|last=Hogan|first=Aidan|last2=Blomqvist|first2=Eva|last3=Cochez|first3=Michael|last4=D’amato|first4=Claudia|last5=Melo|first5=Gerard De|last6=Gutierrez|first6=Claudio|last7=Kirrane|first7=Sabrina|last8=Gayo|first8=José Emilio Labra|last9=Navigli|first9=Roberto|date=2022-05-31|title=Knowledge Graphs|url=https://dl.acm.org/doi/10.1145/3447772|journal=ACM Computing Surveys|language=en|volume=54|issue=4|pages=1–37|doi=10.1145/3447772|issn=0360-0300}}</ref> A knowledge graph consists of a graph or network of interconnected data points, where each data point represents a piece of information or a concept, and the relationships between them are explicitly defined. Knowledge graphs organize and store data in a format that facilitates information retrieval, data analysis, and reasoning."<ref>{{Cite journal|last=Meijer|first=David|last2=Beniddir|first2=Mehdi A.|last3=Coley|first3=Connor W.|last4=Mejri|first4=Yassine M.|last5=Öztürk|first5=Meltem|last6=Hooft|first6=Justin J. J. van der|last7=Medema|first7=Marnix H.|last8=Skiredj|first8=Adam|date=2025-04-16|title=Empowering natural product science with AI: leveraging multimodal data and knowledge graphs|url=https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k|journal=Natural Product Reports|language=en|volume=42|issue=4|pages=654–662|doi=10.1039/D4NP00008K|issn=1460-4752}}</ref> == Building a corpus and enriching bibliographic metadata == === Database search === We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. {| class="wikitable" |+ !Keywords search !Database !Search date !Filters !Number of results |- |(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) |Web of Science (all databases, all dates) |December 2025 |Document type: Review Article |362 |} === Article screening === Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were * Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...) * Not literature reviews (e.g. review of policies, initiatives, cases, review notes, book review...) * Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors or included justice consideration only in “future research” suggestions * Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy * Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper The files resulting from this step are available at : https://doi.org/10.5281/zenodo.20749973 === Importing selected articles into Wikidata === To import the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in the database. Next we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the "export to Wikidata QuickStatements" function of Zotero and use the QuickStatements tool to add them to Wikidata. Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|"Merge" gadget]] on Wikidata. We checked manually for duplicated statments in those items. === Article classification through meta-data enrichement ===<!-- Add : What is meta-data enrichement --> Existing review try to classify existing articles according to various criteria such as industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt).<ref name=":5" /> We selected the most relevant properties in Wikidata to reflect these classifications : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe its geographical focus. ==== Main subjects ==== We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : {| class="wikitable" |+ !Qid !Main topic !Description |- |[[d:Q42377797|Q42377797]] |acceptability |characteristic of a thing being subject to acceptance for some purpose |- |[[d:Q2798912|Q2798912]] |accountability |concept of responsibility in ethics, governance and decision-making |- |[[d:Q421953|Q421953]] |actor–network theory |theory within social science |- |[[d:Q84459973|Q84459973]] |affordability | |- |[[d:Q185836|Q185836]] |age of a person |time elapsed since a person was born |- |[[d:Q4764988|Q4764988]] |animal studies |field in which animals are studied in a variety of cross-disciplinary ways |- |[[d:Q4338318|Q4338318]] |awareness |state or ability to perceive, to feel, or to be conscious of events, objects, or sensory patterns |- |[[d:Q4930066|Q4930066]] |blue carbon |carbon captured by the world's coastal ocean ecosystems |- |[[d:Q430460|Q430460]] |capability approach |economic theory |- |[[d:Q7569|Q7569]] |child |human between birth and puberty |- |[[d:Q4116870|Q4116870]] |civic engagement |individual or group activity addressing issues of public concern |- |[[d:Q125928|Q125928]] |climate change |human-caused changes to climate on Earth |- |[[d:Q260607|Q260607]] |climate change adaptation |process of adjustment to actual or expected climate change and its effects, seeking to moderate or avoid harm or exploit beneficial opportunities |- |[[d:Q1291678|Q1291678]] |climate justice |term linking the climate crisis with environmental and social justice |- |[[d:Q2270945|Q2270945]] |co-creation |product or service design process in which input from consumers plays a central role |- |[[d:Q16972712|Q16972712]] |co-design |approach to design attempting to actively involve all stakeholders |- |[[d:Q16324410|Q16324410]] |coproduction |product or service design process in which input from consumers plays a central role |- |[[d:Q11024|Q11024]] |communication |act of conveying intended meaning |- |[[d:Q177634|Q177634]] |community |social unit of human organisms who share common values |- |[[d:Q5154673|Q5154673]] |community choice aggregation |alternative energy supply system |- |[[d:Q113514984|Q113514984]] |community energy |delivery of community-led renewable energy, energy demand reduction and energy supply projects |- |[[d:Q65807646|Q65807646]] |community participation |The taking part by members of a community in decisionmaking processes related to the development of their community |- |[[d:Q188843|Q188843]] |cosmopolitanism |ideology that all human beings belong to a single community, based on a shared morality |- |[[d:Q11693783|Q11693783]] |decarbonization |change of economy, especially of energy industries, towards lower carbon dioxide emissions |- |[[d:Q284289|Q284289]] |deliberative democracy |form of democracy focusing on consensus |- |[[d:Q7174|Q7174]] |democracy |form of government |- |[[d:Q552284|Q552284]] |distributive justice |concept of the socially just allocation of goods |- |[[d:Q1230584|Q1230584]] |diversity |concept in sociology and political studies |- |[[d:Q1049066|Q1049066]] |ecological economics |research field on the interdependence of human economies and natural ecosystems |- |[[d:Q8134|Q8134]] |economics |social science that studies the production, distribution, and consumption of goods and services |- |[[d:Q868575|Q868575]] |empowerment |providing increased autonomy |- |[[d:Q295865|Q295865]] |ecosystem service |benefits created by nature, forests and environmental systems |- |[[d:Q138359220|Q138359220]] |energy citizenship |involvement of citizens in energy-related decisions |- |[https://www.wikidata.org/w/index.php?title=Q131444737&redirect=no Q131444737] |community energy |[redirection] |- |[[d:Q16869822|Q16869822]] |energy consumption |amount of energy or power used |- |[[d:Q1358789|Q1358789]] |senior |elderly person |- |[[d:Q14944319|Q14944319]] |energy democracy |concept in environmental justice movement |- |[[d:Q192704|Q192704]] |energy efficiency |ratio between the useful energy output and the input of a machine |- |[[d:Q24965464|Q24965464]] |energy modeling |process of building computer models of energy systems in order to analyze them |- |[[d:Q1805337|Q1805337]] |energy policy |policy addressing energy issues |- |[[d:Q1341244|Q1341244]] |energy poverty |lack of access to modern energy services |- |[[d:Q3406659|Q3406659]] |energy production |conversion of energy from a primary source into a form useful to humans |- |[[d:Q117091181|Q117091181]] |energy justice |subconcept of economic equality |- |[[d:Q3456219|Q3456219]] |energy renovation |building works aimed at reducing energy consumption and decarbonising the energy sources used |- |[[d:Q2700433|Q2700433]] |energy security |national security considerations of energy availability |- |[[d:Q837718|Q837718]] |energy storage |capture of energy produced at one time for use at a later time |- |[[d:Q795757|Q795757]] |energy transition |long-term structural change towards sustainable energy systems |- |[[d:Q1479527|Q1479527]] |environmental justice |system of fairness |- |[[d:Q771773|Q771773]] |fairness |concept in sociology and generally the interaction of society |- |[[d:Q56395513|Q56395513]] |farming system |method of agricultural production defined by its physical practices and economic characteristics |- |[[d:Q5465532|Q5465532]] |food system |all processes and infrastructure involved in feeding a population |- |[[d:Q4421|Q4421]] |forest |dense collection of trees covering a relatively large area |- |[[d:Q48277|Q48277]] |gender |social concept which distinguish the different gender categories |- |[[d:Q1553864|Q1553864]] |governance |all of the processes of governing, whether undertaken by a government, market or network, whether over a family, tribe, formal or informal organization or territory and whether through the laws, norms, power or language of an organized society |- |[[d:Q8458|Q8458]] |human rights |inalienable fundamental rights to which a person is inherently entitled |- |[[d:Q11376059|Q11376059]] |human rights violation |act or omission which contravene the principles of human rights |- |[[d:Q103817|Q103817]] |indigenous people |first inhabitants of an area and their descendants |- |[[d:Q113561794|Q113561794]] |indigenous science |indigenous knowledge applied to the scientific method |- |[[d:Q770480|Q770480]] |injustice |quality relating to unfairness or undeserved outcomes |- |[[d:Q17142211|Q17142211]] |interactional justice |the perceived appropriateness of interpersonal treatment |- |[[d:Q1516555|Q1516555]] |intersectionnality |theoretical framework of multidimensional oppression |- |[[d:Q6316391|Q6316391]] |just transition |Framework developed by the trade union movement to encompass wide range of social interventions needed to secure decent work opportunities and a greener economy. |- |[[d:Q366139|Q366139]] |legitimation |the process of making something acceptable and normative to a group |- |[[d:Q3027857|Q3027857]] |living lab |user-centered, open innovation ecosystem integrating research and innovation in real life communities |- |[[d:Q59679511|Q59679511]] |low income |home with little money |- |[[d:Q43619|Q43619]] |natural environment |all living and non-living things occurring naturally on Earth or some region thereof |- |[[d:Q127514833|Q127514833]] |nature-positive |global goal to halt and reverse nature loss by 2030 |- |[[d:Q13023682|Q13023682]] |non-human |organism not in the genus Homo |- |[[d:Q728646|Q728646]] |partnership |arrangement in which parties agree to cooperate to advance their mutual interests |- |[[d:Q3907287|Q3907287]] |policy making |the act of developing policy |- |[[d:Q9357091|Q9357091]] |political theory |class of theory |- |[[d:Q265425|Q265425]] |postcolonialism |academic discipline |- |[[d:Q25107|Q25107]] |power |ability to influence the behavior of others |- |[[d:Q442100|Q442100]] |procedural justice |fairness in the processes that resolve disputes and allocate resources |- |[[d:Q7249406|Q7249406]] |project governance |management framework |- |[[d:Q7257735|Q7257735]] |public engagement |Policy-making practice |- |[[d:Q541936|Q541936]] |public participation |participation of citizens in various policy decisions and planning processes |- |[[d:Q6142016|Q6142016]] |recognition justice |social philosophy theory |- |[[d:Q10509953|Q10509953]] |renewable electricity |electricity from renweable sources |- |[[d:Q12705|Q12705]] |renewable energy |energy collected from renewable resources |- |[[d:Q56510941|Q56510941]] |renewable energy policy | |- |[[d:Q1165392|Q1165392]] |restorative justice |approach to justice where victims and perpetrators mediate a restitution agreement |- |[[d:Q4414036|Q4414036]] |rural population |inhabitants of rural areas or of small towns classified as rural |- |[[d:Q17152351|Q17152351]] |smart system |adaptive intelligent systems |- |[[d:Q187588|Q187588]] |social class |group of people categorized in a hierarchy based on socioeconomic factors |- |[[d:Q264892|Q264892]] |social justice |concept that discrimination recognized in society should be remedied |- |[[d:Q34749|Q34749]] |social science |academic disciplines concerned with society and the relationships between individuals in society |- |[[d:Q2930198|Q2930198]] |stakeholder participation |involvement of groups or individuals affected by the actions of an entity |- |[[d:Q125359881|Q125359881]] |sustainability transition | |- |[[d:Q219416|Q219416]] |sustainability |ability of human civilization to coexist with the biosphere in a steady state |- |[[d:Q131201|Q131201]] |sustainable development |mode of human development that meets current demands without compromising the needs of future generations |- |[[d:Q7649586|Q7649586]] |Sustainable Development Goals |set of United Nations-defined global development goals and climate change |- |[[d:Q69883|Q69883]] |urban planning |technical and political process concerned with the use of land and design of the urban environment |- |[[d:Q920600|Q920600]] |urban renewal |program of land redevelopment in cities, often where there is urban decay |- |[[d:Q3376054|Q3376054]] |vulnerable population |group of persons whose range of options is severely limited, are subjected to coercion, or who may be compromised in their ability to give informed consent |- |[[d:Q107389921|Q107389921]] |water-management | |- |[[d:Q7981051|Q7981051]] |well-being |measure of how well life is to someone or a group with factors such as health, happiness and satisfaction |- |[[d:Q467|Q467]] |woman |female adult human |- |[[d:Q188867|Q188867]] |future studies |study of possible, probable, and preferable social, technological and political futures |- |[[d:Q1038171|Q1038171]] |participatory design |active involvement of all stakeholders in the design process |} <!-- include all below items using the wikidata link template --> Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords. ==== Study types ==== Our review included only litterature reviews. We first read abstracts to identify all the [https://angryloki.github.io/wikidata-graph-builder/?item=Q2412849&property=P279&mode=reverse different types of litterature reviews] present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved these method items using the methodological references cited in the reviewed papers. The types of reviews were : {| class="wikitable" |+ !Qid !Study type !Description |- |[[d:Q603441|Q603441]] |bibliometrics |statistical analysis of written publications, such as books or articles |- |[[d:Q472342|Q472342]] |scientometrics |study of measuring and analysing science, technology and innovation |- |[[d:Q815382|Q815382]] |meta-analysis |statistical method that summarizes data from multiple sources |- |[[d:Q1504425|Q1504425]] |systematic review |publication type, study that gathers, analyzes, and communicates the results of research and information on a topic |- |[[d:Q2412849|Q2412849]] |literature review |process of information search and text of a review article (Q7318358), which includes the current knowledge including substantive findings, as well as theoretical and methodological contributions to a particular topic |- |[[d:Q6822263|Q6822263]] |meta-regression |statistical tool used in meta-analyses |- |[[d:Q7301211|Q7301211]] |realist evaluation |[...] |- |[[d:Q17007303|Q17007303]] |combinatorial meta-analysis |[...] |- |[[d:Q70470634|Q70470634]] |network meta-analysis |meta-analysis of randomized trials in which estimates of comparative treatment effects are visualized and interpreted from a network of interventions |- |[[d:Q101116078|Q101116078]] |scoping review |search for concepts by mapping the language and data which surrounds those concepts and adjusting the search method iteratively to synthesize evidence and assess the scope of an area of inquiry |- |[[d:Q110665014|Q110665014]] |narrative review |type of literature review, without structured method of retrieval and analysis |- |[[d:Q137174203|Q137174203]] |conceptual review |academic research aiming to review existing concepts and definitions in the litterature |- |[[d:Q137174450|Q137174450]] |critical review |type of literature review analysing strenghts, major contributions, mistakes and neglected issues in an academic field of research |- |[[d:Q137209848|Q137209848]] |integrative literature review |type of literature review |- |[[d:Q110665014|Q137211242]] |narrative review |type of literature review, without structured method of retrieval and analysis |}Then, we added the {{Wikidata entity link|P8363}} of each articles based on the abstract and method sections. In case of doubt, we compared our interpretation. ==== Research site ==== When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article "{{Wikidata entity link|Q137901202}}" focused on {{Wikidata entity link|Q132959}}. ==== Results ==== The table listing all the papers in the sample can be visualized [https://tabernacle.toolforge.org/?#/tab/manual/Q137211155%0A%0A%0A%0A%0A%0A%0AQ114306483%0A%0A%0A%0A%0AQ137901181%0A%0A%0A%0AQ137901182%0A%0A%0A%0A%0A%0A%0A%0AQ137901183%0A%0A%0AQ114306476%0A%0A%0A%0A%0AQ137901184%0A%0A%0A%0A%0AQ137901185%0A%0A%0A%0A%0A%0AQ137901186%0A%0A%0A%0A%0A%0A%0AQ137901187%0A%0A%0A%0A%0A%0A%0AQ137901188%0A%0A%0A%0A%0AQ137210566%0A%0A%0A%0A%0AQ114306511%0A%0A%0A%0A%0A%0AQ137901191%0A%0A%0A%0A%0AQ137901192%0A%0A%0A%0A%0AQ137901193%0A%0A%0A%0A%0AQ135979013%0A%0A%0A%0A%0A%0A%0A%0AQ137901195%0A%0A%0A%0A%0A%0AQ137901196%0A%0A%0A%0A%0A%0A%0AQ137901197%0A%0A%0A%0A%0AQ136447761%0A%0A%0A%0AQ137901199%0A%0A%0A%0A%0A%0A%0AQ129652515%0A%0A%0A%0A%0A%0A%0AQ137901201%0A%0A%0A%0A%0A%0AQ137901202%0A%0A%0A%0A%0AQ137901203%0A%0A%0A%0AQ137901204%0A%0A%0A%0A%0A%0A%0A%0AQ137901205%0A%0A%0A%0A%0AQ137901206%0A%0A%0A%0A%0A%0A%0A%0A%0AQ137901207%0A%0A%0A%0A%0AQ129203992%0A%0A%0A%0A%0A%0A%0AQ114197507%0A%0A%0A%0AQ137901161%0A%0A%0A%0A%0A%0A%0A%0AQ137901209%0A%0A%0A%0A%0A%0AQ137901210%0A%0A%0A%0A%0A%0AQ137901211%0A%0A%0A%0A%0AQ11420462%0A%0AQ137901213%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0AQ104887325%0A%0A%0A%0A%0A%0AQ137901162%0A%0A%0AQ137901163%0A%0A%0A%0A%0AQ137901164%0A%0A%0A%0A%0A%0AQ137901215%0A%0A%0A%0A%0AQ137901216%0A%0A%0A%0A%0A%0A%0A%0A%0AQ137901217%0A%0A%0A%0A%0AQ115448818%0A%0A%0A%0A%0AQ137901218%0A%0A%0A%0AQ137901219%0A%0A%0A%0A%0AQ137901220%0A%0A%0A%0A%0A%0AQ137901221%0A%0A%0A%0A%0A%0AQ137901222%0A%0A%0A%0A%0AQ137901223%0A%0A%0AQ137901224%0A%0A%0A%0AQ137901225%0A%0A%0A%0A%0A%0A%0AQ137901226%0A%0A%0A%0AQ137901227%0A%0A%0AQ137901182/Len%3BP921%3BP6153%3BP8363%3BP50 here] (be careful if you are logged into Wikidata as the table is editable). == Modelling knowledge == Concept maps can be a powerful literature review tool<ref>{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}</ref> allowing to synthetize theoretical statements about relationship between concepts<ref>{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}</ref>. In the present study, we explored how concept map can be used to model the knowledge present in the paper we selected. [define knowledge modelling] ==== Wikidata ontology ==== Wikidata "supports multiple coexisting classification" and allow multiple ontological frameworks to coexist.<ref name=":8">{{Cite web|url=https://arxiv.org/abs/2512.12260v1|title=A Multi-Axial Mindset for Ontology Design Lessons from Wikidata's Polyhierarchical Structure|last=Doğan|first=Ege Atacan|last2=Patel-Schneider|first2=Peter F.|date=2025-12-13|website=arXiv.org|language=en|access-date=2026-05-26}}</ref> It also supports epistemic pluralism : different worldviews can be represented in wikidata, even though scientific knowledge is preferred.<ref name=":8" /> See more on membership properties : https://www.wikidata.org/wiki/Help:Basic_membership_properties See the discussion on cause modelling : https://www.wikidata.org/wiki/Help:Modeling_causes/en ==== Conceptual modelling ==== We first reflected on what kind of wikidata properties could be used to represent concepts and theories in wikidata. Capturing the content of a concept is not straightforward and there are various approaches coming from psychology and philosophy on the matter<ref>{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}</ref> we summarize these approaches below and examine which wikidata properties exist to represent them. * Definition: the content of a concept can be formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}... * Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}}. * Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}. * Essence: the content of a concept is "something" deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can create it before really knowing what all its properties will be. * Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}. ==== Categorization and conceptualisation practices in management sciences ==== In management sciences « systematic categorizing is the best and perhaps only method for clearing up semantic confusion, management scholars never take the classical approaches to categorizing that facilitated tremendous progress in the physical sciences, and seldomly build on extant categorial schemes. »<ref>{{Cite journal|last=Pierce|first=Jason R.|date=2025-01|title=Categorizing Concepts and Phenomena in Management Research: A Four-Phase Integrative Review and Recommendations|url=http://journals.aom.org/doi/full/10.5465/annals.2023.0052|journal=Academy of Management Annals|language=en|volume=19|issue=1|page=28|pages=9–37|doi=10.5465/annals.2023.0052|issn=1941-6520}}</ref>. Some scholars discussed how conceptualization should be done<ref>{{Cite journal|last=Podsakoff|first=Philip M.|last2=MacKenzie|first2=Scott B.|last3=Podsakoff|first3=Nathan P.|date=2016-04|title=Recommendations for Creating Better Concept Definitions in the Organizational, Behavioral, and Social Sciences|url=https://journals.sagepub.com/doi/10.1177/1094428115624965|journal=Organizational Research Methods|language=en|volume=19|issue=2|pages=159–203|doi=10.1177/1094428115624965|issn=1094-4281}}</ref>,<ref>{{Cite journal|last=Makowski|first=Piotr Tomasz|date=2021-10|title=Optimizing Concepts: Conceptual Engineering in the Field of Management—The Case of Routines Research|url=http://journals.aom.org/doi/full/10.5465/amr.2019.0252|journal=Academy of Management Review|language=en|volume=46|issue=4|pages=702–724|doi=10.5465/amr.2019.0252|issn=0363-7425}}</ref>. ==== Thematic networks ==== [[File:Thematic network example.jpg|thumb|547x547px|Structure of a thematic network (Source: Attride-Stirling 2001)]] A thematic network is “simply a way of organizing a thematic analysis of qualitative data”<ref name=":7">{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}</ref>. It is compatible with classical coding strategies such as [[grounded theory]]<ref>{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}</ref>. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data<ref name=":7" />. The principle is to assemble basic themes into more general themes. Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding (e.g. grounded theory) to identify themes and sub-themes. However, the nature of the relationship between these various themes and sub-themes is often not specified. * ==== Causal networks ==== The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers sometime present models with boxes and arrows representing correlations and/or causations<ref>{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}</ref>. In qualitative research, building grounded theory models is about "[accounting] for not only all the major emergent concepts, themes, and dimensions, but also for their dynamic interrelationships. Speaking in classic boxes-and-arrows terms, this process amounts to assembling the constellation of boxes with a special focus on the arrows."<ref>{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}</ref> Researchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can "feedback" (two variables can influence each other)<ref>{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}</ref>. Wikidata includes several properties to describe causal relationships: * {{Wikidata entity link|P828}} * {{Wikidata entity link|P1542}} * {{Wikidata entity link|P1537}} * {{Wikidata entity link|P1479}} : it is difficult to identify single causes for social phenomenons, many factors having an effect on the subject item will likely be contributing factors == Testing concept modelling on {{Wikidata entity link|Q14944319}} == We started by experimenting the modelling of concept by focusing on the concept of {{Wikidata entity link|Q14944319}}. We selected a subset of papers which had energy democracy as main topic : * {{Wikidata entity link|Q137901202}} * {{Wikidata entity link|Q137901196}} * {{Wikidata entity link|Q137901182}} * {{Wikidata entity link|Q136447761}} * {{Wikidata entity link|Q129652515}} * {{Wikidata entity link|Q114306483}} We read each paper and used them as source to enter statements in the item {{Wikidata entity link|Q14944319}}. For example, "Energy democracy is both an ideal and a process"<ref>{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research & Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}</ref>, we thus entered the wikidata statement {{Wikidata entity link|Q14944319}} is an {{Wikidata entity link|P31}} {{Wikidata entity link|Q840396}}, using the paper as source. The result of this first step is visible in the archival version of the item (22 May 2026) here https://www.wikidata.org/w/index.php?title=Q14944319&oldid=2495982191. Ontology challenges: *{{Wikidata entity link|P31}}: concepts may have a dual nature because they designate at the same time an idea and the entity that this idea represent. Energy democracy is a concept, an ideal, a process and an outcome. *'''Process versus outcome :''' For material processes, the distinction between process and outcome is rather simple. For example, in Wikidata, {{Wikidata entity link|Q11629}} (practice of applying paint) is different from {{Wikidata entity link|Q3305213}} (visual artwork), and this distinction is based on the criterion "{{Wikidata entity link|Q127270577}}". However, this distinction is less straightforward for social processes that do not have an end. Such processes are ongoing and outcomes cannot be separated as clearly. * '''Ideal versus reality :''' Concepts do not have goals in themselves, but the reality they represent can have goals. To distinguish goals from the process to reach it, we used {{Wikidata entity link|P3712}} to describe ideals and {{Wikidata entity link|P2670}} to describe processes. * '''Phenomenon versus theory :''' Wikidata current items are not really suited to model "meta-research" statements. For example, modelling the idea tha the literature on energy democracy is fragmented would require creating an item representing the energy democracy literature, not just energy democracy in general. Similarly, it can be difficult to model the chronological evolution of the definition of an idea (although it could be technically possible). It is hard to represent in Wikidata affirmations related to missing knowlege, propositions of untested hypothesis, critique of existing research or research agenda recommandations * '''Origin of discourses versus origin of practices :''' To distinguish the causes of the concepts/discourses and the causes of the phenomenon itself, we used {{Wikidata entity link|P3938}} to indicate the origins of the concept or the movements promoting it. Some of the statements we added may seem contradictory. However, Wikidata supports "because statements essentially point to referenceable sources of information and different sources may provide contradicting information, it's possible to represent a plurality of perspectives on Wikidata"<ref>{{Cite web|url=https://www.wikidata.org/wiki/Help:Statements#Plurality_and_consensus|title=Help:Statements - Wikidata|website=www.wikidata.org|language=en|access-date=2026-06-08}}</ref>. The {{Wikidata entity link|Q14944319}} concept could be split into more precise concepts to distinguish the social movement advocating for it, the political concept theorizing it and the concrete initiatives implementing it. However, the current sources do not make this distinction for now. Other challenges * Wikidata does not seem to be the best tool to model quantitative statements, for example, the paper {{Wikidata entity link|Q137901196}} states that "9.8% of the final energy consumed in developing countries comes from modern renewable energy sources". Including energy data in Wikidata require using or creating specific properties (e.g. {{Wikidata entity link|P6826}}) * When concepts are not precisely defined, statements cannot be modelled correctly. For example, in the sentence "management of social affairs by voluntary and self-governing associations is deemed to ensure that both citizen choice and public welfare are best served"<ref>{{Cite journal|last=Veelen|first=Bregje van|last2=Horst|first2=Dan van der|date=2018-12-01|title=What is energy democracy? Connecting social science energy research and political theory|url=https://www.wikidata.org/wiki/Q129652515|journal=Energy Research & Social Science|language=English|volume=46|pages=19–28|doi=10.1016/J.ERSS.2018.06.010}}</ref>, "choice" could refer to {{Wikidata entity link|Q111986453}}, {{Wikidata entity link|Q1331926}}, or {{Wikidata entity link|Q12888920}} as "choice" can refer to the availability of different options, or the decision process to chose among them. Advantages : * Link toward unique identifiers for concepts, but also laws (e.g. {{Wikidata entity link|Q139764294}}) == Interactions with the Wikidata community == * Some Wikidata contributors added labels for {{Wikidata entity link|Q14944319}} in other languages such as Armenian or Slovenian. == Data visualisation == === Filter statements === * Visualize only statements using a specitic source. Example : https://w.wiki/PFqH * Visualize only items which are part to the present project (require that all items of the project include the statement {{Wikidata entity link|P6104}} {{Wikidata entity link|Q134545539}}). === Mapping a concept === Scholia request "topic in context" : 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Example with Energy democracy] === Mapping sources consensus === Visualise graphs and use the number of references to determine edge thickness/weight. == Writing == To cite articles we used the [[Template:Cite Q|Cite Q template.]] Each reference is an item in Wikidata and the template retrieve the necessary data to generate the citation references below. == Future research == The analysis of knowledge graph could in theory allow to make logical deduction to generate new data<ref name=":9" />. Reflect on the future of scholarly communication : https://hal.science/hal-03277615/file/OPERAS_Future_of_Scholarly_Communication_06.2021.pdf == Data == {| class="wikitable sortable" ! QID !! Year !! DOI !! Title |- | [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review |- | [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review |- | [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review |- | [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter? |- | [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset. |- | [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies? |- | [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection |- | [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development |- | [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research |- | [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition |- | [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning |- | [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review |- | [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view |- | [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory |- | [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries |- | [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review |- | [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions |- | [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies |- | [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes |- | [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation |- | [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives |- | [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies |- | [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda |- | [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice? |- | [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review |- | [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research |- | [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape |- | [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models |- | [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review |- | [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions |- | [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions |- | [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation |- | [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings |- | [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda |- | [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review |- | [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework |- | [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende |- | [[d:Q137901185|Q137901185]] || 2024 || [https://doi.org/10.1016/J.ESD.2024.101546 10.1016/J.ESD.2024.101546] || Characterizing 'injustices' in clean energy transitions in Africa |- | [[d:Q137901226|Q137901226]] || 2024 || [https://doi.org/10.1016/J.JCLEPRO.2024.143470 10.1016/J.JCLEPRO.2024.143470] || Energy justice and sustainable urban renewal: A systematic review of low-income old town communities |- | [[d:Q137901222|Q137901222]] || 2024 || [https://doi.org/10.1016/J.JENVMAN.2024.120804 10.1016/J.JENVMAN.2024.120804] || Forest, climate, and policy literature lacks acknowledgement of environmental justice, diversity, equity, and inclusion |- | [[d:Q115441381|Q115441381]] || 2021 || [https://doi.org/10.1016/J.RSER.2021.111504 10.1016/J.RSER.2021.111504] || Participatory methods in energy system modelling and planning – A review |- | [[d:Q137901205|Q137901205]] || 2025 || [https://doi.org/10.1016/J.RSER.2025.115892 10.1016/J.RSER.2025.115892] || A systematic review of the intersection between energy justice and human rights |- | [[d:Q137901225|Q137901225]] || 2024 || [https://doi.org/10.1017/SUS.2024.24 10.1017/SUS.2024.24] || Blue carbon as just transition? A structured literature review |- | [[d:Q137901220|Q137901220]] || 2025 || [https://doi.org/10.1017/SUS.2025.2 10.1017/SUS.2025.2] || Toward an intersectional equity approach in social-ecological transformations |- | [[d:Q137901203|Q137901203]] || 2024 || [https://doi.org/10.1080/14693062.2023.2256697 10.1080/14693062.2023.2256697] || Exploring the democracy-climate nexus: a review of correlations between democracy and climate policy performance |- | [[d:Q137901164|Q137901164]] || 2022 || [https://doi.org/10.1111/GEC3.12662 10.1111/GEC3.12662] || Creating fairer futures for sustainability transitions |- | [[d:Q137901227|Q137901227]] || 2025 || [https://doi.org/10.1139/ER-2024-0018 10.1139/ER-2024-0018] || Community engagement in nature-positive food systems programming and research in East and Southern Africa: a review |- | [[d:Q119955266|Q119955266]] || 2019 || [https://doi.org/10.1146/ANNUREV-ENVIRON-101718-033103 10.1146/ANNUREV-ENVIRON-101718-033103] || Co-Producing Sustainability: Reordering the Governance of Science, Policy, and Practice |- | [[d:Q137901206|Q137901206]] || 2023 || [https://doi.org/10.1146/ANNUREV-ENVIRON-112621-063400 10.1146/ANNUREV-ENVIRON-112621-063400] || Metrics for Decision-Making in Energy Justice |- | [[d:Q137901213|Q137901213]] || 2022 || [https://doi.org/10.1186/S13705-021-00330-4 10.1186/S13705-021-00330-4] || Mapping emergent public engagement in societal transitions: a scoping review |- | [[d:Q137901163|Q137901163]] || 2025 || [https://doi.org/10.17573/CEPAR.2025.2.09 10.17573/CEPAR.2025.2.09] || From Co-Creation to Circular Cities: Exploring Living Labs in EU Governance Frameworks - A Literature Review |- | [[d:Q137901197|Q137901197]] || 2024 || [https://doi.org/10.3390/EN17143512 10.3390/EN17143512] || A Systematic Review on the Path to Inclusive and Sustainable Energy Transitions |- | [[d:Q104887325|Q104887325]] || 2019 || [https://doi.org/10.3390/SU11041023 10.3390/SU11041023] || Deliberation and the Promise of a Deeply Democratic Sustainability Transition |- | [[d:Q137901202|Q137901202]] || 2021 || [https://doi.org/10.3390/SU13042128 10.3390/SU13042128] || A Review of Energy Communities in Sub-Saharan Africa as a Transition Pathway to Energy Democracy |- | [[d:Q137901210|Q137901210]] || 2023 || [https://doi.org/10.3390/SU15032441 10.3390/SU15032441] || Sustainable Project Governance: Scientometric Analysis and Emerging Trends |- | [[d:Q137901224|Q137901224]] || 2024 || [https://doi.org/10.3390/SU16198700 10.3390/SU16198700] || Empowering Communities to Act for a Change: A Review of the Community Empowerment Programs towards Sustainability and Resilience |} == References == {{References}} 67kruh47lkdchgdmr8dn8wtb5vi0i35 2816230 2816229 2026-06-18T16:54:02Z Jeanne Noiraud 1366702 /* Research site */ 2816230 wikitext text/x-wiki == Contributors == {| class="wikitable" |+ !Name !Affiliation !ORCID !Contribution |- |Adélie Ranville |IAE de Grenoble, CERAG lab (https://ror.org/0509qp208) |https://orcid.org/0000-0002-3993-6135 |Research design, database search, article screening, knowledge modelling |- |Amélie Pereira | | |Meta-data enrichement |- | | | | |} Contribution statistics are visible here : https://xtools.wmcloud.org/pageinfo/en.wikiversity.org/Just_sustainability_transitions:_a_living_review == Introduction == === Definition of living review === The concept of living systematic reviews is recent (2014), so the definition has been regularly reworked<ref name="Why1">{{Cite Q |Q40040379 }}</ref>. Living systematic reviews complement the older concept of [[literature review]]. Its objective is the same : obtain an accurate overview of the state of scientific knowledge on a subject<ref name="Why1" /><ref name="Why4">{{Cite journal |last=Akl |first=Elie A. |last2=Meerpohl |first2=Joerg J. |last3=Elliott |first3=Julian |last4=Kahale |first4=Lara A. |last5=Schünemann |first5=Holger J. |last6=Agoritsas |first6=Thomas |last7=Hilton |first7=John |last8=Perron |first8=Caroline |last9=Akl |first9=Elie |last10=Hodder |first10=Rebecca |last11=Pestridge |first11=Charlotte |last12=Albrecht |first12=Lauren |last13=Horsley |first13=Tanya |last14=Platt |first14=Joanne |last15=Armstrong |first15=Rebecca |date=2017-11 |title=Living systematic reviews: 4. Living guideline recommendations |url=https://www.wikidata.org/wiki/Q50084143 |journal=Journal of Clinical Epidemiology |language=en |volume=91 |pages=47–53 |doi=10.1016/j.jclinepi.2017.08.009}}</ref><ref name=":6">{{Citation|title=Living Systematic Reviews|url=https://doi.org/10.1007/978-1-0716-1566-9_7|publisher=Springer US|work=Meta-Research: Methods and Protocols|date=2022|access-date=2026-01-16|place=New York, NY|isbn=978-1-0716-1566-9|pages=121–134|doi=10.1007/978-1-0716-1566-9_7|language=en|first=Mark|last=Simmonds|first2=Julian H.|last2=Elliott|first3=Anneliese|last3=Synnot|first4=Tari|last4=Turner|editor-first=Evangelos|editor-last=Evangelou|editor2-first=Areti Angeliki|editor2-last=Veroniki}}</ref>. A traditional review may be obsolete by the time it is published, as new studies have emerged between the submission of the manuscript and its publication<ref name="Why1"/><ref name="Why4" /><ref name=":6" />. Living systematic reviews exists to address this common problem<ref name="Why1" /><ref name="Why4" /><ref name=":6" /><ref name=":2">https://blogs.lse.ac.uk/impactofsocialsciences/2019/05/14/the-death-of-the-literature-review-and-the-rise-of-the-dynamic-knowledge-map/</ref>. It is therefore particularly useful in rapidly evolving fields of research<ref name="Why1" /><ref name=":6" />, such as just transition. [[wikidata:Q33002955|Knowledge graphs]], a structured representation of knowledge in the form of a graph, linked together by relationships that encode explicit meanings between these entities, are very suitable for conducting living systematic reviews<ref name=":2" /><ref name="Fotopoulou">{{Cite journal|first1=Eleni |last1=Fotopoulou|first2=Ioanna|last2=Mandilara|first3=Anastasios|last3=Zafeiropoulos|first4=Chrysi|last4=Laspidou|first5=Giannis |last5=Adamos|first6=Phoebe|last6=Koundouri|first7=Symeon|last7=Papavassiliou|title=SustainGraph: A knowledge graph for tracking the progress and the interlinking among the sustainable development goals’ targets|journal=Frontiers in environmental science, Frontiers|volume=10|date=2022-10-26|issn=2296-665X|doi=10.3389/FENVS.2022.1003599|url=https://www.wikidata.org/wiki/Q117837999}}.</ref>. Advances in AI could render certain older methodological types of living systematic reviews obsoletes<ref>{{Cite journal|last=Krlev|first=Gorgi|last2=Hannigan|first2=Tim|last3=Spicer|first3=André|date=2025-01|title=What Makes a Good Review Article? Empirical Evidence From Management and Organization Research|url=https://journals.aom.org/doi/abs/10.5465/annals.2021.0051|journal=Academy of Management Annals|volume=19|issue=1|pages=376–403|doi=10.5465/annals.2021.0051|issn=1941-6520}}</ref>, as IA are useful to extract, filter and classify datas<ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref><ref>{{Cite web|url=https://arxiv.org/abs/2504.20276v1|title=Enhancing Systematic Reviews with Large Language Models: Using GPT-4 and Kimi|last=Kaptur|first=Dandan Chen|last2=Huang|first2=Yue|date=2025-04-28|website=arXiv.org|language=en|doi=10.48550/arXiv.2504.20276|access-date=2026-01-21|last3=Ji|first3=Xuejun Ryan|last4=Guo|first4=Yanhui|last5=Kaptur|first5=Bradley}}</ref>. [[Large language models]] (LLM) are "on the rise" (2025), but "not yet ready for use"<ref>{{Cite journal |last=Lieberum |first=Judith-Lisa |last2=Toews |first2=Markus |last3=Metzendorf |first3=Maria-Inti |last4=Heilmeyer |first4=Felix |last5=Siemens |first5=Waldemar |last6=Haverkamp |first6=Christian |last7=Böhringer |first7=Daniel |last8=Meerpohl |first8=Joerg J. |last9=Eisele-Metzger |first9=Angelika |date=2025-05 |title=Large language models for conducting systematic reviews: on the rise, but not yet ready for use—a scoping review |url=https://www.wikidata.org/wiki/Q134545593|journal=Journal of Clinical Epidemiology |language=en |volume=181 |pages=111746 |doi=10.1016/j.jclinepi.2025.111746}}</ref>. The living review method relevant for just transition because it includes topic such as energy democracy which necessitate transdisciplinarity and consolidation of fragmented literature<ref>{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research & Social Science|volume=86|doi=10.1016/J.ERSS.2021.102444}}</ref>. === Definitions of just transition : === * «a fair and equitable process of moving towards a post-carbon society’. »<ref name=":0">{{Cite journal|last=McCauley|first=Darren|last2=Heffron|first2=Raphael|date=2018-08-01|title=Just transition: Integrating climate, energy and environmental justice|url=https://www.wikidata.org/wiki/Q129947262|journal=Energy Policy|language=English|volume=119|pages=1–7|doi=10.1016/J.ENPOL.2018.04.014}}</ref>. The concept of just transition originated from global trade unions in the 1980s to promote green jobs creation as a key element of sustainability transitions<ref name=":0" />. However, scholars have broadened the use of this term to develop frameworks for analysing issues of fairness in these transitions<ref name=":0" />. The concept of just transition can be used to bridge various bodies of scholarship : climate justice, environmental justiceand energy justice<ref name=":3">{{Cite journal|last=Wang|first=Xinxin|last2=Lo|first2=Kevin|date=2021-12-01|title=Just transition: A conceptual review|url=https://www.wikidata.org/wiki/Q137209041|journal=Energy Research & Social Science|volume=82|pages=102291|doi=10.1016/J.ERSS.2021.102291}}</ref><ref name=":1">{{Cite book|url=https://www.wikidata.org/wiki/Q134545572|title=What is the “Just Transition”?|last=Heffron|first=Raphael J.|date=2021-01-01|pages=9–19|language=English}}</ref> and take into account various aspects of justice including distributional justice, procedural justice, restorative justice, recognition justice<ref name=":0" /><ref name=":3" /><ref name=":1" /><ref name=":4">{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. === Definition of Procedural justice === Procedural justice is about the fairness of decision-making processes related to transitions<ref name=":4" /> such as the inclusion of those impacted by these decisions<ref name=":5">{{Cite journal|last=Stark|first=Anthony|last2=Gale|first2=Fred|last3=Murphy-Gregory|first3=Hannah|date=2023-05-05|title=Just Transitions’ Meanings: A Systematic Review|url=https://www.wikidata.org/wiki/Q137210229|journal=Society and Natural Resources|volume=36|issue=10|pages=1277–1297|doi=10.1080/08941920.2023.2207166}}</ref>. Procedural justice can include issues of community and citizen participation in decision making, their political representation their consultation or the integration of their knowledge, with a focus on neglected population (indigenous people, women, gender and ethnic minorities<ref>{{Cite journal|last=Jenkins|first=Kirsten|last2=McCauley|first2=Darren|last3=Heffron|first3=Raphael|last4=Stephan|first4=Hannes|last5=Rehner|first5=Robert|date=2016-01-01|title=Energy justice: A conceptual review|url=https://www.wikidata.org/wiki/Q137210566|journal=Energy Research & Social Science|volume=11|pages=174–182|doi=10.1016/J.ERSS.2015.10.004}}</ref>. For example, the participation of affected communities in decisions related to the construction of new infrastructures<ref name=":0" />. == Methodology == === Wikidata and the semantic web ===<!-- Add introduction to what wikidata is and how the triplet works in a pedagogical manner Example of good description here : https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k#fig1 --> "A knowledge graph is a structured representation of knowledge that captures information in a machine-readable format.<ref name=":9">{{Cite journal|last=Hogan|first=Aidan|last2=Blomqvist|first2=Eva|last3=Cochez|first3=Michael|last4=D’amato|first4=Claudia|last5=Melo|first5=Gerard De|last6=Gutierrez|first6=Claudio|last7=Kirrane|first7=Sabrina|last8=Gayo|first8=José Emilio Labra|last9=Navigli|first9=Roberto|date=2022-05-31|title=Knowledge Graphs|url=https://dl.acm.org/doi/10.1145/3447772|journal=ACM Computing Surveys|language=en|volume=54|issue=4|pages=1–37|doi=10.1145/3447772|issn=0360-0300}}</ref> A knowledge graph consists of a graph or network of interconnected data points, where each data point represents a piece of information or a concept, and the relationships between them are explicitly defined. Knowledge graphs organize and store data in a format that facilitates information retrieval, data analysis, and reasoning."<ref>{{Cite journal|last=Meijer|first=David|last2=Beniddir|first2=Mehdi A.|last3=Coley|first3=Connor W.|last4=Mejri|first4=Yassine M.|last5=Öztürk|first5=Meltem|last6=Hooft|first6=Justin J. J. van der|last7=Medema|first7=Marnix H.|last8=Skiredj|first8=Adam|date=2025-04-16|title=Empowering natural product science with AI: leveraging multimodal data and knowledge graphs|url=https://pubs.rsc.org/en/content/articlelanding/2025/np/d4np00008k|journal=Natural Product Reports|language=en|volume=42|issue=4|pages=654–662|doi=10.1039/D4NP00008K|issn=1460-4752}}</ref> == Building a corpus and enriching bibliographic metadata == === Database search === We conducted preliminary searches in various databases including Web of science, Go Triple, Dimensions and OpenAlex. Web of Science was the database offering the most relevant restults and included the possibility to filter results to display only litterature reviews. Articles metadata were exported (in .ris format) and then imported into the reference manager software Zotero. {| class="wikitable" |+ !Keywords search !Database !Search date !Filters !Number of results |- |(((TS=(procedural justice OR procedural fairness OR democracy OR participation OR participatory)) AND TS=(sustainability OR energy OR climate)) AND TS=(transition OR transitions)) AND TS=(review OR reviews) |Web of Science (all databases, all dates) |December 2025 |Document type: Review Article |362 |} === Article screening === Articles abstract were then screened and we selected only articles which were litterature reviews focusing on concepts related to procedural justice as their main topics. We excluded article which were * Not related to sustainability transition (e.g. sustainable shift in..., hard science papers...) * Not literature reviews (e.g. review of policies, initiatives, cases, review notes, book review...) * Not related to procedural justice but to participation into markets, participation in eco-friendly behaviors or included justice consideration only in “future research” suggestions * Discussing participatory research methodologies (e.g. participatory modelling) without approaching it as an issue of justice, power or democracy * Discussing procedural justice concepts as key variables or key results without it being the main focus of the paper The files resulting from this step are available at : https://doi.org/10.5281/zenodo.20749973 === Importing selected articles into Wikidata === To import the selected articles meta-data into Wikidata, we first ran [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 a script] to check if any article was already present in the database. Next we used [https://gist.github.com/zuphilip/90acdc3eac4109830db1b3ab855fcb24 another script] that checks the ISSN of the publication in Wikidata and add P-Q-pairs in the extra field of Zotero. Then we exported the articles data using the "export to Wikidata QuickStatements" function of Zotero and use the QuickStatements tool to add them to Wikidata. Next we used the [[wikidata:Wikidata:Zotero/Cita|Cita]] (V1.0.0-beta.17) Zotero add-on to add articles QID in Zotero. At this point we identified that duplicates had been created in Wikidata (possibly because the initial [https://gist.github.com/zuphilip/aa9f59271fcb0807fb20c7d0110d26e4 script] did not work that well because of the recent [[wikidata:Wikidata:SPARQL_query_service/WDQS_graph_split|Graph Split]] on Wikidata). We merged duplicates on wikidata using the [[wikidata:Help:Merge|"Merge" gadget]] on Wikidata. We checked manually for duplicated statments in those items. === Article classification through meta-data enrichement ===<!-- Add : What is meta-data enrichement --> Existing review try to classify existing articles according to various criteria such as industry focus, academic discipline, geography of research sites (countries), stakeholder focus (community, consumer, worker...), type of study (case study, theory development) or methodology (quantitative, qualitative, mixt).<ref name=":5" /> We selected the most relevant properties in Wikidata to reflect these classifications : {{Wikidata entity link|P921}} to describe what the article is about, {{Wikidata entity link|P8363}} to describe its main methodology/research design and {{Wikidata entity link|P6153}} to describe its geographical focus. ==== Main subjects ==== We first read the articles abstracts and listed relevant topics and their Wikidata ID in a shared spreadsheet. These topics were : {| class="wikitable" |+ !Qid !Main topic !Description |- |[[d:Q42377797|Q42377797]] |acceptability |characteristic of a thing being subject to acceptance for some purpose |- |[[d:Q2798912|Q2798912]] |accountability |concept of responsibility in ethics, governance and decision-making |- |[[d:Q421953|Q421953]] |actor–network theory |theory within social science |- |[[d:Q84459973|Q84459973]] |affordability | |- |[[d:Q185836|Q185836]] |age of a person |time elapsed since a person was born |- |[[d:Q4764988|Q4764988]] |animal studies |field in which animals are studied in a variety of cross-disciplinary ways |- |[[d:Q4338318|Q4338318]] |awareness |state or ability to perceive, to feel, or to be conscious of events, objects, or sensory patterns |- |[[d:Q4930066|Q4930066]] |blue carbon |carbon captured by the world's coastal ocean ecosystems |- |[[d:Q430460|Q430460]] |capability approach |economic theory |- |[[d:Q7569|Q7569]] |child |human between birth and puberty |- |[[d:Q4116870|Q4116870]] |civic engagement |individual or group activity addressing issues of public concern |- |[[d:Q125928|Q125928]] |climate change |human-caused changes to climate on Earth |- |[[d:Q260607|Q260607]] |climate change adaptation |process of adjustment to actual or expected climate change and its effects, seeking to moderate or avoid harm or exploit beneficial opportunities |- |[[d:Q1291678|Q1291678]] |climate justice |term linking the climate crisis with environmental and social justice |- |[[d:Q2270945|Q2270945]] |co-creation |product or service design process in which input from consumers plays a central role |- |[[d:Q16972712|Q16972712]] |co-design |approach to design attempting to actively involve all stakeholders |- |[[d:Q16324410|Q16324410]] |coproduction |product or service design process in which input from consumers plays a central role |- |[[d:Q11024|Q11024]] |communication |act of conveying intended meaning |- |[[d:Q177634|Q177634]] |community |social unit of human organisms who share common values |- |[[d:Q5154673|Q5154673]] |community choice aggregation |alternative energy supply system |- |[[d:Q113514984|Q113514984]] |community energy |delivery of community-led renewable energy, energy demand reduction and energy supply projects |- |[[d:Q65807646|Q65807646]] |community participation |The taking part by members of a community in decisionmaking processes related to the development of their community |- |[[d:Q188843|Q188843]] |cosmopolitanism |ideology that all human beings belong to a single community, based on a shared morality |- |[[d:Q11693783|Q11693783]] |decarbonization |change of economy, especially of energy industries, towards lower carbon dioxide emissions |- |[[d:Q284289|Q284289]] |deliberative democracy |form of democracy focusing on consensus |- |[[d:Q7174|Q7174]] |democracy |form of government |- |[[d:Q552284|Q552284]] |distributive justice |concept of the socially just allocation of goods |- |[[d:Q1230584|Q1230584]] |diversity |concept in sociology and political studies |- |[[d:Q1049066|Q1049066]] |ecological economics |research field on the interdependence of human economies and natural ecosystems |- |[[d:Q8134|Q8134]] |economics |social science that studies the production, distribution, and consumption of goods and services |- |[[d:Q868575|Q868575]] |empowerment |providing increased autonomy |- |[[d:Q295865|Q295865]] |ecosystem service |benefits created by nature, forests and environmental systems |- |[[d:Q138359220|Q138359220]] |energy citizenship |involvement of citizens in energy-related decisions |- |[https://www.wikidata.org/w/index.php?title=Q131444737&redirect=no Q131444737] |community energy |[redirection] |- |[[d:Q16869822|Q16869822]] |energy consumption |amount of energy or power used |- |[[d:Q1358789|Q1358789]] |senior |elderly person |- |[[d:Q14944319|Q14944319]] |energy democracy |concept in environmental justice movement |- |[[d:Q192704|Q192704]] |energy efficiency |ratio between the useful energy output and the input of a machine |- |[[d:Q24965464|Q24965464]] |energy modeling |process of building computer models of energy systems in order to analyze them |- |[[d:Q1805337|Q1805337]] |energy policy |policy addressing energy issues |- |[[d:Q1341244|Q1341244]] |energy poverty |lack of access to modern energy services |- |[[d:Q3406659|Q3406659]] |energy production |conversion of energy from a primary source into a form useful to humans |- |[[d:Q117091181|Q117091181]] |energy justice |subconcept of economic equality |- |[[d:Q3456219|Q3456219]] |energy renovation |building works aimed at reducing energy consumption and decarbonising the energy sources used |- |[[d:Q2700433|Q2700433]] |energy security |national security considerations of energy availability |- |[[d:Q837718|Q837718]] |energy storage |capture of energy produced at one time for use at a later time |- |[[d:Q795757|Q795757]] |energy transition |long-term structural change towards sustainable energy systems |- |[[d:Q1479527|Q1479527]] |environmental justice |system of fairness |- |[[d:Q771773|Q771773]] |fairness |concept in sociology and generally the interaction of society |- |[[d:Q56395513|Q56395513]] |farming system |method of agricultural production defined by its physical practices and economic characteristics |- |[[d:Q5465532|Q5465532]] |food system |all processes and infrastructure involved in feeding a population |- |[[d:Q4421|Q4421]] |forest |dense collection of trees covering a relatively large area |- |[[d:Q48277|Q48277]] |gender |social concept which distinguish the different gender categories |- |[[d:Q1553864|Q1553864]] |governance |all of the processes of governing, whether undertaken by a government, market or network, whether over a family, tribe, formal or informal organization or territory and whether through the laws, norms, power or language of an organized society |- |[[d:Q8458|Q8458]] |human rights |inalienable fundamental rights to which a person is inherently entitled |- |[[d:Q11376059|Q11376059]] |human rights violation |act or omission which contravene the principles of human rights |- |[[d:Q103817|Q103817]] |indigenous people |first inhabitants of an area and their descendants |- |[[d:Q113561794|Q113561794]] |indigenous science |indigenous knowledge applied to the scientific method |- |[[d:Q770480|Q770480]] |injustice |quality relating to unfairness or undeserved outcomes |- |[[d:Q17142211|Q17142211]] |interactional justice |the perceived appropriateness of interpersonal treatment |- |[[d:Q1516555|Q1516555]] |intersectionnality |theoretical framework of multidimensional oppression |- |[[d:Q6316391|Q6316391]] |just transition |Framework developed by the trade union movement to encompass wide range of social interventions needed to secure decent work opportunities and a greener economy. |- |[[d:Q366139|Q366139]] |legitimation |the process of making something acceptable and normative to a group |- |[[d:Q3027857|Q3027857]] |living lab |user-centered, open innovation ecosystem integrating research and innovation in real life communities |- |[[d:Q59679511|Q59679511]] |low income |home with little money |- |[[d:Q43619|Q43619]] |natural environment |all living and non-living things occurring naturally on Earth or some region thereof |- |[[d:Q127514833|Q127514833]] |nature-positive |global goal to halt and reverse nature loss by 2030 |- |[[d:Q13023682|Q13023682]] |non-human |organism not in the genus Homo |- |[[d:Q728646|Q728646]] |partnership |arrangement in which parties agree to cooperate to advance their mutual interests |- |[[d:Q3907287|Q3907287]] |policy making |the act of developing policy |- |[[d:Q9357091|Q9357091]] |political theory |class of theory |- |[[d:Q265425|Q265425]] |postcolonialism |academic discipline |- |[[d:Q25107|Q25107]] |power |ability to influence the behavior of others |- |[[d:Q442100|Q442100]] |procedural justice |fairness in the processes that resolve disputes and allocate resources |- |[[d:Q7249406|Q7249406]] |project governance |management framework |- |[[d:Q7257735|Q7257735]] |public engagement |Policy-making practice |- |[[d:Q541936|Q541936]] |public participation |participation of citizens in various policy decisions and planning processes |- |[[d:Q6142016|Q6142016]] |recognition justice |social philosophy theory |- |[[d:Q10509953|Q10509953]] |renewable electricity |electricity from renweable sources |- |[[d:Q12705|Q12705]] |renewable energy |energy collected from renewable resources |- |[[d:Q56510941|Q56510941]] |renewable energy policy | |- |[[d:Q1165392|Q1165392]] |restorative justice |approach to justice where victims and perpetrators mediate a restitution agreement |- |[[d:Q4414036|Q4414036]] |rural population |inhabitants of rural areas or of small towns classified as rural |- |[[d:Q17152351|Q17152351]] |smart system |adaptive intelligent systems |- |[[d:Q187588|Q187588]] |social class |group of people categorized in a hierarchy based on socioeconomic factors |- |[[d:Q264892|Q264892]] |social justice |concept that discrimination recognized in society should be remedied |- |[[d:Q34749|Q34749]] |social science |academic disciplines concerned with society and the relationships between individuals in society |- |[[d:Q2930198|Q2930198]] |stakeholder participation |involvement of groups or individuals affected by the actions of an entity |- |[[d:Q125359881|Q125359881]] |sustainability transition | |- |[[d:Q219416|Q219416]] |sustainability |ability of human civilization to coexist with the biosphere in a steady state |- |[[d:Q131201|Q131201]] |sustainable development |mode of human development that meets current demands without compromising the needs of future generations |- |[[d:Q7649586|Q7649586]] |Sustainable Development Goals |set of United Nations-defined global development goals and climate change |- |[[d:Q69883|Q69883]] |urban planning |technical and political process concerned with the use of land and design of the urban environment |- |[[d:Q920600|Q920600]] |urban renewal |program of land redevelopment in cities, often where there is urban decay |- |[[d:Q3376054|Q3376054]] |vulnerable population |group of persons whose range of options is severely limited, are subjected to coercion, or who may be compromised in their ability to give informed consent |- |[[d:Q107389921|Q107389921]] |water-management | |- |[[d:Q7981051|Q7981051]] |well-being |measure of how well life is to someone or a group with factors such as health, happiness and satisfaction |- |[[d:Q467|Q467]] |woman |female adult human |- |[[d:Q188867|Q188867]] |future studies |study of possible, probable, and preferable social, technological and political futures |- |[[d:Q1038171|Q1038171]] |participatory design |active involvement of all stakeholders in the design process |} <!-- include all below items using the wikidata link template --> Then, for each article, we inferred what the {{Wikidata entity link|P921}} was from the abstracts and author provided keywords. ==== Study types ==== Our review included only litterature reviews. We first read abstracts to identify all the [https://angryloki.github.io/wikidata-graph-builder/?item=Q2412849&property=P279&mode=reverse different types of litterature reviews] present in the corpus and created wikidata items which did not exist, for example {{Wikidata entity link|Q137209848}} and {{Wikidata entity link|Q137174203}}. We improved these method items using the methodological references cited in the reviewed papers. The types of reviews were : {| class="wikitable" |+ !Qid !Study type !Description |- |[[d:Q603441|Q603441]] |bibliometrics |statistical analysis of written publications, such as books or articles |- |[[d:Q472342|Q472342]] |scientometrics |study of measuring and analysing science, technology and innovation |- |[[d:Q815382|Q815382]] |meta-analysis |statistical method that summarizes data from multiple sources |- |[[d:Q1504425|Q1504425]] |systematic review |publication type, study that gathers, analyzes, and communicates the results of research and information on a topic |- |[[d:Q2412849|Q2412849]] |literature review |process of information search and text of a review article (Q7318358), which includes the current knowledge including substantive findings, as well as theoretical and methodological contributions to a particular topic |- |[[d:Q6822263|Q6822263]] |meta-regression |statistical tool used in meta-analyses |- |[[d:Q7301211|Q7301211]] |realist evaluation |[...] |- |[[d:Q17007303|Q17007303]] |combinatorial meta-analysis |[...] |- |[[d:Q70470634|Q70470634]] |network meta-analysis |meta-analysis of randomized trials in which estimates of comparative treatment effects are visualized and interpreted from a network of interventions |- |[[d:Q101116078|Q101116078]] |scoping review |search for concepts by mapping the language and data which surrounds those concepts and adjusting the search method iteratively to synthesize evidence and assess the scope of an area of inquiry |- |[[d:Q110665014|Q110665014]] |narrative review |type of literature review, without structured method of retrieval and analysis |- |[[d:Q137174203|Q137174203]] |conceptual review |academic research aiming to review existing concepts and definitions in the litterature |- |[[d:Q137174450|Q137174450]] |critical review |type of literature review analysing strenghts, major contributions, mistakes and neglected issues in an academic field of research |- |[[d:Q137209848|Q137209848]] |integrative literature review |type of literature review |- |[[d:Q110665014|Q137211242]] |narrative review |type of literature review, without structured method of retrieval and analysis |}Then, we added the {{Wikidata entity link|P8363}} of each articles based on the abstract and method sections. In case of doubt, we compared our interpretation. ==== Research site ==== When an article had a specific geographical focus, we used the property {{Wikidata entity link|P6153}} to describe it. For example, the article "{{Wikidata entity link|Q137901202}}" focused on {{Wikidata entity link|Q132959}}. ==== Authors ==== We used the [https://author-disambiguator.toolforge.org/ Author Disambiguator] tool to create Wikidata items for researchers who did not yet have one. This tool helps to minimise errors caused by homonyms among researchers: following a query, it categorises scientific publications into thematic groups. It also automatically searches for [[d:Wikidata:ORCIDator|ORCID]], ResearchGate and VIAF pages. ==== Results ==== The table listing all the papers in the sample can be visualized [https://tabernacle.toolforge.org/?#/tab/manual/Q137211155%0A%0A%0A%0A%0A%0A%0AQ114306483%0A%0A%0A%0A%0AQ137901181%0A%0A%0A%0AQ137901182%0A%0A%0A%0A%0A%0A%0A%0AQ137901183%0A%0A%0AQ114306476%0A%0A%0A%0A%0AQ137901184%0A%0A%0A%0A%0AQ137901185%0A%0A%0A%0A%0A%0AQ137901186%0A%0A%0A%0A%0A%0A%0AQ137901187%0A%0A%0A%0A%0A%0A%0AQ137901188%0A%0A%0A%0A%0AQ137210566%0A%0A%0A%0A%0AQ114306511%0A%0A%0A%0A%0A%0AQ137901191%0A%0A%0A%0A%0AQ137901192%0A%0A%0A%0A%0AQ137901193%0A%0A%0A%0A%0AQ135979013%0A%0A%0A%0A%0A%0A%0A%0AQ137901195%0A%0A%0A%0A%0A%0AQ137901196%0A%0A%0A%0A%0A%0A%0AQ137901197%0A%0A%0A%0A%0AQ136447761%0A%0A%0A%0AQ137901199%0A%0A%0A%0A%0A%0A%0AQ129652515%0A%0A%0A%0A%0A%0A%0AQ137901201%0A%0A%0A%0A%0A%0AQ137901202%0A%0A%0A%0A%0AQ137901203%0A%0A%0A%0AQ137901204%0A%0A%0A%0A%0A%0A%0A%0AQ137901205%0A%0A%0A%0A%0AQ137901206%0A%0A%0A%0A%0A%0A%0A%0A%0AQ137901207%0A%0A%0A%0A%0AQ129203992%0A%0A%0A%0A%0A%0A%0AQ114197507%0A%0A%0A%0AQ137901161%0A%0A%0A%0A%0A%0A%0A%0AQ137901209%0A%0A%0A%0A%0A%0AQ137901210%0A%0A%0A%0A%0A%0AQ137901211%0A%0A%0A%0A%0AQ11420462%0A%0AQ137901213%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0A%0AQ104887325%0A%0A%0A%0A%0A%0AQ137901162%0A%0A%0AQ137901163%0A%0A%0A%0A%0AQ137901164%0A%0A%0A%0A%0A%0AQ137901215%0A%0A%0A%0A%0AQ137901216%0A%0A%0A%0A%0A%0A%0A%0A%0AQ137901217%0A%0A%0A%0A%0AQ115448818%0A%0A%0A%0A%0AQ137901218%0A%0A%0A%0AQ137901219%0A%0A%0A%0A%0AQ137901220%0A%0A%0A%0A%0A%0AQ137901221%0A%0A%0A%0A%0A%0AQ137901222%0A%0A%0A%0A%0AQ137901223%0A%0A%0AQ137901224%0A%0A%0A%0AQ137901225%0A%0A%0A%0A%0A%0A%0AQ137901226%0A%0A%0A%0AQ137901227%0A%0A%0AQ137901182/Len%3BP921%3BP6153%3BP8363%3BP50 here] (be careful if you are logged into Wikidata as the table is editable). == Modelling knowledge == Concept maps can be a powerful literature review tool<ref>{{Cite journal|last=Lewis|first=John Kennedy|date=2016|title=Using ATLAS.ti to Facilitate Data Analysis for a Systematic Review of Leadership Competencies in the Completion of a Doctoral Dissertation|url=https://www.ssrn.com/abstract=2850726|journal=SSRN Electronic Journal|language=en|doi=10.2139/ssrn.2850726|issn=1556-5068}}</ref> allowing to synthetize theoretical statements about relationship between concepts<ref>{{Cite journal|last=Panniers|first=Teresa L|last2=Feuerbach|first2=Renee Daiuta|last3=Soeken|first3=Karen L|date=2003-08-01|title=Methods in informatics: using data derived from a systematic review of health care texts to develop a concept map for use in the neonatal intensive care setting|url=https://www.sciencedirect.com/science/article/pii/S1532046403000911|journal=Journal of Biomedical Informatics|series=Building Nursing Knowledge through Informatics: From Concept Representation to Data Mining|volume=36|issue=4|pages=232–239|doi=10.1016/j.jbi.2003.09.010|issn=1532-0464}}</ref>. In the present study, we explored how concept map can be used to model the knowledge present in the paper we selected. [define knowledge modelling] ==== Wikidata ontology ==== Wikidata "supports multiple coexisting classification" and allow multiple ontological frameworks to coexist.<ref name=":8">{{Cite web|url=https://arxiv.org/abs/2512.12260v1|title=A Multi-Axial Mindset for Ontology Design Lessons from Wikidata's Polyhierarchical Structure|last=Doğan|first=Ege Atacan|last2=Patel-Schneider|first2=Peter F.|date=2025-12-13|website=arXiv.org|language=en|access-date=2026-05-26}}</ref> It also supports epistemic pluralism : different worldviews can be represented in wikidata, even though scientific knowledge is preferred.<ref name=":8" /> See more on membership properties : https://www.wikidata.org/wiki/Help:Basic_membership_properties See the discussion on cause modelling : https://www.wikidata.org/wiki/Help:Modeling_causes/en ==== Conceptual modelling ==== We first reflected on what kind of wikidata properties could be used to represent concepts and theories in wikidata. Capturing the content of a concept is not straightforward and there are various approaches coming from psychology and philosophy on the matter<ref>{{Cite book|title=The Origin of Concepts|last=Carey|first=Susan|date=2011|publisher=Oxford University Press USA - OSO|isbn=978-0-19-536763-8|series=Oxford Series in Cognitive Development Ser|location=Cary}}</ref> we summarize these approaches below and examine which wikidata properties exist to represent them. * Definition: the content of a concept can be formed by its decomposition into other concepts. Many Wikidata properties can be relevant to model definitions, for example: {{Wikidata entity link|P1269}}, {{Wikidata entity link|P361}}/{{Wikidata entity link|P527}}, {{Wikidata entity link|P2670}}, {{Wikidata entity link|P1552}}/{{Wikidata entity link|P6477}}, {{Wikidata entity link|P3712}}... * Categorization: the content of a concept is formed by its illustration by an exemplar (a [[wikipedia:Prototype_theory|prototype]]) that best represent the concept. Apart from the inclusion of images to illustrate an item, Wikidata structure do not highlight exemplars. However, properties signifying relations of categorizations are among the most used with {{Wikidata entity link|P31}} and {{Wikidata entity link|P279}}. * Theory: the content of a concept is formed by its role in providing explanation of the world. Wikidata includes several properties to describe causal relationships: {{Wikidata entity link|P828}}/{{Wikidata entity link|P1542}}, {{Wikidata entity link|P1537}}/{{Wikidata entity link|P1479}}. * Essence: the content of a concept is "something" deep explaning the entity's existence and its properties. We can use concepts before knowing what they mean, and this is what allows us to revise our knowledge about it. The idea of essence is well represented by the QID of Wikidata entities: it is independent of language and definitions and we can create it before really knowing what all its properties will be. * Origin: the content of the concept is determined causally by social and historial factors (e.g. someone inventing the concept and introducing its use in a language community). This can be represented by the property {{Wikidata entity link|P3938}}. ==== Categorization and conceptualisation practices in management sciences ==== In management sciences « systematic categorizing is the best and perhaps only method for clearing up semantic confusion, management scholars never take the classical approaches to categorizing that facilitated tremendous progress in the physical sciences, and seldomly build on extant categorial schemes. »<ref>{{Cite journal|last=Pierce|first=Jason R.|date=2025-01|title=Categorizing Concepts and Phenomena in Management Research: A Four-Phase Integrative Review and Recommendations|url=http://journals.aom.org/doi/full/10.5465/annals.2023.0052|journal=Academy of Management Annals|language=en|volume=19|issue=1|page=28|pages=9–37|doi=10.5465/annals.2023.0052|issn=1941-6520}}</ref>. Some scholars discussed how conceptualization should be done<ref>{{Cite journal|last=Podsakoff|first=Philip M.|last2=MacKenzie|first2=Scott B.|last3=Podsakoff|first3=Nathan P.|date=2016-04|title=Recommendations for Creating Better Concept Definitions in the Organizational, Behavioral, and Social Sciences|url=https://journals.sagepub.com/doi/10.1177/1094428115624965|journal=Organizational Research Methods|language=en|volume=19|issue=2|pages=159–203|doi=10.1177/1094428115624965|issn=1094-4281}}</ref>,<ref>{{Cite journal|last=Makowski|first=Piotr Tomasz|date=2021-10|title=Optimizing Concepts: Conceptual Engineering in the Field of Management—The Case of Routines Research|url=http://journals.aom.org/doi/full/10.5465/amr.2019.0252|journal=Academy of Management Review|language=en|volume=46|issue=4|pages=702–724|doi=10.5465/amr.2019.0252|issn=0363-7425}}</ref>. ==== Thematic networks ==== [[File:Thematic network example.jpg|thumb|547x547px|Structure of a thematic network (Source: Attride-Stirling 2001)]] A thematic network is “simply a way of organizing a thematic analysis of qualitative data”<ref name=":7">{{Cite journal|last=Attride-Stirling|first=Jennifer|date=2001-12|title=Thematic networks: an analytic tool for qualitative research|url=https://journals.sagepub.com/doi/10.1177/146879410100100307|journal=Qualitative Research|language=en|volume=1|issue=3|pages=385–405|doi=10.1177/146879410100100307|issn=1468-7941}}</ref>. It is compatible with classical coding strategies such as [[grounded theory]]<ref>{{Cite journal|last=Corbin|first=Juliet|last2=Strauss|first2=Anselm|date=1990-12-01|title=Grounded Theory Research: Procedures, Canons and Evaluative Criteria|url=https://www.degruyter.com/document/doi/10.1515/zfsoz-1990-0602/html|journal=Zeitschrift für Soziologie|language=en|volume=19|issue=6|pages=418–427|doi=10.1515/zfsoz-1990-0602|issn=2366-0325}}</ref>. Thematic networks can be used to visualise the data structure after identifying themes and help structure and interpret the data<ref name=":7" />. The principle is to assemble basic themes into more general themes. Qualitative researchers usually use {{Wikidata entity link|Q4550939}} and qualitative coding (e.g. grounded theory) to identify themes and sub-themes. However, the nature of the relationship between these various themes and sub-themes is often not specified. * ==== Causal networks ==== The use of diagrams to represent causal relationship exist in various research practices. In statistics, researchers sometime present models with boxes and arrows representing correlations and/or causations<ref>{{Cite book|url=https://mirror.vcu.edu/pub/mx/doc/mxmang10.pdf|title=Statistical Modeling|last=Neale|first=Michael C.|last2=Boker|first2=Steven M.|last3=Xie|first3=Gary|last4=Maes|first4=Hermine H.|publisher=Richmond, VA: Department of Psychiatry|year=1999|location=Virginia Commonwealth University}}</ref>. In qualitative research, building grounded theory models is about "[accounting] for not only all the major emergent concepts, themes, and dimensions, but also for their dynamic interrelationships. Speaking in classic boxes-and-arrows terms, this process amounts to assembling the constellation of boxes with a special focus on the arrows."<ref>{{Cite journal|last=Gioia|first=Dennis A.|last2=Corley|first2=Kevin G.|last3=Hamilton|first3=Aimee L.|date=2013-01|title=Seeking Qualitative Rigor in Inductive Research: Notes on the Gioia Methodology|url=https://journals.sagepub.com/doi/10.1177/1094428112452151|journal=Organizational Research Methods|language=en|volume=16|issue=1|pages=15–31|doi=10.1177/1094428112452151|issn=1094-4281}}</ref> Researchers relying on system theory also use causal loop diagram where boxes represent variables and arrows represent causal influence (positive or negative), causal relationship can "feedback" (two variables can influence each other)<ref>{{Cite book|url=https://link.springer.com/10.1007/978-3-031-01919-7_4|title=Causal Loop Diagrams|last=Barbrook-Johnson|first=Pete|last2=Penn|first2=Alexandra S.|date=2022|publisher=Springer International Publishing|isbn=978-3-031-01833-6|location=Cham|pages=47–59|language=en|doi=10.1007/978-3-031-01919-7_4}}</ref>. Wikidata includes several properties to describe causal relationships: * {{Wikidata entity link|P828}} * {{Wikidata entity link|P1542}} * {{Wikidata entity link|P1537}} * {{Wikidata entity link|P1479}} : it is difficult to identify single causes for social phenomenons, many factors having an effect on the subject item will likely be contributing factors == Testing concept modelling on {{Wikidata entity link|Q14944319}} == We started by experimenting the modelling of concept by focusing on the concept of {{Wikidata entity link|Q14944319}}. We selected a subset of papers which had energy democracy as main topic : * {{Wikidata entity link|Q137901202}} * {{Wikidata entity link|Q137901196}} * {{Wikidata entity link|Q137901182}} * {{Wikidata entity link|Q136447761}} * {{Wikidata entity link|Q129652515}} * {{Wikidata entity link|Q114306483}} We read each paper and used them as source to enter statements in the item {{Wikidata entity link|Q14944319}}. For example, "Energy democracy is both an ideal and a process"<ref>{{Cite journal|last=Droubi|first=Sufyan|last2=Heffron|first2=Raphael|last3=McCauley|first3=Darren|date=2022-04-01|title=A critical review of energy democracy: A failure to deliver justice?|url=https://www.wikidata.org/wiki/Q137901182|journal=Energy Research & Social Science|volume=86|pages=4|doi=10.1016/J.ERSS.2021.102444}}</ref>, we thus entered the wikidata statement {{Wikidata entity link|Q14944319}} is an {{Wikidata entity link|P31}} {{Wikidata entity link|Q840396}}, using the paper as source. The result of this first step is visible in the archival version of the item (22 May 2026) here https://www.wikidata.org/w/index.php?title=Q14944319&oldid=2495982191. Ontology challenges: *{{Wikidata entity link|P31}}: concepts may have a dual nature because they designate at the same time an idea and the entity that this idea represent. Energy democracy is a concept, an ideal, a process and an outcome. *'''Process versus outcome :''' For material processes, the distinction between process and outcome is rather simple. For example, in Wikidata, {{Wikidata entity link|Q11629}} (practice of applying paint) is different from {{Wikidata entity link|Q3305213}} (visual artwork), and this distinction is based on the criterion "{{Wikidata entity link|Q127270577}}". However, this distinction is less straightforward for social processes that do not have an end. Such processes are ongoing and outcomes cannot be separated as clearly. * '''Ideal versus reality :''' Concepts do not have goals in themselves, but the reality they represent can have goals. To distinguish goals from the process to reach it, we used {{Wikidata entity link|P3712}} to describe ideals and {{Wikidata entity link|P2670}} to describe processes. * '''Phenomenon versus theory :''' Wikidata current items are not really suited to model "meta-research" statements. For example, modelling the idea tha the literature on energy democracy is fragmented would require creating an item representing the energy democracy literature, not just energy democracy in general. Similarly, it can be difficult to model the chronological evolution of the definition of an idea (although it could be technically possible). It is hard to represent in Wikidata affirmations related to missing knowlege, propositions of untested hypothesis, critique of existing research or research agenda recommandations * '''Origin of discourses versus origin of practices :''' To distinguish the causes of the concepts/discourses and the causes of the phenomenon itself, we used {{Wikidata entity link|P3938}} to indicate the origins of the concept or the movements promoting it. Some of the statements we added may seem contradictory. However, Wikidata supports "because statements essentially point to referenceable sources of information and different sources may provide contradicting information, it's possible to represent a plurality of perspectives on Wikidata"<ref>{{Cite web|url=https://www.wikidata.org/wiki/Help:Statements#Plurality_and_consensus|title=Help:Statements - Wikidata|website=www.wikidata.org|language=en|access-date=2026-06-08}}</ref>. The {{Wikidata entity link|Q14944319}} concept could be split into more precise concepts to distinguish the social movement advocating for it, the political concept theorizing it and the concrete initiatives implementing it. However, the current sources do not make this distinction for now. Other challenges * Wikidata does not seem to be the best tool to model quantitative statements, for example, the paper {{Wikidata entity link|Q137901196}} states that "9.8% of the final energy consumed in developing countries comes from modern renewable energy sources". Including energy data in Wikidata require using or creating specific properties (e.g. {{Wikidata entity link|P6826}}) * When concepts are not precisely defined, statements cannot be modelled correctly. For example, in the sentence "management of social affairs by voluntary and self-governing associations is deemed to ensure that both citizen choice and public welfare are best served"<ref>{{Cite journal|last=Veelen|first=Bregje van|last2=Horst|first2=Dan van der|date=2018-12-01|title=What is energy democracy? Connecting social science energy research and political theory|url=https://www.wikidata.org/wiki/Q129652515|journal=Energy Research & Social Science|language=English|volume=46|pages=19–28|doi=10.1016/J.ERSS.2018.06.010}}</ref>, "choice" could refer to {{Wikidata entity link|Q111986453}}, {{Wikidata entity link|Q1331926}}, or {{Wikidata entity link|Q12888920}} as "choice" can refer to the availability of different options, or the decision process to chose among them. Advantages : * Link toward unique identifiers for concepts, but also laws (e.g. {{Wikidata entity link|Q139764294}}) == Interactions with the Wikidata community == * Some Wikidata contributors added labels for {{Wikidata entity link|Q14944319}} in other languages such as Armenian or Slovenian. == Data visualisation == === Filter statements === * Visualize only statements using a specitic source. Example : https://w.wiki/PFqH * Visualize only items which are part to the present project (require that all items of the project include the statement {{Wikidata entity link|P6104}} {{Wikidata entity link|Q134545539}}). === Mapping a concept === Scholia request "topic in context" : 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Example with Energy democracy] === Mapping sources consensus === Visualise graphs and use the number of references to determine edge thickness/weight. == Writing == To cite articles we used the [[Template:Cite Q|Cite Q template.]] Each reference is an item in Wikidata and the template retrieve the necessary data to generate the citation references below. == Future research == The analysis of knowledge graph could in theory allow to make logical deduction to generate new data<ref name=":9" />. Reflect on the future of scholarly communication : https://hal.science/hal-03277615/file/OPERAS_Future_of_Scholarly_Communication_06.2021.pdf == Data == {| class="wikitable sortable" ! QID !! Year !! DOI !! Title |- | [[d:Q137901191|Q137901191]] || 2025 || [https://doi.org/10.1002/GEO2.70040 10.1002/GEO2.70040] || Place-Based Sustainability Transformations for Just Futures: A Systematic Review |- | [[d:Q137901187|Q137901187]] || 2025 || [https://doi.org/10.1002/WCC.932 10.1002/WCC.932] || Public Communication of Climate and Justice: A Scoping Review |- | [[d:Q135979013|Q135979013]] || 2025 || [https://doi.org/10.1007/S13280-025-02202-Z 10.1007/S13280-025-02202-Z] || Participatory approaches to climate adaptation, resilience, and mitigation: A systematic review |- | [[d:Q137901223|Q137901223]] || 2022 || [https://doi.org/10.1007/S13412-021-00726-W 10.1007/S13412-021-00726-W] || A review of stakeholder participation studies in renewable electricity and water: does the resource context matter? |- | [[d:Q137901184|Q137901184]] || 2021 || [https://doi.org/10.1007/S40518-021-00184-6 10.1007/S40518-021-00184-6] || Energy Storage as an Equity Asset. |- | [[d:Q114204627|Q114204627]] || 2021 || [https://doi.org/10.1007/S43621-021-00024-Z 10.1007/S43621-021-00024-Z] || Can public awareness, knowledge and engagement improve climate change adaptation policies? |- | [[d:Q137901209|Q137901209]] || 2026 || [https://doi.org/10.1016/J.AGSY.2025.104512 10.1016/J.AGSY.2025.104512] || Designing with non-humans for agricultural systems transformation: An interdisciplinary review and framework for reflection |- | [[d:Q137901201|Q137901201]] || 2025 || [https://doi.org/10.1016/J.COPSYC.2024.101987 10.1016/J.COPSYC.2024.101987] || Individual and community catalysts for Renewable Energy Communities (RECs) development |- | [[d:Q114197507|Q114197507]] || 2022 || [https://doi.org/10.1016/J.CRM.2022.100438 10.1016/J.CRM.2022.100438] || Advancements of sustainable development goals in co-production for climate change adaptation research |- | [[d:Q129203992|Q129203992]] || 2024 || [https://doi.org/10.1016/J.EGYR.2024.01.040 10.1016/J.EGYR.2024.01.040] || Empowering energy citizenship: Exploring dimensions and drivers in citizen engagement during the energy transition |- | [[d:Q137901216|Q137901216]] || 2026 || [https://doi.org/10.1016/J.EIAR.2025.108187 10.1016/J.EIAR.2025.108187] || From participation to partnership: A systematic review of public engagement in sustainable urban planning |- | [[d:Q137210566|Q137210566]] || 2016 || [https://doi.org/10.1016/J.ERSS.2015.10.004 10.1016/J.ERSS.2015.10.004] || Energy justice: A conceptual review |- | [[d:Q115448818|Q115448818]] || 2016 || [https://doi.org/10.1016/J.ERSS.2016.04.001 10.1016/J.ERSS.2016.04.001] || Stakeholder involvement in sustainability science—A critical view |- | [[d:Q129652515|Q129652515]] || 2018 || [https://doi.org/10.1016/J.ERSS.2018.06.010 10.1016/J.ERSS.2018.06.010] || What is energy democracy? Connecting social science energy research and political theory |- | [[d:Q137901196|Q137901196]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101716 10.1016/J.ERSS.2020.101716] || Of renewable energy, energy democracy, and sustainable development: A roadmap to accelerate the energy transition in developing countries |- | [[d:Q136447761|Q136447761]] || 2020 || [https://doi.org/10.1016/J.ERSS.2020.101768 10.1016/J.ERSS.2020.101768] || Energy democracy as a process, an outcome and a goal: A conceptual review |- | [[d:Q137901204|Q137901204]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101834 10.1016/J.ERSS.2020.101834] || Identities, innovation, and governance: A systematic review of co-creation in wind energy transitions |- | [[d:Q137901183|Q137901183]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101837 10.1016/J.ERSS.2020.101837] || Renewable energy for whom? A global systematic review of the environmental justice implications of renewable energy technologies |- | [[d:Q137901207|Q137901207]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101871 10.1016/J.ERSS.2020.101871] || Rethinking community empowerment in the energy transformation: A critical review of the definitions, drivers and outcomes |- | [[d:Q137901215|Q137901215]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101876 10.1016/J.ERSS.2020.101876] || Co-production in the wind energy sector: A systematic literature review of public engagement beyond invited stakeholder participation |- | [[d:Q114306511|Q114306511]] || 2021 || [https://doi.org/10.1016/J.ERSS.2020.101907 10.1016/J.ERSS.2020.101907] || From consultation toward co-production in science and policy: A critical systematic review of participatory climate and energy initiatives |- | [[d:Q137901221|Q137901221]] || 2021 || [https://doi.org/10.1016/J.ERSS.2021.102257 10.1016/J.ERSS.2021.102257] || The challenges of engaging island communities: Lessons on renewable energy from a review of 17 case studies |- | [[d:Q137901218|Q137901218]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102333 10.1016/J.ERSS.2021.102333] || The (in)justices of smart local energy systems: A systematic review, integrated framework, and future research agenda |- | [[d:Q137901182|Q137901182]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102444 10.1016/J.ERSS.2021.102444] || A critical review of energy democracy: A failure to deliver justice? |- | [[d:Q114306483|Q114306483]] || 2022 || [https://doi.org/10.1016/J.ERSS.2021.102482 10.1016/J.ERSS.2021.102482] || The role of energy democracy and energy citizenship for participatory energy transitions: A comprehensive review |- | [[d:Q114306476|Q114306476]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102714 10.1016/J.ERSS.2022.102714] || What about citizens? A literature review of citizen engagement in sustainability transitions research |- | [[d:Q137901193|Q137901193]] || 2022 || [https://doi.org/10.1016/J.ERSS.2022.102862 10.1016/J.ERSS.2022.102862] || When energy justice is contested: A systematic review of a decade of research on Sweden?s conflicted energy landscape |- | [[d:Q137901219|Q137901219]] || 2023 || [https://doi.org/10.1016/J.ERSS.2022.102913 10.1016/J.ERSS.2022.102913] || Can we optimise for justice? Reviewing the inclusion of energy justice in energy system optimisation models |- | [[d:Q137901186|Q137901186]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103010 10.1016/J.ERSS.2023.103010] || Analysing intersections of justice with energy transitions in India- A systematic literature review |- | [[d:Q137901181|Q137901181]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103053 10.1016/J.ERSS.2023.103053] || Fostering justice through engagement: A literature review of public engagement in energy transitions |- | [[d:Q137211155|Q137211155]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103213 10.1016/J.ERSS.2023.103213] || A fairway to fairness: Toward a richer conceptualization of fairness perceptions for just energy transitions |- | [[d:Q137901217|Q137901217]] || 2023 || [https://doi.org/10.1016/J.ERSS.2023.103221 10.1016/J.ERSS.2023.103221] || Powering just energy transitions: A review of the justice implications of community choice aggregation |- | [[d:Q137901199|Q137901199]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104016 10.1016/J.ERSS.2025.104016] || Making energy renovations equitable: A literature review of decision-making criteria for a just energy transition in residential buildings |- | [[d:Q137901188|Q137901188]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104036 10.1016/J.ERSS.2025.104036] || Community energy justice: A review of origins, convergence, and a research agenda |- | [[d:Q137901211|Q137901211]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104067 10.1016/J.ERSS.2025.104067] || Psychological and social factors driving citizen involvement in renewable energy communities: A systematic review |- | [[d:Q137901192|Q137901192]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104149 10.1016/J.ERSS.2025.104149] || Assessing social impacts and Energy Justice along green hydrogen supply chains: a capability-based framework |- | [[d:Q137901195|Q137901195]] || 2025 || [https://doi.org/10.1016/J.ERSS.2025.104422 10.1016/J.ERSS.2025.104422] || Out of place, scale and time? Navigating injustices across mission arenas of the German Energiewende |- | [[d:Q137901185|Q137901185]] || 2024 || [https://doi.org/10.1016/J.ESD.2024.101546 10.1016/J.ESD.2024.101546] || Characterizing 'injustices' in clean energy transitions in Africa |- | [[d:Q137901226|Q137901226]] || 2024 || [https://doi.org/10.1016/J.JCLEPRO.2024.143470 10.1016/J.JCLEPRO.2024.143470] || Energy justice and sustainable urban renewal: A systematic review of low-income old town communities |- | [[d:Q137901222|Q137901222]] || 2024 || [https://doi.org/10.1016/J.JENVMAN.2024.120804 10.1016/J.JENVMAN.2024.120804] || Forest, climate, and policy literature lacks acknowledgement of environmental justice, diversity, equity, and inclusion |- | [[d:Q115441381|Q115441381]] || 2021 || [https://doi.org/10.1016/J.RSER.2021.111504 10.1016/J.RSER.2021.111504] || Participatory methods in energy system modelling and planning – A review |- | [[d:Q137901205|Q137901205]] || 2025 || [https://doi.org/10.1016/J.RSER.2025.115892 10.1016/J.RSER.2025.115892] || A systematic review of the intersection between energy justice and human rights |- | [[d:Q137901225|Q137901225]] || 2024 || [https://doi.org/10.1017/SUS.2024.24 10.1017/SUS.2024.24] || Blue carbon as just transition? A structured literature review |- | [[d:Q137901220|Q137901220]] || 2025 || [https://doi.org/10.1017/SUS.2025.2 10.1017/SUS.2025.2] || Toward an intersectional equity approach in social-ecological transformations |- | [[d:Q137901203|Q137901203]] || 2024 || [https://doi.org/10.1080/14693062.2023.2256697 10.1080/14693062.2023.2256697] || Exploring the democracy-climate nexus: a review of correlations between democracy and climate policy performance |- | [[d:Q137901164|Q137901164]] || 2022 || [https://doi.org/10.1111/GEC3.12662 10.1111/GEC3.12662] || Creating fairer futures for sustainability transitions |- | [[d:Q137901227|Q137901227]] || 2025 || [https://doi.org/10.1139/ER-2024-0018 10.1139/ER-2024-0018] || Community engagement in nature-positive food systems programming and research in East and Southern Africa: a review |- | [[d:Q119955266|Q119955266]] || 2019 || [https://doi.org/10.1146/ANNUREV-ENVIRON-101718-033103 10.1146/ANNUREV-ENVIRON-101718-033103] || Co-Producing Sustainability: Reordering the Governance of Science, Policy, and Practice |- | [[d:Q137901206|Q137901206]] || 2023 || [https://doi.org/10.1146/ANNUREV-ENVIRON-112621-063400 10.1146/ANNUREV-ENVIRON-112621-063400] || Metrics for Decision-Making in Energy Justice |- | [[d:Q137901213|Q137901213]] || 2022 || [https://doi.org/10.1186/S13705-021-00330-4 10.1186/S13705-021-00330-4] || Mapping emergent public engagement in societal transitions: a scoping review |- | [[d:Q137901163|Q137901163]] || 2025 || [https://doi.org/10.17573/CEPAR.2025.2.09 10.17573/CEPAR.2025.2.09] || From Co-Creation to Circular Cities: Exploring Living Labs in EU Governance Frameworks - A Literature Review |- | [[d:Q137901197|Q137901197]] || 2024 || [https://doi.org/10.3390/EN17143512 10.3390/EN17143512] || A Systematic Review on the Path to Inclusive and Sustainable Energy Transitions |- | [[d:Q104887325|Q104887325]] || 2019 || [https://doi.org/10.3390/SU11041023 10.3390/SU11041023] || Deliberation and the Promise of a Deeply Democratic Sustainability Transition |- | [[d:Q137901202|Q137901202]] || 2021 || [https://doi.org/10.3390/SU13042128 10.3390/SU13042128] || A Review of Energy Communities in Sub-Saharan Africa as a Transition Pathway to Energy Democracy |- | [[d:Q137901210|Q137901210]] || 2023 || [https://doi.org/10.3390/SU15032441 10.3390/SU15032441] || Sustainable Project Governance: Scientometric Analysis and Emerging Trends |- | [[d:Q137901224|Q137901224]] || 2024 || [https://doi.org/10.3390/SU16198700 10.3390/SU16198700] || Empowering Communities to Act for a Change: A Review of the Community Empowerment Programs towards Sustainability and Resilience |} == References == {{References}} 50uqn706atf78ktc9jycdnusf2go3dm 0 326533 2816223 2815855 2026-06-18T14:37:15Z Bluerasberry 125661 /* Method */ image 2816223 wikitext text/x-wiki {{Article info |journal = WikiJournal Preprints <!-- WikiJournal of Medicine, Science, or Humanities --> |last1 = Rasberry |first1 = Lane |last2 = |first2 = |last3 = |first3 = |last4 = |first4 = <!-- up to 9 authors can be added in this above format --> |et_al = <!-- if there are >9 authors, hyperlink to the list here --> |affiliations = [[:en:wikipedia:School of Data Science|School of Data Science]] at the [[:en:wikipedia:University of Virginia|University of Virginia]] |correspondence = rasberry@virginia.edu |keywords = {{Q|Q1324130}}, {{Q|Q9687}}, {{Q|Q17386807}}, {{Q|Q136709591}}, {{Q|Q913302}}, {{Q|Q5250686}} |license = <!-- default is CC-BY --> |abstract = This case study demonstrates the use of civic technology to reduce wildlife-vehicle collisions in a region of the United States. Wildlife-vehicle collisions, also called roadkill, occur when humans in cars strike animals who are crossing roads. When community organizations and government agencies collaborate to analyze available data, then they can gain insights which inform decision making and reduce the problem. Code for Charlottesville, a regional civic tech organization, convened wildlife lovers, technologists, students, government officials, and nonprofit organization representatives to collaborate in applying civic tech to reduce vehicle collisions. The process included accessing and analyzing data describing past collisions, using technology to create data visualizations and other storytelling devices to understand the challenge and propose possible solutions, then collectively presenting recommendations for helpful infrastructure developments. Values which enable this process include favoring inclusive crowdsourced participation; using free and open source software and open datasets; negotiating multi-institutional partnerships among organizers, nonprofit organizations, and government agencies; and creating incentives to encourage participation and cooperation. }} ==Introduction== [[File:Reh im Feld mit Auto im Hintergrund.jpg|thumb|left | Preventing [[:en:wikipedia:Deer–vehicle collisions|deer–vehicle collisions]] is possible with infrastructure development (attribution: Morbakka, [https://creativecommons.org/licenses/by/4.0/deed.en CC-BY 4.0]) ]] A wildlife-vehicle collision (WVC) occurs when an automobile hits wildlife. Although often fatal and called "roadkill", the collisions can happen in ways that do not end the animals life. State departments of transportation concern themselves with road safety, and because their scope includes carcass removal and because larger animals are greater risk of damage to drivers, their records focus larger animal collisions where cars are at risk. For WVCs which involve small animals and non-fatal collisions, a driver may collect and transport the injured animal to a wildlife clinic for triage and medical assessment. When this happens, the clinic collects data including species, time, and location. It is uncommon for state transportation agencies to have records of small animals, and it is uncommon for citizens to personally transport large animals to clinics. Both wildlife advocates and transportation interests would like to reduce wildlife-vehicle collisions. To get more complete understanding, this project sought to collect data on WVCs from multiple sources, combine it all, and identify the places and circumstances where WVCs occur. The intent of presenting an analysis of this data is to inform decision making so that the design of transportation infrastructure can reduce WVCs, making systems safer for drivers and wildlife. The general study of the relation between wildlife and transportation infrastructure is [[:en:wikipedia:Road ecology|road ecology]]. A major topic in that field is [[:en:wikipedia:Habitat fragmentation|habitat fragmentation]], which is the recognition that wildlife require territory to live and that human development splits the living space. When wildlife cross roads to travel in their home territory, then [[:en:wikipedia:Roadkill|WCVs occur]]. When transportation designers can identify problem areas for WVCs, then they can place infrastructure to mitigate damage, such as by using signage, wildlife road barriers, [[:en:wikipedia:Wildlife corridor|wildlife corridors]], or [[:en:wikipedia:Wildlife crossing|wildlife crossings]]. {{Q|Q107227054}} is a civic tech organization based in [[:en:wikipedia:Charlottesville, Virginia|Charlottesville, Virginia]] which seeks to solve community problems with [[:en:wikipedia:Civic technology|civic technology]] through volunteer [[:en:wikipedia|Open collaboration|open collaboration]]. {{Q|Q137544492}} is a regional nonprofit organization which seeks to protect wildlife, and which issued the challenge of reducing wildlife collisions to Code for Cville. Wild Virginia referred project members to the wildlife veterinary hospital {{Q|Q54556122}}, which had been collecting data about wildlife collisions from the public. Along with these organizations, community participants volunteered to contribute labor to the project for their own interests in protecting wildlife from collisions. ==Method== The project's goal was to collect data about WVCs in Virginia, analyze this data, then share the result as open data in ways that are useful for visualization or reuse in other existing systems. The overall steps for this are data collection, data processing, and data sharing. The results of the data sharing process include a map overlay which visualize the locations on road were WVCs are more likely to occur, and also a public repository where anyone can access processed data or reuse the tools which this project developed for the project. ;Data collection [[File:Inaturalist - observations - 34322085 - Virginia opossum.jpg|thumb|right|example of photo with accompanying research observation record from {{Q|16958215}} (attribution: Daniel Mietchen, [https://creativecommons.org/publicdomain/zero/1.0/deed.en CC0])]] Wild Virginia requested the project from Code for Charlottesville, and they assisted in collecting data from other organizations. The data for wildlife collisions is not sensitive, but also it is unpublished, and is only available by request for research. This project only sought public data, and after processing, all project outcomes were open civic data. The inclusion criteria for all records in this study were as follows: #Must be a record of wildlife #Must have verification that animal suffered a vehicle collision #Must give the location of the collision These criteria exclude, for example, reports including vehicle collisions where there is no record of an animal, veterinary records which do not confirm that the animal experienced a collision, and WVCs without location data. In addition to the required data fields, when available, this project also collected species data, time of collision, and metadata for the provenance of the information. [[File:Virginia Wildlife Connectivity and Collision Concept Map.png|thumb|right|This image is an AI-generated map approximation is based on Department of Transportation wildlife corridor and wildlife-vehicle collision data. (attribution: AI based on open data, [https://creativecommons.org/publicdomain/zero/1.0/deed.en CC0])]] There were three sources of data: the Wildlife Center of Virginia, the {{Q|Q140234153}}, and the {{Q|Q7934247}}. The Wildlife Center of Virginia is an animal hospital which only treats wild animals. They have their own system and database for patient records, and are prominent as an animal care clinic. The Wildlife Rehabilitation Medical Database (WRMD) is a software application which the nonprofit organization {{Q|140234379}} develops and hosts as a tool which veterinary clinics can use to manage electronic health records for animals. They provide non-sensitive data from this database by requests from researchers. The Virginia Department of Transportation does not arrange veterinary care, but they had datasets for collisions, typically for larger animals where there was damage or danger to automobiles or roads, and they also provided records of animal carcass removal following reports of roadkill. ;Data processing The data processing tasks were cleaning ambiguous data and combining the datasets. Data cleaning included disambiguation of unclear locations and verifying in the patient notes that there was a WVC. The different data sources had different systems for reporting the location of the collision. The Department of Transportation had exact location coordinates, which is best. Other systems allowed for plaintext data entry. Sometimes this was an address, sometimes an intersection which needed to be clarified, sometimes just one road, and sometimes the location was a description. An entry like "on the interstate headed toward the gas station" made sense for people in the community at a particular clinic, but this project excluded such descriptions for lack of local knowledge clarification of the collision location. For clinics many records were originally entered on paper forms, then later had a human office worker type them into a digital form as data entry. The patient notes were similar, as some had fields were the veterinarian clearly indicated vehicle collision, but ambiguous records where the animal had collision-like injuries but no veterinary confirmation of vehicle collision were not included. Once the team cleaned the data from the sources and verified all the entries to meet inclusion criteria, then it was combined into a single dataset. The tools for accomplishing all this were {{Q|Q105099901}}, {{Q|Q15967387}}, {{Q|Q197520}}, and {{Q|Q513297}}, all of which are popular free and open source software. The Jupyter Notebook was the working environment of the project; pandas structures the prose and combines the datasets; NumPy handles the logic of verifying data compliance and managing gaps; and ArcGIS confirms locations. ;Data sharing Code for Cville made the results available as an open dataset in their [https://github.com/code-for-charlottesville/wildlife_collisions public repository]. The Virginia Department of Transportation data was map-ready, and with other data processed, it was combined to create a map overlay of where and how many WVCs occurred. ==Results== Based on analysis, it was determined that collisions were more likely to occur in certain places. {| class="wikitable sortable" |+ Count of most received species at Wildlife Center of Virginia, 2014-2023 ! Image !! Common Name !! Wikidata !! Count !! Image Credit |- | [[File:Opossum 2.jpg|100px]] | Virginia Opossum | {{Q|Q147267}} | 377 | [[:en:Wikipedia:User:Cody.pope|Cody Pope]], [https://creativecommons.org/licenses/by-sa/2.5/deed.en CC By-SA 2.5] |- | [[File:Eastern box turtle.jpg|100px]] | Eastern Box Turtle | {{Q|Q3768639}} | 292 | [https://www.flickr.com/photos/furryscalyman/ Matt Reinbold], [https://creativecommons.org/licenses/by-sa/2.0/deed.en CC By-SA 2.0] |- | [[File:Eastern Screech Owl.jpg|100px]] | Eastern Screech-owl | {{Q|Q251939}} | 121 | [[:commons:User:Wwcsig|Wolfgang Wander]], [https://creativecommons.org/licenses/by-sa/3.0/deed.en CC By-SA 3.0] |- | [[File:Eastern Cottontail (Sylvilagus floridanus).JPG|100px]] | Eastern Cottontail | {{Q|Q774716}} | 75 | [[User:The High Fin Sperm Whale|The High Fin Sperm Whale]], [https://creativecommons.org/licenses/by-sa/3.0/deed.en CC By-SA 3.0] |- | [[File:EasternGraySquirrel GAm.jpg|100px]] | Eastern Gray Squirrel | {{Q|Q468500}} | 69 | JeffreyGammon, [https://creativecommons.org/licenses/by-sa/4.0/deed.en CC By-SA 4.0] |} ==Discussion== While ideally it would be possible to build infrastructure along all roads to prevent wildlife vehicle collisions, the expense for comprehensive prevention is prohibitively high, and the affordable solution is to intervene in cost effective ways. Data analysis showed that collisions are more likely to occur at particular places on particular roads. Given this insight, building preventative infrastructure in the places were road crossing is most dangerous for animals is recommended as a way to reduce wildlife collisions while also being selective in spending available budgets. ==Conclusion== ==Acknowledgements== *{{Q|Q137544492}} identified the challenge of protecting wildlife from vehicle collisions and was the stakeholder organization concerned with creating knowledge to address the problem. *{{Q|Q54556122}} contributed data which their clinic compiled on wildlife-vehicle collisions. Their veterinarians also triage and rehabilitate animals experiencing collisions. *{{Q|Q140234379}} manages the {{Q|Q140234153}}, and they provided research access to patient records from Virginia veterinary clinics which make reports there. *{{Q|Q7934247}} collects data on WVCs and provides it as open government data through their website *{{Q|Q107227054}} is the civic tech organization which recruited and managed community members, technical developers, data scientists, and students to volunteer the labor to address the challenge. ==Competing interests== The project team identified no conflicts or competing interests. ==Ethics statement== The project team identified no ethical issues which needed disclosure. This project is an analysis of public data. This project accessed parts of veterinary electronic health records, only for wildlife, and not with personal information of any associated humans. While research access to this data was required, this was because the clinics lack capacity to distribute these non-sensitive records of wild animal rehabilitation, and not because the data contained anything to put someone at risk. ==References== {{reflist|35em}} [[Category:Virginia]] 0p3ubsdft41u8rrmbhja9e8kz46jl6m 2816224 2816223 2026-06-18T14:37:43Z Bluerasberry 125661 /* Method */ fix 2816224 wikitext text/x-wiki {{Article info |journal = WikiJournal Preprints <!-- WikiJournal of Medicine, Science, or Humanities --> |last1 = Rasberry |first1 = Lane |last2 = |first2 = |last3 = |first3 = |last4 = |first4 = <!-- up to 9 authors can be added in this above format --> |et_al = <!-- if there are >9 authors, hyperlink to the list here --> |affiliations = [[:en:wikipedia:School of Data Science|School of Data Science]] at the [[:en:wikipedia:University of Virginia|University of Virginia]] |correspondence = rasberry@virginia.edu |keywords = {{Q|Q1324130}}, {{Q|Q9687}}, {{Q|Q17386807}}, {{Q|Q136709591}}, {{Q|Q913302}}, {{Q|Q5250686}} |license = <!-- default is CC-BY --> |abstract = This case study demonstrates the use of civic technology to reduce wildlife-vehicle collisions in a region of the United States. Wildlife-vehicle collisions, also called roadkill, occur when humans in cars strike animals who are crossing roads. When community organizations and government agencies collaborate to analyze available data, then they can gain insights which inform decision making and reduce the problem. Code for Charlottesville, a regional civic tech organization, convened wildlife lovers, technologists, students, government officials, and nonprofit organization representatives to collaborate in applying civic tech to reduce vehicle collisions. The process included accessing and analyzing data describing past collisions, using technology to create data visualizations and other storytelling devices to understand the challenge and propose possible solutions, then collectively presenting recommendations for helpful infrastructure developments. Values which enable this process include favoring inclusive crowdsourced participation; using free and open source software and open datasets; negotiating multi-institutional partnerships among organizers, nonprofit organizations, and government agencies; and creating incentives to encourage participation and cooperation. }} ==Introduction== [[File:Reh im Feld mit Auto im Hintergrund.jpg|thumb|left | Preventing [[:en:wikipedia:Deer–vehicle collisions|deer–vehicle collisions]] is possible with infrastructure development (attribution: Morbakka, [https://creativecommons.org/licenses/by/4.0/deed.en CC-BY 4.0]) ]] A wildlife-vehicle collision (WVC) occurs when an automobile hits wildlife. Although often fatal and called "roadkill", the collisions can happen in ways that do not end the animals life. State departments of transportation concern themselves with road safety, and because their scope includes carcass removal and because larger animals are greater risk of damage to drivers, their records focus larger animal collisions where cars are at risk. For WVCs which involve small animals and non-fatal collisions, a driver may collect and transport the injured animal to a wildlife clinic for triage and medical assessment. When this happens, the clinic collects data including species, time, and location. It is uncommon for state transportation agencies to have records of small animals, and it is uncommon for citizens to personally transport large animals to clinics. Both wildlife advocates and transportation interests would like to reduce wildlife-vehicle collisions. To get more complete understanding, this project sought to collect data on WVCs from multiple sources, combine it all, and identify the places and circumstances where WVCs occur. The intent of presenting an analysis of this data is to inform decision making so that the design of transportation infrastructure can reduce WVCs, making systems safer for drivers and wildlife. The general study of the relation between wildlife and transportation infrastructure is [[:en:wikipedia:Road ecology|road ecology]]. A major topic in that field is [[:en:wikipedia:Habitat fragmentation|habitat fragmentation]], which is the recognition that wildlife require territory to live and that human development splits the living space. When wildlife cross roads to travel in their home territory, then [[:en:wikipedia:Roadkill|WCVs occur]]. When transportation designers can identify problem areas for WVCs, then they can place infrastructure to mitigate damage, such as by using signage, wildlife road barriers, [[:en:wikipedia:Wildlife corridor|wildlife corridors]], or [[:en:wikipedia:Wildlife crossing|wildlife crossings]]. {{Q|Q107227054}} is a civic tech organization based in [[:en:wikipedia:Charlottesville, Virginia|Charlottesville, Virginia]] which seeks to solve community problems with [[:en:wikipedia:Civic technology|civic technology]] through volunteer [[:en:wikipedia|Open collaboration|open collaboration]]. {{Q|Q137544492}} is a regional nonprofit organization which seeks to protect wildlife, and which issued the challenge of reducing wildlife collisions to Code for Cville. Wild Virginia referred project members to the wildlife veterinary hospital {{Q|Q54556122}}, which had been collecting data about wildlife collisions from the public. Along with these organizations, community participants volunteered to contribute labor to the project for their own interests in protecting wildlife from collisions. ==Method== The project's goal was to collect data about WVCs in Virginia, analyze this data, then share the result as open data in ways that are useful for visualization or reuse in other existing systems. The overall steps for this are data collection, data processing, and data sharing. The results of the data sharing process include a map overlay which visualize the locations on road were WVCs are more likely to occur, and also a public repository where anyone can access processed data or reuse the tools which this project developed for the project. ;Data collection [[File:Inaturalist - observations - 34322085 - Virginia opossum.jpg|thumb|right|example of photo with accompanying research observation record from {{Q|16958215}} (attribution: Daniel Mietchen, [https://creativecommons.org/publicdomain/zero/1.0/deed.en CC0])]] Wild Virginia requested the project from Code for Charlottesville, and they assisted in collecting data from other organizations. The data for wildlife collisions is not sensitive, but also it is unpublished, and is only available by request for research. This project only sought public data, and after processing, all project outcomes were open civic data. The inclusion criteria for all records in this study were as follows: #Must be a record of wildlife #Must have verification that animal suffered a vehicle collision #Must give the location of the collision These criteria exclude, for example, reports including vehicle collisions where there is no record of an animal, veterinary records which do not confirm that the animal experienced a collision, and WVCs without location data. In addition to the required data fields, when available, this project also collected species data, time of collision, and metadata for the provenance of the information. [[File:Virginia Wildlife Connectivity and Collision Concept Map.png|thumb|right|This image is an AI-generated map approximation is based on Department of Transportation wildlife corridor and wildlife-vehicle collision data. (attribution: AI, [https://creativecommons.org/publicdomain/zero/1.0/deed.en CC0])]] There were three sources of data: the Wildlife Center of Virginia, the {{Q|Q140234153}}, and the {{Q|Q7934247}}. The Wildlife Center of Virginia is an animal hospital which only treats wild animals. They have their own system and database for patient records, and are prominent as an animal care clinic. The Wildlife Rehabilitation Medical Database (WRMD) is a software application which the nonprofit organization {{Q|140234379}} develops and hosts as a tool which veterinary clinics can use to manage electronic health records for animals. They provide non-sensitive data from this database by requests from researchers. The Virginia Department of Transportation does not arrange veterinary care, but they had datasets for collisions, typically for larger animals where there was damage or danger to automobiles or roads, and they also provided records of animal carcass removal following reports of roadkill. ;Data processing The data processing tasks were cleaning ambiguous data and combining the datasets. Data cleaning included disambiguation of unclear locations and verifying in the patient notes that there was a WVC. The different data sources had different systems for reporting the location of the collision. The Department of Transportation had exact location coordinates, which is best. Other systems allowed for plaintext data entry. Sometimes this was an address, sometimes an intersection which needed to be clarified, sometimes just one road, and sometimes the location was a description. An entry like "on the interstate headed toward the gas station" made sense for people in the community at a particular clinic, but this project excluded such descriptions for lack of local knowledge clarification of the collision location. For clinics many records were originally entered on paper forms, then later had a human office worker type them into a digital form as data entry. The patient notes were similar, as some had fields were the veterinarian clearly indicated vehicle collision, but ambiguous records where the animal had collision-like injuries but no veterinary confirmation of vehicle collision were not included. Once the team cleaned the data from the sources and verified all the entries to meet inclusion criteria, then it was combined into a single dataset. The tools for accomplishing all this were {{Q|Q105099901}}, {{Q|Q15967387}}, {{Q|Q197520}}, and {{Q|Q513297}}, all of which are popular free and open source software. The Jupyter Notebook was the working environment of the project; pandas structures the prose and combines the datasets; NumPy handles the logic of verifying data compliance and managing gaps; and ArcGIS confirms locations. ;Data sharing Code for Cville made the results available as an open dataset in their [https://github.com/code-for-charlottesville/wildlife_collisions public repository]. The Virginia Department of Transportation data was map-ready, and with other data processed, it was combined to create a map overlay of where and how many WVCs occurred. ==Results== Based on analysis, it was determined that collisions were more likely to occur in certain places. {| class="wikitable sortable" |+ Count of most received species at Wildlife Center of Virginia, 2014-2023 ! Image !! Common Name !! Wikidata !! Count !! Image Credit |- | [[File:Opossum 2.jpg|100px]] | Virginia Opossum | {{Q|Q147267}} | 377 | [[:en:Wikipedia:User:Cody.pope|Cody Pope]], [https://creativecommons.org/licenses/by-sa/2.5/deed.en CC By-SA 2.5] |- | [[File:Eastern box turtle.jpg|100px]] | Eastern Box Turtle | {{Q|Q3768639}} | 292 | [https://www.flickr.com/photos/furryscalyman/ Matt Reinbold], [https://creativecommons.org/licenses/by-sa/2.0/deed.en CC By-SA 2.0] |- | [[File:Eastern Screech Owl.jpg|100px]] | Eastern Screech-owl | {{Q|Q251939}} | 121 | [[:commons:User:Wwcsig|Wolfgang Wander]], [https://creativecommons.org/licenses/by-sa/3.0/deed.en CC By-SA 3.0] |- | [[File:Eastern Cottontail (Sylvilagus floridanus).JPG|100px]] | Eastern Cottontail | {{Q|Q774716}} | 75 | [[User:The High Fin Sperm Whale|The High Fin Sperm Whale]], [https://creativecommons.org/licenses/by-sa/3.0/deed.en CC By-SA 3.0] |- | [[File:EasternGraySquirrel GAm.jpg|100px]] | Eastern Gray Squirrel | {{Q|Q468500}} | 69 | JeffreyGammon, [https://creativecommons.org/licenses/by-sa/4.0/deed.en CC By-SA 4.0] |} ==Discussion== While ideally it would be possible to build infrastructure along all roads to prevent wildlife vehicle collisions, the expense for comprehensive prevention is prohibitively high, and the affordable solution is to intervene in cost effective ways. Data analysis showed that collisions are more likely to occur at particular places on particular roads. Given this insight, building preventative infrastructure in the places were road crossing is most dangerous for animals is recommended as a way to reduce wildlife collisions while also being selective in spending available budgets. ==Conclusion== ==Acknowledgements== *{{Q|Q137544492}} identified the challenge of protecting wildlife from vehicle collisions and was the stakeholder organization concerned with creating knowledge to address the problem. *{{Q|Q54556122}} contributed data which their clinic compiled on wildlife-vehicle collisions. Their veterinarians also triage and rehabilitate animals experiencing collisions. *{{Q|Q140234379}} manages the {{Q|Q140234153}}, and they provided research access to patient records from Virginia veterinary clinics which make reports there. *{{Q|Q7934247}} collects data on WVCs and provides it as open government data through their website *{{Q|Q107227054}} is the civic tech organization which recruited and managed community members, technical developers, data scientists, and students to volunteer the labor to address the challenge. ==Competing interests== The project team identified no conflicts or competing interests. ==Ethics statement== The project team identified no ethical issues which needed disclosure. This project is an analysis of public data. This project accessed parts of veterinary electronic health records, only for wildlife, and not with personal information of any associated humans. While research access to this data was required, this was because the clinics lack capacity to distribute these non-sensitive records of wild animal rehabilitation, and not because the data contained anything to put someone at risk. ==References== {{reflist|35em}} [[Category:Virginia]] gr264ulpp8rkbweo84fyfu9lvscwdex 2816225 2816224 2026-06-18T14:40:15Z Bluerasberry 125661 /* Method */ fix 2816225 wikitext text/x-wiki {{Article info |journal = WikiJournal Preprints <!-- WikiJournal of Medicine, Science, or Humanities --> |last1 = Rasberry |first1 = Lane |last2 = |first2 = |last3 = |first3 = |last4 = |first4 = <!-- up to 9 authors can be added in this above format --> |et_al = <!-- if there are >9 authors, hyperlink to the list here --> |affiliations = [[:en:wikipedia:School of Data Science|School of Data Science]] at the [[:en:wikipedia:University of Virginia|University of Virginia]] |correspondence = rasberry@virginia.edu |keywords = {{Q|Q1324130}}, {{Q|Q9687}}, {{Q|Q17386807}}, {{Q|Q136709591}}, {{Q|Q913302}}, {{Q|Q5250686}} |license = <!-- default is CC-BY --> |abstract = This case study demonstrates the use of civic technology to reduce wildlife-vehicle collisions in a region of the United States. Wildlife-vehicle collisions, also called roadkill, occur when humans in cars strike animals who are crossing roads. When community organizations and government agencies collaborate to analyze available data, then they can gain insights which inform decision making and reduce the problem. Code for Charlottesville, a regional civic tech organization, convened wildlife lovers, technologists, students, government officials, and nonprofit organization representatives to collaborate in applying civic tech to reduce vehicle collisions. The process included accessing and analyzing data describing past collisions, using technology to create data visualizations and other storytelling devices to understand the challenge and propose possible solutions, then collectively presenting recommendations for helpful infrastructure developments. Values which enable this process include favoring inclusive crowdsourced participation; using free and open source software and open datasets; negotiating multi-institutional partnerships among organizers, nonprofit organizations, and government agencies; and creating incentives to encourage participation and cooperation. }} ==Introduction== [[File:Reh im Feld mit Auto im Hintergrund.jpg|thumb|left | Preventing [[:en:wikipedia:Deer–vehicle collisions|deer–vehicle collisions]] is possible with infrastructure development (attribution: Morbakka, [https://creativecommons.org/licenses/by/4.0/deed.en CC-BY 4.0]) ]] A wildlife-vehicle collision (WVC) occurs when an automobile hits wildlife. Although often fatal and called "roadkill", the collisions can happen in ways that do not end the animals life. State departments of transportation concern themselves with road safety, and because their scope includes carcass removal and because larger animals are greater risk of damage to drivers, their records focus larger animal collisions where cars are at risk. For WVCs which involve small animals and non-fatal collisions, a driver may collect and transport the injured animal to a wildlife clinic for triage and medical assessment. When this happens, the clinic collects data including species, time, and location. It is uncommon for state transportation agencies to have records of small animals, and it is uncommon for citizens to personally transport large animals to clinics. Both wildlife advocates and transportation interests would like to reduce wildlife-vehicle collisions. To get more complete understanding, this project sought to collect data on WVCs from multiple sources, combine it all, and identify the places and circumstances where WVCs occur. The intent of presenting an analysis of this data is to inform decision making so that the design of transportation infrastructure can reduce WVCs, making systems safer for drivers and wildlife. The general study of the relation between wildlife and transportation infrastructure is [[:en:wikipedia:Road ecology|road ecology]]. A major topic in that field is [[:en:wikipedia:Habitat fragmentation|habitat fragmentation]], which is the recognition that wildlife require territory to live and that human development splits the living space. When wildlife cross roads to travel in their home territory, then [[:en:wikipedia:Roadkill|WCVs occur]]. When transportation designers can identify problem areas for WVCs, then they can place infrastructure to mitigate damage, such as by using signage, wildlife road barriers, [[:en:wikipedia:Wildlife corridor|wildlife corridors]], or [[:en:wikipedia:Wildlife crossing|wildlife crossings]]. {{Q|Q107227054}} is a civic tech organization based in [[:en:wikipedia:Charlottesville, Virginia|Charlottesville, Virginia]] which seeks to solve community problems with [[:en:wikipedia:Civic technology|civic technology]] through volunteer [[:en:wikipedia|Open collaboration|open collaboration]]. {{Q|Q137544492}} is a regional nonprofit organization which seeks to protect wildlife, and which issued the challenge of reducing wildlife collisions to Code for Cville. Wild Virginia referred project members to the wildlife veterinary hospital {{Q|Q54556122}}, which had been collecting data about wildlife collisions from the public. Along with these organizations, community participants volunteered to contribute labor to the project for their own interests in protecting wildlife from collisions. ==Method== The project's goal was to collect data about WVCs in Virginia, analyze this data, then share the result as open data in ways that are useful for visualization or reuse in other existing systems. The overall steps for this are data collection, data processing, and data sharing. The results of the data sharing process include a map overlay which visualize the locations on road were WVCs are more likely to occur, and also a public repository where anyone can access processed data or reuse the tools which this project developed for the project. ;Data collection [[File:Inaturalist - observations - 34322085 - Virginia opossum.jpg|thumb|right|example of photo with accompanying research observation record from {{Q|16958215}} (attribution: Daniel Mietchen, [https://creativecommons.org/publicdomain/zero/1.0/deed.en CC0])]] Wild Virginia requested the project from Code for Charlottesville, and they assisted in collecting data from other organizations. The data for wildlife collisions is not sensitive, but also it is unpublished, and is only available by request for research. This project only sought public data, and after processing, all project outcomes were open civic data. The inclusion criteria for all records in this study were as follows: #Must be a record of wildlife #Must have verification that animal suffered a vehicle collision #Must give the location of the collision These criteria exclude, for example, reports including vehicle collisions where there is no record of an animal, veterinary records which do not confirm that the animal experienced a collision, and WVCs without location data. In addition to the required data fields, when available, this project also collected species data, time of collision, and metadata for the provenance of the information. [[File:Virginia Wildlife Connectivity and Collision Concept Map.png|thumb|right|AI-map approximation based on Department of Transportation wildlife corridor and wildlife-vehicle collision data. (attribution: AI, [https://creativecommons.org/publicdomain/zero/1.0/deed.en CC0])]] There were three sources of data: the Wildlife Center of Virginia, the {{Q|Q140234153}}, and the {{Q|Q7934247}}. The Wildlife Center of Virginia is an animal hospital which only treats wild animals. They have their own system and database for patient records, and are prominent as an animal care clinic. The Wildlife Rehabilitation Medical Database (WRMD) is a software application which the nonprofit organization {{Q|140234379}} develops and hosts as a tool which veterinary clinics can use to manage electronic health records for animals. They provide non-sensitive data from this database by requests from researchers. The Virginia Department of Transportation does not arrange veterinary care, but they had datasets for collisions, typically for larger animals where there was damage or danger to automobiles or roads, and they also provided records of animal carcass removal following reports of roadkill. ;Data processing The data processing tasks were cleaning ambiguous data and combining the datasets. Data cleaning included disambiguation of unclear locations and verifying in the patient notes that there was a WVC. The different data sources had different systems for reporting the location of the collision. The Department of Transportation had exact location coordinates, which is best. Other systems allowed for plaintext data entry. Sometimes this was an address, sometimes an intersection which needed to be clarified, sometimes just one road, and sometimes the location was a description. An entry like "on the interstate headed toward the gas station" made sense for people in the community at a particular clinic, but this project excluded such descriptions for lack of local knowledge clarification of the collision location. For clinics many records were originally entered on paper forms, then later had a human office worker type them into a digital form as data entry. The patient notes were similar, as some had fields were the veterinarian clearly indicated vehicle collision, but ambiguous records where the animal had collision-like injuries but no veterinary confirmation of vehicle collision were not included. Once the team cleaned the data from the sources and verified all the entries to meet inclusion criteria, then it was combined into a single dataset. The tools for accomplishing all this were {{Q|Q105099901}}, {{Q|Q15967387}}, {{Q|Q197520}}, and {{Q|Q513297}}, all of which are popular free and open source software. The Jupyter Notebook was the working environment of the project; pandas structures the prose and combines the datasets; NumPy handles the logic of verifying data compliance and managing gaps; and ArcGIS confirms locations. ;Data sharing Code for Cville made the results available as an open dataset in their [https://github.com/code-for-charlottesville/wildlife_collisions public repository]. The Virginia Department of Transportation data was map-ready, and with other data processed, it was combined to create a map overlay of where and how many WVCs occurred. ==Results== Based on analysis, it was determined that collisions were more likely to occur in certain places. {| class="wikitable sortable" |+ Count of most received species at Wildlife Center of Virginia, 2014-2023 ! Image !! Common Name !! Wikidata !! Count !! Image Credit |- | [[File:Opossum 2.jpg|100px]] | Virginia Opossum | {{Q|Q147267}} | 377 | [[:en:Wikipedia:User:Cody.pope|Cody Pope]], [https://creativecommons.org/licenses/by-sa/2.5/deed.en CC By-SA 2.5] |- | [[File:Eastern box turtle.jpg|100px]] | Eastern Box Turtle | {{Q|Q3768639}} | 292 | [https://www.flickr.com/photos/furryscalyman/ Matt Reinbold], [https://creativecommons.org/licenses/by-sa/2.0/deed.en CC By-SA 2.0] |- | [[File:Eastern Screech Owl.jpg|100px]] | Eastern Screech-owl | {{Q|Q251939}} | 121 | [[:commons:User:Wwcsig|Wolfgang Wander]], [https://creativecommons.org/licenses/by-sa/3.0/deed.en CC By-SA 3.0] |- | [[File:Eastern Cottontail (Sylvilagus floridanus).JPG|100px]] | Eastern Cottontail | {{Q|Q774716}} | 75 | [[User:The High Fin Sperm Whale|The High Fin Sperm Whale]], [https://creativecommons.org/licenses/by-sa/3.0/deed.en CC By-SA 3.0] |- | [[File:EasternGraySquirrel GAm.jpg|100px]] | Eastern Gray Squirrel | {{Q|Q468500}} | 69 | JeffreyGammon, [https://creativecommons.org/licenses/by-sa/4.0/deed.en CC By-SA 4.0] |} ==Discussion== While ideally it would be possible to build infrastructure along all roads to prevent wildlife vehicle collisions, the expense for comprehensive prevention is prohibitively high, and the affordable solution is to intervene in cost effective ways. Data analysis showed that collisions are more likely to occur at particular places on particular roads. Given this insight, building preventative infrastructure in the places were road crossing is most dangerous for animals is recommended as a way to reduce wildlife collisions while also being selective in spending available budgets. ==Conclusion== ==Acknowledgements== *{{Q|Q137544492}} identified the challenge of protecting wildlife from vehicle collisions and was the stakeholder organization concerned with creating knowledge to address the problem. *{{Q|Q54556122}} contributed data which their clinic compiled on wildlife-vehicle collisions. Their veterinarians also triage and rehabilitate animals experiencing collisions. *{{Q|Q140234379}} manages the {{Q|Q140234153}}, and they provided research access to patient records from Virginia veterinary clinics which make reports there. *{{Q|Q7934247}} collects data on WVCs and provides it as open government data through their website *{{Q|Q107227054}} is the civic tech organization which recruited and managed community members, technical developers, data scientists, and students to volunteer the labor to address the challenge. ==Competing interests== The project team identified no conflicts or competing interests. ==Ethics statement== The project team identified no ethical issues which needed disclosure. This project is an analysis of public data. This project accessed parts of veterinary electronic health records, only for wildlife, and not with personal information of any associated humans. While research access to this data was required, this was because the clinics lack capacity to distribute these non-sensitive records of wild animal rehabilitation, and not because the data contained anything to put someone at risk. ==References== {{reflist|35em}} [[Category:Virginia]] ecf9uzcupljrt46dv9p1zqzx3f16a35 2 326765 2816277 2816128 2026-06-19T02:18:46Z Dc.samizdat 2856930 /* The 600-cell */ 2816277 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>r_{8}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 15 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the edges of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{10}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{10}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{10}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{10}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_{13}=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its <math>r_{3}</math> edges, over <math>r_{13}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_{13}</math> chords. The <math>r_{4}</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_{13}</math> chords are edges of different 24-cells. The rotational curve over each <math>r_{13}</math> chord makes two 30° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_{13}</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell; it takes 24-cells to a non-disjoint 24-cell and 16-cells to a 16-cell in another 24-cell. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} 83aqax5tslix188imn4nax65mitxgf1 2816278 2816277 2026-06-19T02:35:06Z Dc.samizdat 2856930 /* The 600-cell */ 2816278 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>r_{8}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing vertices of the short chord {30}-gon, over isocline chords of the long chord {30}-gon, the rotation's Clifford polygon. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{10}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{10}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{10}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{10}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_{13}=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its <math>r_{3}</math> edges, over <math>r_{13}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_{13}</math> chords. The <math>r_{4}</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_{13}</math> chords are edges of different 24-cells. The rotational curve over each <math>r_{13}</math> chord makes two 30° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_{13}</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell; it takes 24-cells to a non-disjoint 24-cell and 16-cells to a 16-cell in another 24-cell. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} duvs7chswyx83hfizzmamxyxfharksw 2816279 2816278 2026-06-19T02:39:32Z Dc.samizdat 2856930 /* The 600-cell */ 2816279 wikitext text/x-wiki = Golden chords of the 120-cell = {{align|center|David Brooks Christie}} {{align|center|dc@samizdat.org}} {{align|center|Draft in progress}} {{align|center|January 2026 - June 2026}} <blockquote>Steinbach discovered the formula for the ratios of diagonal to side in the regular polygons. Fontaine and Hurley extended this result, discovering a formula for the reciprocal of a regular polygon chord derived geometrically from the chord's star polygon. We observe that these findings in plane geometry apply more generally, to polytopes of any dimensionality. Fontaine and Hurley's geometric procedure for finding the reciprocals of the chords of a regular polygon from their star polygons also finds the rotational geodesics of any polytope of any dimensionality.</blockquote> == Introduction == Steinbach discovered the Diagonal Product Formula and the Golden Fields family of ratios of diagonal to side in the regular polygons. He showed how this family extends beyond the pentagon {5} with its well-known golden bisection proportional to 𝜙, finding that the heptagon {7} has an analogous trisection, the nonagon {9} has an analogous quadrasection, and the hendecagon {11} has an analogous pentasection, an extended family of golden proportions with quasiperiodic properties. Kappraff and Adamson extended these findings in plane geometry to a theory of Generalized Fibonacci Sequences, showing that the Golden Fields not only do not end with the hendecagon, they form an infinite number of periodic trajectories when operated on by the Mandelbrot operator. They found a relation between the edges of star polygons and dynamical systems in the state of chaos, revealing a connection between chaos theory, number, and rotations in Coxeter Euclidean geometry. Fontaine and Hurley examined Steinbach's finding that the length of each chord of a regular polygon is both the product of two chords and the sum of a set of smaller chords, so that in rotations to add is to multiply. They illustrated Steinbach's sets of additive chords lying parallel to each other in the plane (pointing in the same direction), and by applying Steinbach's formula more generally they found another summation relation of signed parallel chords (pointing in opposite directions) which relates each chord length to its reciprocal, and relates the summation to a distinct star polygon rotation. We examine these remarkable findings (which stem from study of the chords of humble regular polygons) in higher-dimensional spaces, specifically in the chords, polygons and rotations of the [[120-cell]], the largest four-dimensional regular convex polytope. == Visualizing the 120-cell == {| class="wikitable floatright" width="400" |style="vertical-align:top"|[[File:120-cell.gif|200px]]<br>Orthographic projection of the 600-point 120-cell <small><math>\{5,3,3\}</math></small> performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; "Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]]."}} In this simplified rendering only the 120-cell's own edges are shown; its 29 interior chords are not rendered. Therefore even though it is translucent, only its outer surface is visible. The complex interior parts of the 120-cell, all its inscribed 5-cells, 16-cells, 8-cells, 24-cells, 600-cells and its much larger inventory of polyhedra, are completely invisible in this view, as none of their edges are rendered at all. |style="vertical-align:top"|[[File:Ortho solid 016-uniform polychoron p33-t0.png|200px]]<br>Orthographic projection of the 600-point [[W:Great grand stellated 120-cell|great grand stellated 120-cell]] <small><math>\{\tfrac{5}{2},3,3\}</math></small>.{{Sfn|Ruen: Great grand stellated 120-cell|2007}} The 120-cell is its convex hull. The projection to the left renders only the 120-cell's shortest chord, its 1200 edges. The projection above also renders only one of the 120-cell's 30 chords, the edges of its 120 inscribed regular 5-cells. The 120-cell itself (the convex hull) is invisible in this view, as its edges are not rendered. |} [[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], containing inscribed instances of every regular 1-, 2-, 3-, and 4-polytope, except the regular polygons of more than {15} sides. The 120-cell is the convex hull of a regular [[120-cell#Relationships among interior polytopes|compound of each of the 6 regular convex 4-polytopes]]. They are the [[5-cell|5-point (5-cell) 4-simplex]], the [[16-cell|8-point (16-cell) 4-orthoplex]], the [[W:Tesseract|16-point (8-cell) tesseract]], the [[24-cell|24-point (24-cell)]], the [[600-cell|120-point (600-cell)]], and the [[120-cell|600-point (120-cell)]]. The 120-cell is the convex hull of a compound of 120 disjoint regular 5-cells, of 75 disjoint 16-cells, of 25 disjoint 24-cells, and of 5 disjoint 600-cells. The 120-cell contains an even larger inventory of irregular polytopes, created by the intersection of multiple instances of these component regular 4-polytopes. Many are quite unexpected, because they do not occur as components of any regular polytope smaller than the 120-cell. As just one example among the [[120-cell#Concentric hulls|sections of the 120-cell]], there is an irregular 24-point polyhedron with 16 triangle faces and 4 nonagon {9} faces.{{Sfn|Moxness|}} Most renderings of the 120-cell, like the rotating projection here, only illustrate its outer surface, which is a honeycomb of face-bonded dodecahedral cells. Only the objects in its 3-dimensional surface are rendered, namely the 120 dodecahedra, their pentagon faces, and their edges. Although the 120-cell has chords of 30 distinct lengths, in this kind of simplified rendering only the 120-cell's own edges (its shortest chord) are shown. Its 29 interior chords, the edges of objects in the interior of the 120-cell, are not rendered, so interior objects are not visible at all. Visualizing the complete interior of the 600-vertex 120-cell in a single image is impractical because of its complexity. Only four 120-cell edges are incident at each vertex, but [[120-cell#Chords|600 chords (of all 30 lengths)]] are incident at ''each'' vertex. == Compounds in the 120-cell == The 8-point (16-cell), not the 5-point (5-cell), is the smallest building block; it compounds to every larger regular 4-polytope. The 5-point (5-cell) does compound to the 600-point (120-cell), but it does not fit into any smaller regular 4-polytope. The 8-point (16-cell) compounds by 2 in the 16-point (8-cell), and by 3 in the 24-point (24-cell). The 16-point (8-cell) compounds in the 24-point (24-cell) by 3 non-disjoint instances of itself, with each of the 24 vertices shared by two 16-point (8-cells). The 24-point (24-cell) compounds by 5 disjoint instances of itself in the 120-point (600-cell), and the 120-point (600-cell) compounds by 5 disjoint instances of itself in the 600-point (120-cell). The 24-point (24-cell) also compounds by 5² non-disjoint instances of itself in the 120-point (600-cell); it compounds in 5 disjoint instances of itself, 10 (not 5) different ways. Whichever set of 5 disjoint 24-point (24-cells) are assembled, the resulting 120-point (600-cell) contains 25 distinct 24-point (24-cells), not just 5 (or 10). This implies that 15 disjoint 8-point (16-cells) will construct a 120-point (600-cell), which will contain 75 distinct 8-point (16-cells). The 600-point (120-cell) is 5 disjoint 120-point (600-cells), just 2 different ways (not 5 or 10 ways), so it is 10 distinct 120-point (600-cells). This implies that the 8-point (16-cell) compounds by 3 times 5² (75) disjoint instances of itself in the 600-point (120-cell), which contains 3² times 5² (225) distinct instances of the 24-point (24-cell), and 3³ times 5² (675) distinct instances of the 8-point (16-cell). These facts were discovered painstakingly by various researchers, and no one has found a general rule governing subsumption relations among regular polytopes. The reasons for some of their numeric incidence relations are far from obvious. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cells) ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler & Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler & Claiborne|2020|loc=''The geometry of H4 polytopes''}} So much for the compounds of 16-cells. The 120-cell is also the convex hull of the compound of 120 disjoint regular 5-cells. That stellated compound (without its convex hull of 120-cell edges) is the [[w:Great_grand_stellated_120-cell|great grand stellated 120-cell]] illustrated above, the final regular [[W:Stellation|stellation]] of the 120-cell, and the only [[W:Schläfli-Hess polychoron|regular star 4-polytope]] to have the 120-cell for its convex hull. The edges of the great grand stellated 120-cell are <math>\phi^6</math> as long as those of its 120-cell [[W:List of polyhedral stellations#Stellation process|stellation core]] deep inside. The compound of 120 disjoint 5-point (5-cells) can be seen to be equivalent to the compound of 5 disjoint 120-point (600-cells), as follows. Beginning with a single 120-point (600-cell), expand each vertex into a regular 5-cell, by adding 4 new equidistant vertices, such that the 5 vertices form a regular 5-cell inscribed in the 3-sphere. The 120 5-cells are disjoint, and the 600 vertices form 5 disjoint 120-point (600-cells): a 120-cell. == Thirty distinguished distances == The 30 numbers listed in the table are all-important in Euclidean geometry. A case can be made on symmetry grounds that their squares are the 30 most important numbers between 0 and 4. The 30 rows of the table are the 30 distinct [[120-cell#Geodesic rectangles|chord lengths of the unit-radius 120-cell]], the largest regular convex 4-polytope. Since the 120-cell subsumes all smaller regular polytopes, its 30 chords are the complete chord set of all the regular polytopes that can be constructed in the first four dimensions of Euclidean space, except for regular polygons of more than 15 sides. {| class="wikitable" style="white-space:nowrap;text-align:center" !rowspan=2|<math>c_t</math> !rowspan=2|arc !rowspan=2|<small><math>\left\{\frac{30}{n}\right\}</math></small> !rowspan=2|<math>\left\{p\right\}</math> !rowspan=2|<small><math>m\left\{\frac{k}{d}\right\}</math></small> !rowspan=2|Steinbach roots !colspan=7|Chord lengths of the unit 120-cell |- !colspan=5|unit-radius length <math>c_t</math> !colspan=2|unit-edge length <math>c_t/c_1</math><br>in 120-cell of radius <math>c_8=\sqrt{2}\phi^2</math> |- |<small><math>c_{1,1}</math></small> |<small><math>15.5{}^{\circ}</math></small> |<small><math>\left\{30\right\}</math></small> |<small><math></math></small> |<small><math>\left\{30\right\}</math></small> |<small><math>c_{4,1}-c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7-3 \sqrt{5}}</math></small> |<small><math>0.270091</math></small> |<small><math>\frac{1}{\sqrt{2} \phi ^2}</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^4}}</math></small> |<small><math>\sqrt{0.072949}</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |- |<small><math>c_{2,1}</math></small> |<small><math>25.2{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{2}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{15\right\}</math></small> |<small><math>\frac{1}{2} \left(c_{18,1}-c_{4,1}\right)</math></small> |<small><math>\frac{\sqrt{3-\sqrt{5}}}{2}</math></small> |<small><math>0.437016</math></small> |<small><math>\frac{1}{\sqrt{2} \phi }</math></small> |<small><math>\sqrt{\frac{1}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.190983}</math></small> |<small><math>\phi </math></small> |<small><math>1.61803</math></small> |- |<small><math>c_{3,1}</math></small> |<small><math>36{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{3}\right\}</math></small> |<small><math>\left\{10\right\}</math></small> |<small><math>3 \left\{\frac{10}{3}\right\}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right) c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(\sqrt{5}-1\right)</math></small> |<small><math>0.618034</math></small> |<small><math>\frac{1}{\phi }</math></small> |<small><math>\sqrt{\frac{1}{\phi ^2}}</math></small> |<small><math>\sqrt{0.381966}</math></small> |<small><math>\sqrt{2} \phi </math></small> |<small><math>2.28825</math></small> |- |<small><math>c_{4,1}</math></small> |<small><math>41.4{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{7}\right\}</math></small> |<small><math>\frac{c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>0.707107</math></small> |<small><math>\frac{1}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{1}{2}}</math></small> |<small><math>\sqrt{0.5}</math></small> |<small><math>\phi ^2</math></small> |<small><math>2.61803</math></small> |- |<small><math>c_{5,1}</math></small> |<small><math>44.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{4}\right\}</math></small> |<small><math></math></small> |<small><math>2 \left\{\frac{15}{2}\right\}</math></small> |<small><math>\sqrt{3} c_{2,1}</math></small> |<small><math>\frac{1}{2} \sqrt{9-3 \sqrt{5}}</math></small> |<small><math>0.756934</math></small> |<small><math>\frac{\sqrt{\frac{3}{2}}}{\phi }</math></small> |<small><math>\sqrt{\frac{3}{2 \phi ^2}}</math></small> |<small><math>\sqrt{0.572949}</math></small> |<small><math>\sqrt{3} \phi </math></small> |<small><math>2.80252</math></small> |- |<small><math>c_{6,1}</math></small> |<small><math>49.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{17}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{5-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{5-\sqrt{5}}}{2}</math></small> |<small><math>0.831254</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\frac{1}{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5}}{2 \phi }}</math></small> |<small><math>\sqrt{0.690983}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^3}</math></small> |<small><math>3.07768</math></small> |- |<small><math>c_{7,1}</math></small> |<small><math>56.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{3}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>0.93913</math></small> |<small><math>\frac{\sqrt{\frac{\psi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{0.881966}</math></small> |<small><math>\sqrt{\psi \phi ^3}</math></small> |<small><math>3.47709</math></small> |- |<small><math>c_{8,1}</math></small> |<small><math>60{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{5}\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>\left\{6\right\}</math></small> |<small><math>1</math></small> |<small><math>1</math></small> |<small><math>1.</math></small> |<small><math>1</math></small> |<small><math>\sqrt{1}</math></small> |<small><math>\sqrt{1.}</math></small> |<small><math>\sqrt{2} \phi ^2</math></small> |<small><math>3.70246</math></small> |- |<small><math>c_{9,1}</math></small> |<small><math>66.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{7}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{2 \phi }} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}-\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.09132</math></small> |<small><math>\frac{\sqrt{\frac{\chi }{\phi }}}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\chi }{2 \phi }}</math></small> |<small><math>\sqrt{1.19098}</math></small> |<small><math>\sqrt{\chi \phi ^3}</math></small> |<small><math>4.04057</math></small> |- |<small><math>c_{10,1}</math></small> |<small><math>69.8{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{11}\right\}</math></small> |<small><math>\phi c_{4,1}</math></small> |<small><math>\frac{1+\sqrt{5}}{2 \sqrt{2}}</math></small> |<small><math>1.14412</math></small> |<small><math>\frac{\phi }{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^2}{2}}</math></small> |<small><math>\sqrt{1.30902}</math></small> |<small><math>\phi ^3</math></small> |<small><math>4.23607</math></small> |- |<small><math>c_{11,1}</math></small> |<small><math>72{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{6}\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\left\{5\right\}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{1}{\phi }} c_{8,1}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.17557</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{3-\phi }</math></small> |<small><math>\sqrt{1.38197}</math></small> |<small><math>\sqrt{2} \sqrt{3-\phi } \phi ^2</math></small> |<small><math>4.3525</math></small> |- |<small><math>c_{12,1}</math></small> |<small><math>75.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{24}{5}\right\}</math></small> |<small><math>\sqrt{\frac{3}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>1.22474</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{\frac{3}{2}}</math></small> |<small><math>\sqrt{1.5}</math></small> |<small><math>\sqrt{3} \phi ^2</math></small> |<small><math>4.53457</math></small> |- |<small><math>c_{13,1}</math></small> |<small><math>81.1{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{9-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>1.30038</math></small> |<small><math>\frac{\sqrt{9-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(9-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{1.69098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(9-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>4.8146</math></small> |- |<small><math>c_{14,1}</math></small> |<small><math>84.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{40}{9}\right\}</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi } c_{8,1}}{\sqrt{2}}</math></small> |<small><math>\frac{1}{2} \sqrt[4]{5} \sqrt{1+\sqrt{5}}</math></small> |<small><math>1.345</math></small> |<small><math>\frac{\sqrt[4]{5} \sqrt{\phi }}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\sqrt{5} \phi }{2}}</math></small> |<small><math>\sqrt{1.80902}</math></small> |<small><math>\sqrt[4]{5} \sqrt{\phi ^5}</math></small> |<small><math>4.9798</math></small> |- |<small><math>c_{15,1}</math></small> |<small><math>90.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{7}\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>\left\{4\right\}</math></small> |<small><math>2 c_{4,1}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>1.41421</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2}</math></small> |<small><math>\sqrt{2.}</math></small> |<small><math>2 \phi ^2</math></small> |<small><math>5.23607</math></small> |- |<small><math>c_{16,1}</math></small> |<small><math>95.5{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{29}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>1.4802</math></small> |<small><math>\frac{\sqrt{11-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.19098}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(11-\sqrt{5}\right)} \phi ^2</math></small> |<small><math>5.48037</math></small> |- |<small><math>c_{17,1}</math></small> |<small><math>98.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{31}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>1.51954</math></small> |<small><math>\frac{\sqrt{7+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(7+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.30902}</math></small> |<small><math>\sqrt{\psi \phi ^5}</math></small> |<small><math>5.62605</math></small> |- |<small><math>c_{18,1}</math></small> |<small><math>104.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{8}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{4}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>1.58114</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{\frac{5}{2}}</math></small> |<small><math>\sqrt{2.5}</math></small> |<small><math>\sqrt{5} \sqrt{\phi ^4}</math></small> |<small><math>5.8541</math></small> |- |<small><math>c_{19,1}</math></small> |<small><math>108.0{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{9}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{10}{3}\right\}</math></small> |<small><math>c_{3,1}+c_{8,1}</math></small> |<small><math>\frac{1}{2} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.61803</math></small> |<small><math>\phi </math></small> |<small><math>\sqrt{1+\phi }</math></small> |<small><math>\sqrt{2.61803}</math></small> |<small><math>\sqrt{2} \phi ^3</math></small> |<small><math>5.9907</math></small> |- |<small><math>c_{20,1}</math></small> |<small><math>110.2{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13-\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>1.64042</math></small> |<small><math>\frac{\sqrt{13-\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13-\sqrt{5}\right)}</math></small> |<small><math>\sqrt{2.69098}</math></small> |<small><math>\phi ^2 \sqrt{8-\phi ^2}</math></small> |<small><math>6.07359</math></small> |- |<small><math>c_{21,1}</math></small> |<small><math>113.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{60}{19}\right\}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>1.67601</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{1}{1+\sqrt{5}}}</math></small> |<small><math>\sqrt{2.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\chi }{\phi }}</math></small> |<small><math>6.20537</math></small> |- |<small><math>c_{22,1}</math></small> |<small><math>120{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{10}\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\left\{3\right\}</math></small> |<small><math>\sqrt{3} c_{8,1}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>1.73205</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3}</math></small> |<small><math>\sqrt{3.}</math></small> |<small><math>\sqrt{6} \phi ^2</math></small> |<small><math>6.41285</math></small> |- |<small><math>c_{23,1}</math></small> |<small><math>124.0{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{120}{41}\right\}</math></small> |<small><math>\sqrt{\frac{1}{\phi }+\frac{5}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{5}{2}+\frac{2}{1+\sqrt{5}}}</math></small> |<small><math>1.7658</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{4-\frac{\psi }{2 \phi }}</math></small> |<small><math>\sqrt{3.11803}</math></small> |<small><math>\sqrt{\chi \phi ^5}</math></small> |<small><math>6.53779</math></small> |- |<small><math>c_{24,1}</math></small> |<small><math>130.9{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{20}{7}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{11+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>1.81907</math></small> |<small><math>\frac{\sqrt{11+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(11+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.30902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{\sqrt{5}}{\phi }}</math></small> |<small><math>6.73503</math></small> |- |<small><math>c_{25,1}</math></small> |<small><math>135.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{11}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{1}{2} \sqrt{7+3 \sqrt{5}}</math></small> |<small><math>1.85123</math></small> |<small><math>\frac{\phi ^2}{\sqrt{2}}</math></small> |<small><math>\sqrt{\frac{\phi ^4}{2}}</math></small> |<small><math>\sqrt{3.42705}</math></small> |<small><math>\phi ^4</math></small> |<small><math>6.8541</math></small> |- |<small><math>c_{26,1}</math></small> |<small><math>138.6{}^{\circ}</math></small> |<small><math></math></small> |<small><math></math></small> |<small><math>\left\{\frac{12}{5}\right\}</math></small> |<small><math>\sqrt{\frac{7}{2}} c_{8,1}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>1.87083</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{\frac{7}{2}}</math></small> |<small><math>\sqrt{3.5}</math></small> |<small><math>\sqrt{7} \phi ^2</math></small> |<small><math>6.92667</math></small> |- |<small><math>c_{27,1}</math></small> |<small><math>144{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{12}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{5}{2}\right\}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)} c_{8,1}</math></small> |<small><math>\sqrt{\frac{1}{2} \left(5+\sqrt{5}\right)}</math></small> |<small><math>1.90211</math></small> |<small><math>\sqrt{\phi +2}</math></small> |<small><math>\sqrt{2+\phi }</math></small> |<small><math>\sqrt{3.61803}</math></small> |<small><math>\phi ^2 \sqrt{2 \phi +4}</math></small> |<small><math>7.0425</math></small> |- |<small><math>c_{28,1}</math></small> |<small><math>154.8{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{30}{13}\right\}</math></small> |<small><math>\frac{1}{2} \sqrt{13+\sqrt{5}} c_{8,1}</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>1.95167</math></small> |<small><math>\frac{\sqrt{13+\sqrt{5}}}{2}</math></small> |<small><math>\sqrt{\frac{1}{4} \left(13+\sqrt{5}\right)}</math></small> |<small><math>\sqrt{3.80902}</math></small> |<small><math>\phi ^2 \sqrt{8-\frac{1}{\phi ^2}}</math></small> |<small><math>7.22598</math></small> |- |<small><math>c_{29,1}</math></small> |<small><math>164.5{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{14}\right\}</math></small> |<small><math></math></small> |<small><math>\left\{\frac{15}{7}\right\}</math></small> |<small><math>\phi c_{12,1}</math></small> |<small><math>\frac{1}{2} \sqrt{\frac{3}{2}} \left(1+\sqrt{5}\right)</math></small> |<small><math>1.98168</math></small> |<small><math>\sqrt{\frac{3}{2}} \phi </math></small> |<small><math>\sqrt{\frac{3 \phi ^2}{2}}</math></small> |<small><math>\sqrt{3.92705}</math></small> |<small><math>\sqrt{3} \phi ^3</math></small> |<small><math>7.33708</math></small> |- |<small><math>c_{30,1}</math></small> |<small><math>180{}^{\circ}</math></small> |<small><math>\left\{\frac{30}{15}\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>\left\{2\right\}</math></small> |<small><math>2 c_{8,1}</math></small> |<small><math>2</math></small> |<small><math>2.</math></small> |<small><math>2</math></small> |<small><math>\sqrt{4}</math></small> |<small><math>\sqrt{4.}</math></small> |<small><math>2 \sqrt{2} \phi ^2</math></small> |<small><math>7.40492</math></small> |- |rowspan=4 colspan=6| |rowspan=4 colspan=4| <small><math>\phi</math></small> is the golden ratio:<br> <small><math>\phi ^2-\phi -1=0</math></small><br> <small><math>\frac{1}{\phi }+1=\phi</math></small>, and: <small><math>\phi+1=\phi^2</math></small><br> <small><math>\frac{1}{\phi }::1::\phi ::\phi ^2</math></small><br> <small><math>1/\phi</math></small> and <small><math>\phi</math></small> are the golden sections of <small><math>\sqrt{5}</math></small>:<br> <small><math>\phi +\frac{1}{\phi }=\sqrt{5}</math></small> |colspan=2|<small><math>\phi = (\sqrt{5} + 1)/2</math></small> |<small><math>1.618034</math></small> |- |colspan=2|<small><math>\chi = (3\sqrt{5} + 1)/2</math></small> |<small><math>3.854102</math></small> |- |colspan=2|<small><math>\psi = (3\sqrt{5} - 1)/2</math></small> |<small><math>2.854102</math></small> |- |colspan=2|<small><math>\psi = 11/\chi = 22/(3\sqrt{5} + 1)</math></small> |<small><math>2.854102</math></small> |} == The 5-cell 4-simplex == ... == The 16-cell 4-orthoplex == In 2-space we have the regular 8-point octagon, in 3-space the regular 8-point cube, and in 4-space the regular 8-point [[16-cell]]. A planar octagon with rigid edges of unit length has chords of length: :<math>r_1=1,r_2=\sqrt{2+\sqrt{2}} \approx 1.848,r_3=\sqrt{2}+1 \approx 2.414,r_4=\sqrt{4 + \sqrt{8}} \approx 2.613</math> The chord ratio <math>r_3=\sqrt{2}+1</math> is a geometrical proportion, the [[W:Silver ratio|silver ratio]]. Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_3-r_1-r_1=1/r_3 \approx 0.414</math> Note that <math>r_3-2=1/r_3=\sqrt{2}-1</math>. The procedure rotates counterclockwise over three <math>r_3</math> chords of an {8/3} octagram. Over the first <math>r_3</math> chord the displacement is <math>\sqrt{2}+r_1</math>. Over the second <math>r_3</math> chord it moves in the opposite direction a distance of <math>-r_1</math> . Over the third <math>r_3</math> chord it moves a distance of <math>-r_1</math>. If we embed the planar octagon in 3-space, we can make it skew, repositioning its vertices so that each is one unit-edge length distant from three others instead of two others, at the vertices of a unit-edge cube with chords of length: :<math>r_1=1, r_2=\sqrt{2}, r_3=\sqrt{3}, r_4=\sqrt{2}</math> If we embed this cube in 4-space, we can skew it some more, repositioning its vertices so that each is one unit-edge length distant from six others instead of three others, at the vertices of a unit-edge 4-polytope with chords of length: :<math>r_1=1,r_2=1,r_3=1,r_4=\sqrt{2}</math> All of its chords except its long diameters are the same unit length as its edge. In fact they are its 24 edges, and it is a 16-cell of radius <math>1/\sqrt{2}</math>. [[File:octagon16cell.png|thumb|Orthogonal projection of a regular 16-cell to the [[16-cell#Projections|B₄ Coxeter plane]]. Only its edges are shown; its long diameter chords are not drawn. All 24 edges are the same length and none lie parallel to the projection plane. The octagon circumference is a Petrie polygon. The two disjoint squares lie in completely orthogonal central planes. The blue octagram is a Clifford polygon. ]] The [[16-cell]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{3,3,4\}</math></small>. It has 8 vertices, 24 edges, 32 equilateral triangle faces, and 16 regular tetrahedron cells. It is the [[16-cell#Octahedral dipyramid|four-dimensional analogue of the octahedron]], and each of its four orthogonal central hyperplanes is an octahedron. The only planar regular polygons found in the 16-cell are face triangles and central plane squares, but the 16-cell also contains a skew regular octagon, its [[W:Petrie polygon|Petrie polygon]].{{Efn|name=Petrie polygon of a honeycomb}} The chords of this regular octagon, which lies skew in 4-space, are those given above for the 16-cell, as opposed to those for the cube or the regular octagon in the plane. The 16-cell is a construct of 3 Petrie octagons which share the same 8 vertices but have disjoint sets of 8 edges each. The regular octad has higher symmetry in 4-space than it does in 2-space. The 16-cell is the 4-[[w:Cross-polytope|orthoplex]], the simplest regular 4-polytope after the [[5-cell|4-simplex]]. All the larger regular convex 4-polytopes are compounds of the 16-cell. The regular octagon exhibits this high symmetry only when embedded in 4-space at the vertices of the 16-cell. The 16-cell constitutes an [[W:Orthonormal basis|orthonormal basis]] for the choice of a 4-dimensional Cartesian reference frame, because its vertices define four orthogonal axes. The eight vertices of a unit-radius 16-cell are (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0), (0, 0, 0, ±1). All vertices are connected by <math>\sqrt{2}</math> edges except opposite pairs. The vertex coordinates of the 16-cell form 6 central squares lying in 6 pairwise [[W:Orthogonal|orthogonal]] coordinate planes. Great squares in opposite planes that do not share an axis (e.g. in the ''xy'' and ''wz'' planes) are completely disjoint (they do not intersect at any vertices). These planes are [[W:Completely orthogonal|completely orthogonal]].{{Efn|name=Six orthogonal planes of the Cartesian basis}} Since the unit-radius coordinate system is convenient, let us derive the unit-radius 16-cell by skewing a unit-radius planar octagon, which has chords of length: :<math>r_1=\sqrt{2-\sqrt{2}} \approx 0.765,r_2=\sqrt{2},r_3=\sqrt{2+\sqrt{2}} \approx 1.848,r_4=2</math> We will need a planar octagon with rigid <math>r_2</math> chords, rather than one with rigid <math>r_1</math> edges. The octagon's <math>r_2</math> chords form two disjoint great squares, visible in the orthogonal projection, which we can reposition in 3-space to form a cube by making them parallel, and in 4-space to form a 16-cell by making them completely orthogonal. Since the edges of the 16-cell are all the same length <math>r_1=\sqrt{2},r_2=\sqrt{2},r_3=\sqrt{2}</math>, those chords are distinct only in the context of a rotation. Each chord is a 4-vector with a length and a direction. The rotational curve over each <math>r_i</math> chord makes <math>i</math> 45° turns. [[File:16-cell-orig.gif|thumb|Orthographic projection of the 8-point 16-cell <small><math>\{3,3,4\}</math></small> performing a double rotation.{{Sfn|Hise|2007}}]] [[W:Rotations in 4-dimensional Euclidean space|Rotations in 4-dimensional Euclidean space]] can be seen as the composition of two 2-dimensional rotations in completely orthogonal planes. The general rotation in 4-space is a [[W:SO(4)#Double rotations|double rotation]] in pairs of completely orthogonal planes. Two completely orthogonal planes are called invariant planes of the rotation when all points in the plane rotate on circles that remain in the plane, even as the whole plane tilts sideways (like a coin flipping) into another plane. The two completely orthogonal rotations of each plane (like a wheel, and like a coin flipping) are simultaneous but independent, in that they are not geometrically constrained to turn at the same rate. However, the most circular kind of rotation (as opposed to an elliptical double rotation of a rigid spherical object) occurs when the completely orthogonal planes do rotate through the same angle in the same time interval. Such equi-angled double rotations are called [[w:SO(4)#Isoclinic_rotations|isoclinic]], also [[w:William_Kingdon_Clifford|Clifford]] displacements. The <math>r_1</math> chords of the 16-cell form a Petrie polygon which zig-zags back and forth, in the left and right rotational directions, between two completely orthogonal great squares formed by <math>r_2</math> chords. The <math>r_2</math> chords of two completely orthogonal great squares lie parallel and perpendicular to each other. A ''simple'' rotation of the 16-cell in ''one'' of those two square central planes rotates that square like a wheel, while the other square does not move.{{Efn|name=simple rotations}} The four vertices of the rotating square orbit on a great circle in the plane. The <math>r_3</math> chords of the 16-cell form a circular helix, visible as a blue {8/3} octagram in the orthogonal projection. A ''double'' rotation of the 16-cell, in both of two completely orthogonal invariant <math>r_2</math> square planes at once by equal angles, moves the eight vertices along the circular helix over the <math>r_3</math> chords. The vertex motion is a [[w:Geodesic|geodesic]] circle orbit on the 3-sphere of a special kind: it does not lie in a central plane, its [[w:Winding_number|winding number]] is not 1 (it is 3 in this case), its circumference is not <math>2\pi</math>, and it moves in either a left or right handed circular spiral. We shall refer to such a chiral circle orbit as an ''isocline'', and to the skew polygram of its rotational chords as a ''Clifford polygon''. The 16-cell is the simplest possible frame in which to [[16-cell#Rotations|observe 4-dimensional rotations]] because its characteristic rotations feature a single pair of invariant rotation planes. In the 16-cell an isoclinic rotation by 90° in any pair of invariant completely orthogonal square central planes takes every great square to its completely orthogonal great square in a twisting displacement, as the invariant planes tilt sideways 90° into each other's plane while rotating 90° internally. All the vertices move at once along the same circular helix geodesic isocline of <math>r_3</math> chords, displaced 90° in 8 orthogonal directions, and the rigid 16-cell assumes a new orientation in 4-space. When the 90° isoclinic rotation is continued in the same rotational direction through an additional 90°, each vertex is again displaced 90°, but from the new orientation in a direction orthogonal to its first 90° displacement. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. In 360° of isoclinic rotation over four <math>r_3</math> chords, each vertex makes six 90° turns and reaches its antipodal position. The trajectory of each vertex over each 90° isoclinic rotational displacement is a one-eighth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>6\pi</math> over eight <math>r_3</math> chords, and also traces an ordinary great circle in the plane twice, over the four <math>r_2</math> edges of a great square in one of the two moving invariant rotation planes. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions just once and returns to its original position, and the 16-cell returns to its original orientation. Because this is the isoclinic rotation of the 16-cell in its invariant edge planes we shall refer to it as the ''characteristic rotation of the 16-cell'', and note once again that it is Fontaine and Hurley's rotation over the <math>r_3</math> star polygon which constructs <math>1/r_3</math>. == The 8-cell tesseract == The long diameter of the unit-edge [[W:Hypercube|hypercube]] of dimension <math>n</math> is <math>\sqrt{n}</math>, so the unit-edge [[w:Tesseract|4-hypercube, the 16-point (8-cell) tesseract,]] has chords: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> Uniquely in its 4-dimensional case, the hypercube's edge length equals its radius, like the hexagon. We call such polytopes ''radially equilateral'', because they can be constructed from equilateral triangles which meet at their center, each contributing two radii and an edge. The [[w:Cuboctahedron|cuboctahedron]] and the 24-cell are also radially equilateral. [[File:8-cell.gif|thumb|Orthographic projection of the 16-point (8-cell) tesseract <small><math>\{4,3,3\}</math></small> performing a simple rotation about a plane in 4-space.{{Sfn|Hise|2007}} The stationary plane bisects the figure from front-left to back-right and top to bottom.]] The [[W:Tesseract|tesseract]] is the [[W:Regular convex 4-polytope|regular convex 4-polytope]] with [[W:Schläfli symbol|Schläfli symbol]] <small><math>\{4,3,3\}</math></small>. It has 16 vertices, 32 edges, 24 square faces, and 8 cube cells. It is the four-dimensional analogue of the cube. The 16-point tesseract is the convex hull of a compound of two 8-point 16-cells, in exact dimensional analogy to the way the 8-point cube is the convex hull of a [[W:Stellated octahedron|compound of two 4-point regular tetrahedra]]. The [[W:Demihypercube|demihypercubes]] occupy alternate vertices of the hypercubes. The diagonals of the square faces of the unit-edge, unit-radius tesseract are the <math>\sqrt{2}</math> edges of two unit-radius 16-cells, also the edges of the square central planes. We can rotate the tesseract isoclinically the way we rotated the 16-cell, by 90° in completely orthogonal invariant square central planes, with the same effect on both alternate-position 16-cells. In the course of a 720° isoclinic rotation in invariant square central planes each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cell. The two skew {8/3} octagram Clifford polygons lie on two disjoint parallel isoclines of the same chirality, of circumference <math>6\pi</math> over <math>\sqrt{2}</math> chords. They form a circular double helix which intersects each vertex of the tesseract once. The tesseract is the [[W:Dual polytope|dual polytope]] of the 16-cell. They have the same Petrie polygon, the regular skew octagon, but the tesseract is a construct of 4 Petrie octagons with disjoint sets of 8 tesseract edges each. We can construct the tesseract by skewing two planar octagons. Because the tesseract is radially equilateral (unlike the 16-cell), we use two octagons of unit-edge length to build the unit-radius tesseract. To start we embed the planar octagons in 4-space at the same point and make them completely orthogonal. Then we skew each planar octagon into a cube, so we have a compound of two completely orthogonal cubes, provided we skewed them both in the same direction. The 16 vertices will be the vertices of a tesseract with half its 32 edges missing. Because the tesseract contains two 16-cells in alternate positions it has two sets of 6 orthogonal square central planes. Two angles are required to specify the relationship between two planes in 4-space. Pairs of square central planes within each 16-cell are 90° apart in one angle, and either 0° or 90° apart in the other angle. They are 90° apart in both angles if and only if they are completely orthogonal planes, 90° apart by isoclinic rotation, with no vertices in common. Otherwise they are 0° apart in one of the angles, 90° apart by simple rotation, and they intersect in one axis and lie in a common 3-dimensional hyperplane.{{Efn|A double rotation in which one of the two angles of rotation is 0°, so that one of the completely orthogonal invariant planes does not rotate, is called a simple rotation. Ordinary rotations observed in a 3-dimensional space are simple rotations.|name=simple rotations}} A pair of square central planes from alternate-position 16-cells are 60° apart by isoclinic rotation, with their corresponding vertices 120° apart. The planes are not orthogonal or parallel, so they intersect in a line somewhere, but they have no vertices in common, they have no 3-dimensional hyperplane in common, and they cannot reach each other by simple rotation. Such pairs of objects are called [[W:Clifford parallel|Clifford parallel]] because all their corresponding pairs of vertices are the same distance apart, although they are not parallel in the usual sense, because they have a common center. Not only the alternate-position 16-cells' corresponding square central planes, but also the 16-cells themselves, are Clifford parallel objects. More generally, multiple disjoint instances of a 4-polytope which compound to make a larger 4-polytope are Clifford parallel objects. == The 24-cell == [[File:24-cell vertex geometry.png|thumb|Planar geometry of the radially equilateral 24-cell, showing its 3 great circle polygons and its 4 chord lengths.]] In 2-space we have the radially equilateral 6-point hexagon. In 3-space we have the radially equilateral 12-point cuboctahedron, with 4 hexagonal central planes. In 4-space we have the radially equilateral 24-point 24-cell, with 12 cuboctahedron central hyperplanes and 16 hexagonal central planes. The [[24-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,4,3\}</math></small>. It has 24 vertices, 96 edges, 96 equilateral triangle faces, and 24 octahedron cells. It is the four-dimensional analogue of the cuboctahedron. The 24-cell has the same chord set as the 4-hypercube tesseract: :<math>r_1=\sqrt{1},r_2=\sqrt{2},r_3=\sqrt{3},r_4=\sqrt{4}</math> [[Image:24-cell.gif|thumb|Orthographic projection of the 24-point 24-cell <small><math>\{3,4,3\}</math></small> performing a simple rotation.{{Sfn|Hise|2007}} The 3-dimensional surface made of 24 octahedra is visible.]] The 24-cell is [[W:Dual polytope|self-dual]], like the regular polygons and regular simplexes. It is the maximal regular construct of triangles and squares (with no pentagons). It is the convex hull of a compound of three disjoint 8-point 16-cells, rotated 60° isoclinically with respect to each other. Each of the three pairs of 16-cells is a tesseract. Each 24-cell edge is also a tesseract edge. The corresponding vertices of two 16-cells or two tesseracts are 120° apart by a <math>\sqrt{3}</math> chord. Each tesseract has 8 cube cells, and each cube has four <math>\sqrt{3}</math> long diameters. The <math>\sqrt{3}</math> chords joining the corresponding vertices of two tesseracts belong to the third tesseract as cell long diameters. The 24-cell's Petrie polygon is the regular dodecagon {12}, which has chords: :<math>r_1=\tfrac{\sqrt{3}-1}{\sqrt{2}} \approx 0.518,r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\tfrac{\sqrt{3}+1}{\sqrt{2}} \approx 1.932,r_6=\sqrt{4}</math> Fontaine and Hurley's procedure for obtaining the reciprocal of a chord tells us that: :<math>r_5-r_3+r_1+r_1-r_3=1/r_5</math> when <math>r_1=1</math>. The procedure rotates counterclockwise over five <math>r_5</math> chords of a {12/5} dodecagram. In the system of unit-radius coordinates <math>r_1=1/r_5</math>. The <math>r_1</math> and <math>r_5</math> chords of the planar dodecagon do not occur in the 24-cell, which is a construct of eight skew dodecagons with disjoint sets of twelve <math>\sqrt{1}</math> edges each. In the skew dodecagons the chord lengths are: :<math>r_1=\sqrt{1},r_2=\sqrt{1},r_3=\sqrt{2},r_4=\sqrt{3},r_5=\sqrt{3},r_6=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. [[File:dodecagon24cell.png|thumb|Orthogonal projection of half a 24-cell to the [[24-cell#Geodesics|F₄ Coxeter plane]]. Only one Petrie dodecagon {12} of the 24-cell is shown. In a unit-radius 24-cell, all black lines are 24-cell edges of unit length, also tesseract edges. The two disjoint hexagons lie in Clifford parallel central planes. Blue chords are <math>\sqrt{2}</math> 16-cell edges, also isocline chords in square rotations. Green chords are <math>\sqrt{3}</math> distances between corresponding vertices of two 16-cells, also isocline chords in hexagonal rotations. Note the {12/5} dodecagram.]] [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_3=\sqrt{2}</math></small>]] We can rotate the 24-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in completely orthogonal invariant great square planes, with the same effect on all three 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, but it does not visit the vertex positions of the other 16-cells. The <math>r_3=\sqrt{2}</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Three Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_3</math> chords form a circular triple helix {24/9}=3{8/3} that intersects each 24-cell vertex once. [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_5=\sqrt{3}</math></small> ]] We can also rotate the 24-cell isoclinically in 4 Clifford parallel invariant great hexagon planes containing its vertices, over <math>r_{5}=\sqrt{3}</math> isocline chords. This is the ''characteristic rotation of the 24-cell'' in its invariant edge planes, also Fontaine and Hurley's rotation over the <math>r_5</math> star polygon which constructs <math>1/r_5</math>. A complete hexagonal isoclinic revolution requires 720° like a complete square isoclinic revolution, but it is completed in 12 isoclinic displacements of 60° each rather than 8 isoclinic displacements of 90° each. The rotational curve over each 120° <math>r_5</math> chord makes five 30° turns. Two Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_5</math> chords form a circular double helix {24/10}=2{12/5} that intersects each 24-cell vertex once. In the 24-cell the characteristic isoclinic rotation by 60° in any invariant hexagon central plane takes every great hexagon to a Clifford parallel great hexagon in a twisting displacement, as all the central planes tilt sideways 60° while rotating 60° internally. It also takes every great square to a Clifford parallel great square in another 16-cell; it takes every 16-cell to another 16-cell. The 16-cells revolve within the 24-cell as well as rotating within it. All 24 vertices move at once on two Clifford parallel geodesic isoclines, displaced 120° in different directions. The trajectory of each vertex over each 60° isoclinic rotational displacement is a one-twelfth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>10\pi</math> over twelve <math>\sqrt{3}</math> chords, and also traces an ordinary great circle in the plane twice, over the six <math>\sqrt{1}</math> edges of a great hexagon in a moving invariant rotation plane. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions just once and returns to its original position, and the 24-cell returns to its original orientation. == The 600-cell == [[Image:600-cell.gif|thumb|Orthographic projection of the 120-point 600-cell <small><math>\{3,3,5\}</math></small> performing a simple rotation.{{Sfn|Hise|2011}} The 3-dimensional surface made of 600 tetrahedra is visible. Invisible in this rendering are 25 inscribed instances of the 24-cell (above), which occur in the 600-cell as interior boundary envelopes.]] The [[600-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{3,3,5\}</math></small>. It has 120 vertices, 720 edges, 1200 equilateral triangle faces, and 600 tetrahedron cells. It is the four-dimensional analogue of the icosahedron. The 600-cell rounds out the 24-cell by adding 96 more vertices (four more disjoint 24-cells) between the 24-cell's existing 24 vertices, in effect adding twenty-four more distinct 24-cells inscribed in the 600-cell. The new surface thus formed is a honeycomb of smaller, more numerous cells: tetrahedra of edge length <math>\phi^{-1} \approx 0.618</math> instead of octahedra of edge length <math>\sqrt{1}</math>. It encloses the <math>\sqrt{1}</math> edges of the 24-cells, which become invisible interior chords in the 600-cell, like the <math>\sqrt{2}</math> and <math>\sqrt{3}</math> chords. Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of <math>\phi^{-1}</math>, the inverse golden ratio), the 600-cell is not radially equilateral like the 24-cell and the tesseract. Like them it is radially triangular in a special way, but one in which [[w:Golden_triangle_(mathematics)|golden triangles]] rather than equilateral triangles meet at the center. In 2-space we have the ''radially golden'' [[W:Decagon#The golden ratio in decagon|regular decagon]]. In 3-space we have the radially golden 30-point [[W:icosidodecahedron|icosidodecahedron]], with 6 decagon central planes. In 4-space we have the radially golden 120-point 600-cell, with 60 icosidodecahedron central hyperplanes and 72 decagon central planes. The 600-cell's Petrie polygon is the regular [[w:Triacontagon|triacontagon {30}]]. The unit-radius planar {30}-gon has these distinct chords: :<math>r_1=2 \sin (\tfrac{\pi}{15}/2) \approx 0.209</math> :<math>r_2=2 \sin (\tfrac{2\pi}{15}/2) \approx 0.416</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{4\pi}{15}/2) \approx 0.813</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{7\pi}{15}/2) \approx 1.338</math> :<math>r_8=2 \cos (\tfrac{7\pi}{15}/2) \approx 1.486</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{11}=2 \cos (\tfrac{4\pi}{15}/2) \approx 1.827</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \cos (\tfrac{2\pi}{15}/2) \approx 1.956</math> :<math>r_{14}=2 \cos (\tfrac{\pi}{15}/2) \approx 1.989</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Only the chord lengths <math>r_3</math>, <math>r_5</math>, <math>r_6</math>, <math>\sqrt{2}</math>, <math>r_9</math>, <math>r_{10}</math>, <math>r_{12}</math>, <math>r_{15}</math> occur in the 600-cell, which is a construct of 24 Petrie {30}-gons of edge length <math>r_3</math>, six of which intersect in each icosahedral vertex figure. In the skew {30}-gons the chord lengths are: [[File:600-cell vertex geometry.png|thumb|Planar geometry of the 600-cell, showing its 5 regular great circle polygons and its 8 chord lengths with angles of arc. The golden ratio governs the fractional roots of every other chord, and the radial golden triangles which meet at the center.|400x400px]] :<math>r_1=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_2=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_3=2 \sin (\tfrac{\pi}{5}/2)=\phi^{-1} \approx 0.618</math> :<math>r_4=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}</math> :<math>r_5=2 \sin (\tfrac{\pi}{3}/2)=\sqrt{1}=\text{24-cell-}r_2</math> :<math>r_6=2 \sin (\tfrac{2\pi}{5}/2)=\sqrt{3-\phi} \approx 1.176</math> :<math>r_7=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}</math> :<math>r_8=2 \sin (\tfrac{\pi}{2}/2)=\sqrt{2}=\text{16-cell-}r_3</math> :<math>r_9=2 \sin (\tfrac{3\pi}{5}/2)=\phi \approx 1.618</math> :<math>r_{10}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}=\text{24-cell-}r_5</math> :<math>r_{11}=2 \sin (\tfrac{2\pi}{3}/2)=\sqrt{3}</math> :<math>r_{12}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{13}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{14}=2 \sin (\tfrac{4\pi}{5}/2)=\sqrt{2+\phi} \approx 1.902</math> :<math>r_{15}=2 \sin (\pi/2)=\sqrt{4}</math> Where chords are the same length, they are distinct only in the context of a rotation. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="7" |15 chords (4 distinct 180° pairs) make 4 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>r_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>r_{15}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>r_1</math> |36° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14}=2{15/7} |144° | rowspan="3" |<math>r_{14}</math> |- style="background: palegreen;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: palegreen;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>r_2</math> |36° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |144° | rowspan="3" |<math>r_{13}</math> |- style="background: gainsboro;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: gainsboro;" | |0.618~ |1.902~ |- style="background: yellow;" | | rowspan="3" |<math>r_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" |[[File:V1 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>r_{12}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: palegreen;" | | rowspan="3" |<math>r_4</math> |60° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |120° | rowspan="3" |<math>r_{11}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: palegreen;" | | rowspan="3" |<math>r_5</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" |[[File:V2 dodecahedron.png|100px]]<br>Dodecahedron | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>r_{10}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: yellow;" | | rowspan="3" |<math>r_{6}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" |[[File:V3 icosahedron.png|100px]]<br>Icosahedron | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>r_{9}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>r_{8}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: seashell;" | | rowspan="3" |<math>r_{7}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" |[[File:V4 icosidodecahedron.png|100px]]<br>Icosidodecahedron | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>r_{8}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The list of 600-cell chords <math>r_{i}</math> can be rearranged into a table of 8 rows and 2 columns with a pair of 180° complements in each row. The short chord and long chord each have their characteristic {30}-gon. Each row identifies a discrete isoclinic rotation of the 600-cell in invariant central planes containing the vertices of the short chord {30}-gon, over the isocline chords of the long chord {30}-gon, the rotation's Clifford polygon. Each distinct pair of complementary chord lengths is identified with a distinct [[w:600-cell#Polyhedral sections|polyhedral section of the 600-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 7 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>\phi^{-1}</math> is a icosahedron vertex figure, and the largest section at radial distance <math>\sqrt{2}</math> is an [[W:Icosidodecahedron|icosidodecahedron]] central section bisecting the 600-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>\sqrt{2}</math> the successive complement-radius polyhedra decrease in size, to the antipodal icosahedron vertex figure at distance <math>\sqrt{2+\phi}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 7 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). [[File:Regular_star_figure_3(8,3).svg|thumb|left|150px|{24/9}=3{8/3} <small><math>r_8=\sqrt{2}</math></small>]] We can rotate the 600-cell isoclinically in the characteristic rotation of the 16-cell, by 90° in two completely orthogonal invariant great square planes over <math>r_8=\sqrt{2}</math> isocline chords, with the same effect on 15 disjoint 16-cells. In the course of a 720° isoclinic rotation each vertex departs from all 8 vertex positions of its 16-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_8</math> chord is the 16-cell <math>r_3</math> chord. The rotational curve over each 90° <math>r_3</math> chord makes three 45° turns. Fifteen Clifford parallel {8/3} octagram geodesic isoclines of circumference <math>6\pi</math> over <math>r_8</math> chords form a circular helix of 15 twisted parallel strands 5{24/9}=15{8/3} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_polygon_30-7.svg|thumb|left|150px|{30/7} <small><math>r_7=\sqrt{2}</math></small>]] In the 600-cell there is another distinct 90° isoclinic rotation, over <math>r_7=\sqrt{2}</math> isocline chords. This rotation has period 30 and visits every vertex of a 600-cell Petrie polygon. Each 90° isoclinic rotational displacement takes every great square plane to a great square plane in another 16-cell. The invariant completely orthogonal central planes of this rotation each intersect only one vertex of the 600-cell, which makes seven orbits on a great circle within the moving invariant plane in the course of one complete isoclinic revolution. The rotational curve over each 90° <math>r_7</math> isocline chord makes seven 12° turns. Four Clifford parallel {30/7} geodesic isoclines of circumference <math>14\pi</math> over <math>r_7</math> chords form a circular quadruple helix that intersects each 600-cell vertex once. {{Clear}} [[File:Regular star figure 2(12,5).svg|thumb|left|150px|{24/10}=2{12/5} <small><math>r_{10}=\sqrt{3}</math></small> ]] We can also rotate the 600-cell isoclinically in the characteristic rotation of the 24-cell, by 60° in great hexagon planes over <math>r_{10}=\sqrt{3}</math> isocline chords, with the same effect on 5 disjoint 24-cells. In the course of a 720° isoclinic rotation each vertex departs from 12 vertex positions of its 24-cell just once and returns to its original position, without visiting other vertex positions. The <math>r_{10}</math> chord is the 24-cell <math>r_5</math> chord. The rotational curve over each 60° <math>r_5</math> chord makes five 30° turns. Ten Clifford parallel {12/5} dodecagram geodesic isoclines of circumference <math>10\pi</math> over <math>r_{10}</math> chords form a circular helix of 10 twisted parallel strands 5{24/10}=10{12/5} that intersects each 600-cell vertex once. {{Clear}} [[File:Regular_star_figure_2(15,4).svg|thumb|left|150px|{30/8}=2{15/4} <small><math>r_{13}=\sqrt{1}</math></small>]] We can also rotate the 600-cell isoclinically in 12 Clifford parallel invariant decagon central planes containing its <math>r_{3}</math> edges, over <math>r_{13}=\sqrt{1}</math> isocline chords. This is the ''characteristic rotation of the 600-cell'' in its invariant edge planes. Its Clifford polygon is a skew {15/4} pentadecagram of <math>r_{13}</math> chords. The <math>r_{4}</math> chord is the 24-cell <math>r_2</math> chord. Successive <math>r_{13}</math> chords are edges of different 24-cells. The rotational curve over each <math>r_{13}</math> chord makes two 30° turns. Eight Clifford parallel {15/4} pentadecagon geodesic isoclines of circumference <math>5\pi</math> over <math>r_{13}</math> chords form a circular helix of eight twisted parallel strands 4{30/8}=8{15/4} that intersects each 600-cell vertex once. In the 600-cell the characteristic isoclinic rotation by 36° in any invariant decagon central plane takes every great decagon to a Clifford parallel great decagon in a twisting displacement, as all the central planes tilt sideways 36° while rotating 36° internally. It also takes every great hexagon to a Clifford parallel great hexagon in another 24-cell, and every great square to a Clifford parallel great square in another 16-cell; it takes 24-cells to a non-disjoint 24-cell and 16-cells to a 16-cell in another 24-cell. The 24-cells revolve within the 600-cell, as the 16-cells revolve within the 24-cells. All 120 vertices move at once on eight Clifford parallel geodesic isoclines, displaced 60° in different directions. The trajectory of each vertex over each 36° isoclinic rotational displacement is a one-fifteenth segment of its geodesic orbit. Its entire orbit traces an isocline circle in 4-space of circumference <math>5\pi</math> over 15 <math>r_5</math> chords, and also traces an ordinary great circle in the plane 3 times, over the 5 edges of a great pentagon in a moving invariant rotation plane. In the course of a complete isoclinic revolution each vertex departs from 15 vertex positions just once and returns to its original position, and the 600-cell returns to its original orientation. {{Clear}} [[File:Regular_star_figure_6(5,2).svg|thumb|left|150px|{30/12}=6{5/2} <small><math>r_{12}=\sqrt{3.618\sim}</math></small>]] In the 600-cell there is another distinct isoclinic rotation taking decagon planes to each other, over 144° <math>r_{12}</math> isocline chords. It also takes disjoint 24-cells to each other. This rotation has period 5 and visits every 12th vertex of a 600-cell Petrie polygon. Its Clifford polygon is a skew {5/2} pentagram of <math>r_{12}</math> chords. The invariant central planes of this rotation each intersect only one vertex of the 600-cell, which makes two orbits of a great pentagon within the moving invariant plane in the course of one complete isoclinic revolution of period 5. The rotational curve over each <math>r_{12}</math> chord makes twelve 12° turns. 24 Clifford parallel {5/2} pentagram geodesic isoclines of circumference <math>4\pi</math> over five <math>r_{12}</math> chords form a circular helix of 24 twisted parallel strands 4{30/12}=24{5/2} that intersects each 600-cell vertex once. {{Clear}} == Finally the 120-cell == The [[120-cell]] is the regular convex 4-polytope with Schläfli symbol <small><math>\{5,3,3\}</math></small>. It has 600 vertices, 1200 edges, 720 pentagon faces, and 120 dodecahedron cells. It is the four-dimensional analogue of the dodecahedron. The 120-cell is the [[W:Dual polytope|dual polytope]] of the 600-cell. They have the same Petrie polygon, the regular skew triacontagon {30}, but the 120-cell is a construct of 40 Petrie {30}-gons of edge length <math>c_1</math>, two of which intersect in each tetrahedral vertex figure. {| class="wikitable floatright" style="white-space:nowrap;text-align:center" ! colspan="9" |30 chords (15 180° pairs) make 15 distinct section polyhedra |- ! colspan="3" |Short chord ! Section ! colspan="3" |Long chord |- style="background: palegreen;" | | rowspan="3" |<math>c_0</math> |0° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_15(2,1).svg|100px]]<br>{30/15}=15{2} |180° | rowspan="3" |<math>c_{30}</math> |- style="background: palegreen;" | |{{radic|0}} |{{radic|4}} |- style="background: palegreen;" | |0 |2 |- style="background: palegreen;" | | rowspan="3" |<math>c_1</math> |15.5~° | rowspan="3" |[[File:Regular_polygon_30.svg|100px]]<br>{30/1} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,7).svg|100px]]<br>{30/14} |164.5~° | rowspan="3" |<math>c_{29}</math> |- style="background: palegreen;" | |{{radic|0.073~}} |{{radic|3.927~}} |- style="background: palegreen;" | |0.270~ |1.982~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_2</math> |25.2~° | rowspan="3" |[[File:Regular_star_figure_2(15,1).svg|100px]]<br>{30/2}=2{15} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-13.svg|100px]]<br>{30/13} |154.8~° | rowspan="3" |<math>c_{28}</math> |- style="background: gainsboro;" | |{{radic|0.191~}} |{{radic|3.809~}} |- style="background: gainsboro;" | |0.437~ |1.952~ |- style="background: yellow;" | | rowspan="3" |<math>c_3</math> |36° | rowspan="3" |[[File:Regular_star_figure_3(10,1).svg|100px]]<br>{30/3}=3{10} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_6(5,2).svg|100px]]<br>{30/12}=6{5/2} |144° | rowspan="3" |<math>c_{27}</math> |- style="background: yellow;" | |{{radic|0.382~}} |{{radic|3.618~}} |- style="background: yellow;" | |0.618~ |1.902~ |- style="background: gainsboro;" | | rowspan="3" |<math>c_4</math> |41.4~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |138.6~° | rowspan="3" |<math>c_{26}</math> |- style="background: gainsboro;" | |{{radic|0.5}} |{{radic|3.5}} |- style="background: gainsboro;" | |0.707~ |1.871~ |- style="background: palegreen;" | | rowspan="3" |<math>c_5</math> |44.5~° | rowspan="3" |[[File:Regular_star_figure_2(15,2).svg|100px]]<br>{30/4}=2{15/2} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-11.svg|100px]]<br>{30/11} |135.5~° | rowspan="3" |<math>c_{25}</math> |- style="background: palegreen;" | |{{radic|0.573~}} |{{radic|3.427~}} |- style="background: palegreen;" | |0.757~ |1.851~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_6</math> |49.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |130.9~° | rowspan="3" |<math>c_{24}</math> |- style="background: gainsboro;" | |{{radic|0.691~}} |{{radic|3.309~}} |- style="background: gainsboro;" | |0.831~ |1.819~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_7</math> |56° | rowspan="3" | | rowspan="3" | | rowspan="3" | |124° | rowspan="3" |<math>c_{23}</math> |- style="background: gainsboro;" | |{{radic|0.882~}} |{{radic|3.118~}} |- style="background: gainsboro;" | |0.939~ |1.766~ |- style="background: palegreen;" | | rowspan="3" |<math>c_8</math> |60° | rowspan="3" |[[File:Regular_star_figure_5(6,1).svg|100px]]<br>{30/5}=5{6} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_10(3,1).svg|100px]]<br>{30/10}=10{3} |120° | rowspan="3" |<math>c_{22}</math> |- style="background: palegreen;" | |{{radic|1}} |{{radic|3}} |- style="background: palegreen;" | |1 |1.732~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_9</math> |66.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |113.9~° | rowspan="3" |<math>c_{21}</math> |- style="background: gainsboro;" | |{{radic|1.191~}} |{{radic|2.809~}} |- style="background: gainsboro;" | |1.091~ |1.676~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{10}</math> |69.8~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |110.2~° | rowspan="3" |<math>c_{20}</math> |- style="background: gainsboro;" | |{{radic|1.309~}} |{{radic|2.691~}} |- style="background: gainsboro;" | |1.144~ |1.640~ |- style="background: yellow;" | | rowspan="3" |<math>c_{11}</math> |72° | rowspan="3" |[[File:Regular_star_figure_6(5,1).svg|100px]]<br>{30/6}=6{5} | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_3(10,3).svg|100px]]<br>{30/9}=3{10/3} |108° | rowspan="3" |<math>c_{19}</math> |- style="background: yellow;" | |{{radic|1.382~}} |{{radic|2.618~}} |- style="background: yellow;" | |1.176~ |1.618~ |- style="background: palegreen; height:50px" | | rowspan="3" |<math>c_{12}</math> |75.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" |[[File:Regular_star_figure_2(15,4).svg|100px]]<br>{30/8}=2{15/4} |104.5~° | rowspan="3" |<math>c_{18}</math> |- style="background: palegreen;" | |{{radic|1.5}} |{{radic|2.5}} |- style="background: palegreen;" | |1.224~ |1.581~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{13}</math> |81.1~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |98.9~° | rowspan="3" |<math>c_{17}</math> |- style="background: gainsboro;" | |{{radic|1.691~}} |{{radic|2.309~}} |- style="background: gainsboro;" | |1.300~ |1.520~ |- style="background: gainsboro; height:50px" | | rowspan="3" |<math>c_{14}</math> |84.5~° | rowspan="3" | | rowspan="3" | | rowspan="3" | |95.5~° | rowspan="3" |<math>c_{16}</math> |- style="background: gainsboro;" | |{{radic|0.809~}} |{{radic|2.191~}} |- style="background: gainsboro;" | |1.345~ |1.480~ |- style="background: seashell;" | | rowspan="3" |<math>c_{15}</math> |90° | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} | rowspan="3" | | rowspan="3" |[[File:Regular_star_polygon_30-7.svg|100px]]<br>{30/7} |90° | rowspan="3" |<math>c_{15}</math> |- style="background: seashell;" | |{{radic|2}} |{{radic|2}} |- style="background: seashell;" | |1.414~ |1.414~ |} The [[User:Dc.samizdat/Golden chords of the 120-cell#Thirty distinguished distances|table above]] of 30 chords <math>c_{t}</math> can be rearranged into a table of 16 rows and 2 columns with a pair of 180° complements in each row. This table first appears in [[w:Regular_Polytopes_(book)|''Regular Polytopes'']] (1947),{{Sfn|Coxeter|1973|loc=Table V(v): Simplified sections of {5,3,3} beginning with a vertex|pp=300-301}} where Coxeter identified each row with a distinct [[w:120-cell#Concentric_hulls|polyhedral section of the 120-cell]] beginning with a vertex. In spherical [[w:3-sphere|3-dimensional space <math>\mathbb{S}^3</math>]], every vertex is the center of a set of 29 concentric polyhedra of increasing radii that nest like [[w:Matryoshka_doll|Russian dolls.]] The smallest polyhedral section at radial distance <math>c_1</math> is a tetrahedron vertex figure, and the largest section at radial distance <math>c_{15}</math> is a central section bisecting the 120-cell. Because [[w:3-sphere|<math>\mathbb{S}^3</math>]] is spherical, at radial distances greater than <math>c_{15}</math> the successive complement-radius polyhedra decrease in size, to the antipodal tetrahedron vertex figure at distance <math>c_{29}</math>. In Euclidean 4-dimensional space <math>\mathbb{R}^4</math>, every vertex is the apex of 29 [[w:Hyperpyramid|polyhedral pyramids]], where the pyramid's lateral edge length is the radial distance and its base polyhedron is the section. Each section lies parallel to a congruent complement-radius section (or coincident with it, in the case of the central section). Each section also lies completely orthogonal to a congruent section. Only 8 of the 30 chords in the table occur in the 600-cell and the planar {30)-gon. The 120-cell's additional chords arise originally from the regular 5-cell, in its interaction with the other regular 4-polytopes that compound to make the 120-cell. Since all those polytopes except the 5-cell occur in the 600-cell, and the 600-cell and the 120-cell have the same symmetry group, the 5-cell's symmetry group is what's new in the 120-cell. ... {{Clear}} == Conclusions == Fontaine and Hurley's discovery is more than a geometric formula for the reciprocal of a regular ''n''-polygon diagonal. It also yields the discrete sequence of isocline chords of the characteristic isoclinic rotation of a ''d''-dimensional polytope in its invariant edge planes. The characteristic rotational chord sequence of the ''d''-polytope can be represented geometrically in two dimensions on a distinct star polygon, but it lies on a geodesic circle through ''d''-dimensional space. Fontaine and Hurley discovered the geodesic topology of polytopes generally. Their procedure will reveal the geodesics of arbitrary non-uniform polytopes, since it can be applied to a polytope of any dimensionality and irregularity, by first fitting the polytope to the smallest regular polygon whose chords include its chords. [If what is meant by this is its Petrie polygon, it is not quite necessary or possible with respect to the planar polygon chords, e.g. the planar Petrie polygon of the 600-cell does not contain the <math>\sqrt{2}</math> chord. But perhaps it would work if the fit is to the smallest regular skew polygon in the ''d''-space.] The discovery of a chordal construction for discrete isoclinic rotations generally closes the circuit on Kappraff and Adamson's discovery of a rotational connection between dynamical systems, Steinbach's golden fields, and Coxeter's Euclidean geometry of ''n'' dimensions. Application of the Fontaine and Hurley procedure in the 120-cell demonstrates why the connection exists: because polytope sequences generally, from Steinbach's golden chord sequences in polygons, to sequences of star polygons in isoclinic rotations, to subsumption relations in the sequence of regular 4-polytopes, arise as expressions of the reflections and rotations of distinct Coxeter symmetry groups, when those various groups interact. == Appendix: Sequence of regular 4-polytopes == {{Regular convex 4-polytopes|wiki=W:|columns=7}} == Notes == {{Notelist}} == Citations == {{Reflist}} == References == {{Refbegin}} * {{Cite journal | last=Steinbach | first=Peter | year=1997 | title=Golden fields: A case for the Heptagon | journal=Mathematics Magazine | volume=70 | issue=Feb 1997 | pages=22–31 | doi=10.1080/0025570X.1997.11996494 | jstor=2691048 | ref={{SfnRef|Steinbach|1997}} }} * {{Cite journal | last=Steinbach | first=Peter | year=2000 | title=Sections Beyond Golden| journal=Bridges: Mathematical Connections in Art, Music and Science | issue=2000 | pages=35-44 | url=https://archive.bridgesmathart.org/2000/bridges2000-35.pdf | ref={{SfnRef|Steinbach|2000}}}} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Jablan | first2=Slavik | last3=Adamson | first3=Gary | last4=Sazdanovich | first4=Radmila | year=2004 | title=Golden Fields, Generalized Fibonacci Sequences, and Chaotic Matrices | journal=Forma | volume=19 | pages=367-387 | url=https://archive.bridgesmathart.org/2005/bridges2005-369.pdf | ref={{SfnRef|Kappraff, Jablan, Adamson & Sazdanovich|2004}} }} * {{Cite journal | last1=Kappraff | first1=Jay | last2=Adamson | first2=Gary | year=2004 | title=Polygons and Chaos | journal=Dynamical Systems and Geometric Theories | url=https://archive.bridgesmathart.org/2001/bridges2001-67.pdf | ref={{SfnRef|Kappraff & Adamson|2004}} }} * {{Cite journal | last1=Fontaine | first1=Anne | last2=Hurley | first2=Susan | year=2006 | title=Proof by Picture: Products and Reciprocals of Diagonal Length Ratios in the Regular Polygon | journal=Forum Geometricorum | volume=6 | pages=97-101 | url=https://scispace.com/pdf/proof-by-picture-products-and-reciprocals-of-diagonal-length-1aian8mgp9.pdf }} {{Refend}} aa9nqxztt3nlc9ikme73j6uu994c237 2 327407 2816231 2816047 2026-06-18T17:09:43Z Atcovi 276019 rearrange 2816231 wikitext text/x-wiki ''<small>Will be moved to [[WikiJournal Preprints/Mental health in Sri Lanka]] - I think the page is too long, so it's making editing it burdensome.</small>'' == Future Outlook == Despite significant changes to the mental health environment in Sri Lanka, the current legal framework shaping mental health in the country has not been updated since 1956. A Cambridge University Press article detailed many limitations of the Mental Disease Ordinance of 1956, including discrepancies between the legal provisions of involuntary admissions and modern practices, potential exposure to trauma through extra-legal detentions of the mentally ill, and an absence of legal guidelines addressing the restraint of violent patients (https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4). Participants from Sri Lanka reported in a comparative legislative questionnaire that they felt the mental health laws were "outdated" and descriptions of clinical roles remained ambiguous (https://link.springer.com/article/10.1186/s13033-019-0322-7). A drafted mental health legislation from 2007 includes provisions for human rights, but due to "bureaucratic processes" and a "lack of consensus", an agreement has not been reached for the legislation to be enacted. These limitations pose challenges to the standardization of patient admissions for mental healthcare and may impact the rights of detained patients. Detained patients may have their human rights violated with a lack of an up-to-date legal framework, impeding the identification of such violations. Additionally, with the lack of clarity on clinical roles, clinical responsibilities may not be routinely recognized and observed, leading to role confusion and potential legal ramifications. Stagnation in policy development leaves Sri Lanka without a practical, up-to-date, and comprehensive mental health legislation, which could put both clinicians and patients at risk. Future reforms should include clarification on the treatment and detention process of involuntary admissions of patients and a clear delineation of clinical roles and their responsibilities. ''Criticism of the Mental Disease Ordinance of 1956:'' <ref name=":6">{{Cite journal|last=Hapangama|first=Aruni|last2=Mendis|first2=Jayan|last3=Kuruppuarachchi|first3=K. a. L. A.|date=2023-02|title=Why are we still living in the past? Sri Lanka needs urgent and timely reforms of its archaic mental health laws|url=https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4|journal=BJPsych International|language=en|volume=20|issue=1|pages=4–6|doi=10.1192/bji.2022.26|issn=2056-4740|pmc=9909436|pmid=36812028}}</ref><ref>{{Cite journal|last=Dey|first=Sangeeta|last2=Mellsop|first2=Graham|last3=Diesfeld|first3=Kate|last4=Dharmawardene|first4=Vajira|last5=Mendis|first5=Susitha|last6=Chaudhuri|first6=Sreemanti|last7=Deb|first7=Aniruddha|last8=Huq|first8=Nafisa|last9=Ahmed|first9=Helal Uddin|date=2019-10-24|title=Comparing legislation for involuntary admission and treatment of mental illness in four South Asian countries|url=https://ijmhs.biomedcentral.com/articles/10.1186/s13033-019-0322-7|journal=International Journal of Mental Health Systems|volume=13|issue=1|pages=67|doi=10.1186/s13033-019-0322-7|issn=1752-4458|pmc=6813093|pmid=31666805}}</ref> === Expansion of services for women facing domestic violence === <ref name=":8">{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}</ref> (last paragraph before 4.2; see discussion + conclusion as well) ''so what?'' [finisher] [[Category:Atcovi's Work]] 36pbqx79sfgsl4w3ddrh6ez36jxdne9 2816232 2816231 2026-06-18T17:24:01Z Atcovi 276019 /* Future Outlook */ 2816232 wikitext text/x-wiki ''<small>Will be moved to [[WikiJournal Preprints/Mental health in Sri Lanka]] - I think the page is too long, so it's making editing it burdensome.</small>'' == Future Outlook == Despite significant changes to the mental health environment in Sri Lanka, the current legal framework shaping mental health in the country has not been updated since 1956. A Cambridge University Press article detailed many limitations of the Mental Disease Ordinance of 1956, including discrepancies between the legal provisions of involuntary admissions and modern practices, potential exposure to trauma through extra-legal detentions of the mentally ill, and an absence of legal guidelines addressing the restraint of violent patients (https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4). Participants from Sri Lanka reported in a comparative legislative questionnaire that they felt the mental health laws were "outdated" and descriptions of clinical roles remained ambiguous (https://link.springer.com/article/10.1186/s13033-019-0322-7). A drafted mental health legislation from 2007 includes provisions for human rights, but due to "bureaucratic processes" and a "lack of consensus", an agreement has not been reached for the legislation to be enacted. These limitations pose challenges to the standardization of patient admissions for mental healthcare and may impact the rights of detained patients. Detained patients may have their human rights violated with a lack of an up-to-date legal framework, impeding the identification of such violations. Additionally, with the lack of clarity on clinical roles, clinical responsibilities may not be routinely recognized and observed, leading to role confusion and potential legal ramifications. Lastly, current efforts should increase beyond just addressing poverty-centered matters, but also expanding efforts to domestic violence victims and children with disabilities as facilities and specialized clinicians for both groups are scarce in the country<ref name=":8" />. Stagnation in policy development leaves Sri Lanka without a practical, up-to-date, and comprehensive mental health legislation, which could put both clinicians and patients at risk. Future reforms should include clarification on the treatment and detention process of involuntary admissions of patients and a clear delineation of clinical roles and their responsibilities. ''Criticism of the Mental Disease Ordinance of 1956:'' <ref name=":6">{{Cite journal|last=Hapangama|first=Aruni|last2=Mendis|first2=Jayan|last3=Kuruppuarachchi|first3=K. a. L. A.|date=2023-02|title=Why are we still living in the past? Sri Lanka needs urgent and timely reforms of its archaic mental health laws|url=https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4|journal=BJPsych International|language=en|volume=20|issue=1|pages=4–6|doi=10.1192/bji.2022.26|issn=2056-4740|pmc=9909436|pmid=36812028}}</ref><ref>{{Cite journal|last=Dey|first=Sangeeta|last2=Mellsop|first2=Graham|last3=Diesfeld|first3=Kate|last4=Dharmawardene|first4=Vajira|last5=Mendis|first5=Susitha|last6=Chaudhuri|first6=Sreemanti|last7=Deb|first7=Aniruddha|last8=Huq|first8=Nafisa|last9=Ahmed|first9=Helal Uddin|date=2019-10-24|title=Comparing legislation for involuntary admission and treatment of mental illness in four South Asian countries|url=https://ijmhs.biomedcentral.com/articles/10.1186/s13033-019-0322-7|journal=International Journal of Mental Health Systems|volume=13|issue=1|pages=67|doi=10.1186/s13033-019-0322-7|issn=1752-4458|pmc=6813093|pmid=31666805}}</ref> === Expansion of services for women facing domestic violence === <ref name=":8">{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}</ref> (last paragraph before 4.2; see discussion + conclusion as well) ''so what?'' [finisher] [[Category:Atcovi's Work]] nny70kregfchnh6dt21dd37uzfzvu4r 2816280 2816232 2026-06-19T02:42:08Z Atcovi 276019 /* Future Outlook */ 2816280 wikitext text/x-wiki ''<small>Will be moved to [[WikiJournal Preprints/Mental health in Sri Lanka]] - I think the page is too long, so it's making editing it burdensome.</small>'' == Future Outlook == Despite significant changes to the mental health environment in Sri Lanka, the current legal framework shaping mental health in the country has not been updated since 1956. A Cambridge University Press article detailed many limitations of the Mental Disease Ordinance of 1956, including discrepancies between the legal provisions of involuntary admissions and modern practices, potential exposure to trauma through extra-legal detentions of the mentally ill, and an absence of legal guidelines addressing the restraint of violent patients (https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4). Participants from Sri Lanka reported in a comparative legislative questionnaire that they felt the mental health laws were "outdated" and descriptions of clinical roles remained ambiguous (https://link.springer.com/article/10.1186/s13033-019-0322-7). A drafted mental health legislation from 2007 includes provisions for human rights, but due to "bureaucratic processes" and a "lack of consensus", an agreement has not been reached for the legislation to be enacted. These limitations pose challenges to the standardization of patient admissions for mental healthcare and may impact the rights of detained patients. Detained patients may have their human rights violated due to a lack of an up-to-date legal framework, thereby impeding the identification of such violations. Additionally, with the lack of clarity on clinical roles, clinical responsibilities may not be routinely recognized and observed, leading to role confusion and potential legal ramifications. Lastly, current efforts should increase beyond just addressing poverty-centered matters, but also expand efforts to domestic violence victims and children with disabilities, as facilities and specialized clinicians for both groups are scarce in the country<ref name=":8" />. Stagnation in policy development leaves Sri Lanka without a practical, up-to-date, and comprehensive mental health legislation, which could put both clinicians and patients at risk. Future reforms should include clarification on the treatment and detention process of involuntary admissions of patients and a clear delineation of clinical roles and their responsibilities. Without the necessary reforms to advance Sri Lankan mental health legislation, clinicians and vulnerable patients may suffer from a lack of comprehensive oversight. ''Criticism of the Mental Disease Ordinance of 1956:'' <ref name=":6">{{Cite journal|last=Hapangama|first=Aruni|last2=Mendis|first2=Jayan|last3=Kuruppuarachchi|first3=K. a. L. A.|date=2023-02|title=Why are we still living in the past? Sri Lanka needs urgent and timely reforms of its archaic mental health laws|url=https://www.cambridge.org/core/journals/bjpsych-international/article/why-are-we-still-living-in-the-past-sri-lanka-needs-urgent-and-timely-reforms-of-its-archaic-mental-health-laws/B18B03DC962CC6F09BC6D7877E390EE4|journal=BJPsych International|language=en|volume=20|issue=1|pages=4–6|doi=10.1192/bji.2022.26|issn=2056-4740|pmc=9909436|pmid=36812028}}</ref><ref>{{Cite journal|last=Dey|first=Sangeeta|last2=Mellsop|first2=Graham|last3=Diesfeld|first3=Kate|last4=Dharmawardene|first4=Vajira|last5=Mendis|first5=Susitha|last6=Chaudhuri|first6=Sreemanti|last7=Deb|first7=Aniruddha|last8=Huq|first8=Nafisa|last9=Ahmed|first9=Helal Uddin|date=2019-10-24|title=Comparing legislation for involuntary admission and treatment of mental illness in four South Asian countries|url=https://ijmhs.biomedcentral.com/articles/10.1186/s13033-019-0322-7|journal=International Journal of Mental Health Systems|volume=13|issue=1|pages=67|doi=10.1186/s13033-019-0322-7|issn=1752-4458|pmc=6813093|pmid=31666805}}</ref> === Expansion of services for women facing domestic violence === <ref name=":8">{{Cite journal|last=Augustyniak|first=Nadia|date=2025-06-01|title=Public mental healthcare and economic vulnerability in Sri Lanka|url=https://linkinghub.elsevier.com/retrieve/pii/S2666560324000926|journal=SSM - Mental Health|volume=7|pages=100387|doi=10.1016/j.ssmmh.2024.100387|issn=2666-5603}}</ref> (last paragraph before 4.2; see discussion + conclusion as well) ''so what?'' [finisher]: [[Category:Atcovi's Work]] a9dmahmx5nvu9oed7z7ho1ct8x1pncp 2 328661 2816235 2808118 2026-06-18T18:28:07Z ThinkingScience 3061446 /* Posting on talk pages using "Quasi-AI" depending on user preference */ wikilink fixed 2816235 wikitext text/x-wiki == Main focus: my "idea" == * This is my [[Draft:The Neurodiversity-inspired Idea]]. There goes the "main effort" based on my other smaller effort in various places and also by using the methodology I one day hope I will make. * [[User:ThinkingScience/ND_Inspired_Idea_Notebook|Daily Diary of ND Inspired Idea]] * These are my course notes: [[User:ThinkingScience/Draftspace/Coursera]] === Secondary Goals === These are secondary goals in no particular order: * Contribute to Wikiversity by understanding and improving on content: ** I begin with interpreting the [[Neurodiversity Movement]] by creating this subpage: [[User:ThinkingScience/Neurodiversity_Movement_Interpretation]] * Documenting all "Quasi-AI" inputs and outputs related to a particular action: ** [[User:ThinkingScience/All General AI Prompt History Archive|All General AI Prompt History Archive]] == Taking responsibility for famous people or people to focus on in [[Draft:The Neurodiversity-inspired Idea]] == I need to take responsibility for the choices I make. If any of my choices resulted in a harm to a real person I am responsible whether I agree or not to any blame being put on me. This section may be moved to a sub-page if I think it starts getting too "cluttered" and later into more sub-pages if the list just grows and grows. === T === ==== '''Taylor Swift''' ==== Taylor Swift. Why I chose Taylor Swift. I have watched interviews with her before. She is an interesting person to me. I discovered she is open about her creation process. I value that in human beings and that includes people I meet offline, in the "real world" but it will be a challenge for me to make video notes that are "Do no harm". I may be "way in over my head". Please help me if you think I'm doing something wrong. She has a dedicated follower base which may have a large influence. Maybe I'll suffer for this but keeping my "idea" locked inside "my head" I think will cause greater harm to me than good. I know what risk I am taking...or probably not but I gotta move forward or try to. Perhaps my fears are greater than real risks in reality but who knows? ==== '''Tom Hanks''' ==== I chose Tom Hanks because I need to move on. In my personal notes I have tens of famous people and not. Now that I'm adding people here I need to adhere to the Wikiversity rules and guidelines so the list will be limited, or will it? Maybe it will grow because applying "Do no harm" is a very good foundation considering I am not consciously trying to come up with idea how my comparisons here will do not harm and I think that is healthy for both me and for everyone! == April 20th experiment, "AI Decisions, sure. AI-generation NEVER" == Starting today on April 20th after 08:46 UTC Time(I got UTC time on this computer where I'm so far only using this account), I'll begin by editing Wikiversity resources by being more encouraged by "yeah, do that" comments by Large Language Models. Nothing of it will be "AI-generated" but the decisions I take: the reason for the decisions I take may be because of "AI-generation" but of course I will try to stay away from clear stupidity like if the AI-generation says "jump off a cliff". An extreme example, but I wanted to make a point that I won't take any decision and I will question the "AI/LLM" if it suggests something that to me sounds insane. If you see anything weird please comment on my talk page after you've reverted my edits. When this experiment ends, I don't have a plan for that yet. User input might help. This is where I make notes of decisions that may motivate me to do edits in places. It should include both inputs and outputs and what kind of "version" of "AI"s/LLMs I'm using: * [[/April 20th Experiment Notes|"AI Notes" for motivation purposes in this "experiment"]] === Defining the end goal === The end goal is supposed to make me make useful edits or being able to make edits or contributions on more than just only the projects I'm "mainly" interested in. '''Success Criteria''' * "Did the AI's output lead me to make a useful edit on a page I wouldn't have thought about and/or chosen on my own?" ** This question was modified by myself based on an AI's output(this part was generated by "AI Mode" on May 1, 2026) == Coursera schedule and notes == Today April 16, 2026 my contributions contain a lot of spelling mistakes. They may be present other days too. You'll probably spot spelling mistakes all over. My studying schedule as I've understood it so far(studying with my mother): This schedule is not reliable(cause my studying partner keeps changing the time, which is not necessarily bad): UTC TIME: 07:30 - 09:30 (2 hours a day, 6 hours a week) * Monday * Thursday * Saturday == I'm studying on Coursera and about their Terms of Use == '''Nothing here is legal advice'''. This is very important. Nothing in this "Wikisection" constitutes legal advice! Please don't blindly follow my advice and if someone copies some parts of this text without providing context then they are responsible for what they share! If you have been tricked by scammers that's sad but I am NOT responsible for illegal activities. * web.archive.org/web/20260325233813/https://www.coursera.org/about/terms "When you create your Coursera account, and when you subsequently use certain features, you must provide us with accurate and complete information, and you agree to update your information to keep it accurate and complete." My interpretation of that is that on Coursera I have to provide a real name. There is a field for "Full name"(retrieved 2026-04-09 UTC YYYY-MM-DD). How does that correspond to these terms? It doesn't say "Real name" but even if it did, what if I choose a name for myself and I'd like to call myself ThinkingScience? Is it still accurate? They don't specify what I actually have to do, just based on my quote. It would be nice for me and other Coursera learners to know what is true. Is the privacy on Wikiversity better? I'd say it is because on Coursera we are forced to provide an email address to create an account. We are not forced to do that on Wikiversity, Wikidata etc. == notes about this account == This account is an alternative account on a computer I don't trust. It should never be allowed to vote and if it does please block this account. Doubling down on this today at 2026-04-30!(intent unchanged) It's an alt of [[User:Dekatriofovia]] which unfortunately I have to prove right now despite me being in a hurry...so I'll edit my account at Dekatriofovia at the same time almost and publish at the same time...so you know it's me. The reason for this account is it's on a computer with a bigger screen so I can more easily read books and documents. == a thing I did not regret(modified section title) == This may be blathering but it ends with another Wikilink where I will pass my "idea" through '''Wikiversity:Research ethics''' and through anything else that might be required before anything enters Draft space. The "idea" is "'''The Neurodiversity-inspired idea'''". [[Protoscience]] was an interesting read. I think it will be calming for me if my idea is proven to be pseudoscience cause I can stop worrying about it and leave it behind me. "The Neurodiversity-inspired idea"(in lack for a better name, for now) will not be published in main space, only in draft space. [[Wikiversity:Original research]] made me think "I may be way over my head" (though I stumbled around a bit due to not knowing English at an advanced enough level...this parenthesis is about some unimportant trivia). I'm gonna place everything regarding "The Neurodiversity-inspired idea" into draft space and pass it through '''Wikiversity:Research ethics'''(sorry for repeating myself) and anything else I can find and also ask the community here on Wikiversity what else to place it through. I thought I was gonna create '''User:ThinkingScience/The Neurodiversity-inspired Idea'''(but turns out I was encouraged to create it in Draft: space ... (this paragraph has been modified. Edit history might keep the original). Here are my notes again which I wanted to link to [[User:ThinkingScience/ND Inspired Idea Notebook]] '''Draft:The Neurodiversity-inspired Idea''' that probably is in line with "be bold". === It happened, a small burden has been lifted === I posted to the [[Wikiversity:Colloquium]] https://en.wikiversity.org/w/index.php?title=Wikiversity:Colloquium&oldid=2805080 Thing may be archive in the future. I've lost many things that way.(but also re-discovered many things that landed in the archive that I had posted too!) One week. One small burden lifted. It was the only way forward. I may have been driven insane otherwise or this is just a very bad day I'm having. Full of things that "real life" is demanding of me. More specifically, this is what I posted [[Wikiversity:Colloquium#Advice_needed:_A_Neurodiversity-inspired_Idea/observation]] == Posting on talk pages using "Quasi-AI" depending on user preference == This is a list of users who have told me how they prefer maximum amount of characters/words to be if I post on their talk page using "Quasi-AI"[1](whatever that is since there's no such Q item nor alias on Wikidata right now regarding that term): * [[User:Juandev]] - max amount of words: 100 as long as it respects [[Wikiversity:Artificial intelligence]] == References == [1] - https://en.wikiversity.org/w/index.php?title=Is_the_output_of_ChatGPT_copyrighted%3F&oldid=2711065 {{quote|ChatGPT is an automated quasi-AI technology that allows human-like text console interaction online in which a human can ask questions and ChatGPT returns answers.}} cxxpv4xr3jtoq3dn6effs6enck6wzei 118 329157 2816234 2814682 2026-06-18T18:26:45Z ThinkingScience 3061446 /* Questions that might encourage the development of this idea and its methodology */ 2026-06-18 update 2816234 wikitext text/x-wiki {{Research project|status=draft}} {{AI-generated}} {{Notice|'''Please excuse mistakes and problems''' this is a work in progress and pages may be published which are unfinished and that contain unfinished sentences and repetitions}} == Explanation regarding {{tl|AI-generated}} template presence on this page == * Some questions that can be asked have been generated where there is a note about it. ("AI Mode" by Google) * If something has been generated by an "AI"/LLM then please make a note of that so the reader knows. Also please document the specific "AI name", ie. "AI Mode", "GPT-5 mini" etc. as long as that name is enough to find the AI/LLM on Wikidata or on Wikipedia. For questions that have been "AI-Generated" this section has been created to document the queries/input: * [[Draft:The Neurodiversity-inspired Idea/AI Prompt History for Questions|Please document your input and output here when interacting with an AI/LLM]] == Hypothes-is/-es == Preliminary date for basics of hypothesis/hypotheses being developed: June 16, 2026. === In preparation for Hypothes-is/-es: Core Assumption Example === Google's "AI Mode" was used on 2026-04-29 and the formation of an "example Core Assumption" was made. The following is input sent to "AI Mode": {{quote|This I posted previously: {{quote|Example of a "Core Assumption": That not "Everybody is different" in brain structure but that some humen show so great similarities between each other that one might say "it is the same brain". If that fails, my idea fails completely.}} This I'm thinking about posting now and I want your output based on this input: Example of a "Core Assumption": That not "Everybody is different" in brain structure but that some humen show so great similarities between each other that one might say "it is the same brain". I assume that that my core assumption and "main idea" can be fully broken by measurement of Rapport. I assume that I can predict better rapport between individuals who interact for the first time compared to a random sample. If that fails, my idea fails completely.}} The system in which predictions are made regarding the related hypotheses are assumed to be found in the data. Studying the data should lead to understanding how to make the predictions in the hypotheses. === Hypothesis: Rapport between dating couples or random strangers meeting etc. === This was written in a hurry 100% by a human so there may be multiple grammar, words that don't exist and spelling and other incoherences(this is probably not even a word). This is a "pre-hypothesis". What is meant with that is that this hypothesis purpose is to "prove" or "disprove" that there is anything at all going on...or not going on. If there is nothing going on at all after "control group in, control group out" etc., then this project could be ended/closed, unless the focus changes or something else is tested that hasn't been tested yet. Sometimes something new is discovered or the significance of a parameter is questioned in the beginning and may matter more later. Regardless, speculation is pointless. The very existence of this project was shaky from the beginning and is still shaky and this hypothesis might help in discovering whether there can be a systematic way of finding rapport between people of either friendship nature or of a romantic/dating nature. If this hypothesis is disproven. Though it should be clear how it can be disproven or proven. The rapport...if there is no significant change in rapport in comparison with control group 1 and 2 then the hypothesis is prove wrong, it is assumed. * Control group 1: "normal group" / "random group" of people selected purely by chance alone. Method of how "by chance" is calculated should be disclosed. Random is also which couples should be matched together, that is the distinctive part of control group 1 and all the control groups, namely whether random matchmaking is worse or better in regards to rapport between the couples. * Control group 2: same "normal group" / "random group" of people selected but the couples that are selected are chosen this time. It is not by randomness at all. Control group 2 is where a "core" of this idea happens. Participants can sign up to make the choosing of the hypothesis, practically each participant who signs up conducts their own hypothesis experiment as long as it's in a "Do no harm" way and it is legal in the country/countries where they are doing the experiment. == Data == There is currently no data. == Original Motivations == This section can list motivations by each user who contributes content or questions to this page. * '''User:ThinkingScience''' My motivation is related to perceived limited progress by psychiatry and getting inspired by writers exploring Neurodiversity topics. I don't feel I have a right to have an opinion about psychiatry considering this idea's methodology is being developed during the publishing(and before) of this edit. It is my hope that if this idea develops how I expect it to it will be an "extra parameter" that some other sciences can use, including psychiatry. It is my hope that this idea will thus help psychiatry develop in a great way though my personal hope is it will help sociology more. * Example user x * Example user y etc. == "Do no harm" == This section can list ideas/comments by users who are trying to do no harm while using their methodology(which should be documented on this page, if possible!): * '''User:ThinkingScience''' Considering I am watching videos of famous people in interviews. I am making notes...my goal should be that not only my public notes are following the "Do no harm" but that my private notes do as well. That can be my goal for now. We'll see how this develops... * Example user x * Example user y etc. Do no harm links from [[Wikiversity:Research ethics]] into Wikipedia with no examples for Wikiversity users specifically, yet. == "Research projects must fully document the methods" == {{quote|Safety - Research must be conducted in a safe and lawful manner. Do no harm.}} An idea for the minimum age of the subjects which are "studied" in a "Do no harm" way, if even possible, might be 29 years of age. There is disagreement regarding the exact age of a person when their brain matures and it may benefit the student/contributor if the "starting age" or "minimum age" is high enough so that a person with a "fully matured brain" also has some experience living with that fully matured brain. 💡💡💡'''Suggestion: 29 minimum years of age for anyone participating in this project.'''💡💡💡 for the sake of "Do no harm"? which is described on [[Wikiversity:Research ethics]] that links to Wikipedia. This section needs work. One of the methodologies is to watch a video, ie. a video interview of famous people or footage where the researcher has gotten legal access and specific consent from any person appearing in the video footage. Methodologies need to be developed where data is gathered in a way that adheres to "Do no harm". === Focusing on creating a "Do no harm"-compliant method === This needs to be developed. This sub-page is created so we can make video notes. We can watch a video and then we can make video notes. We must do the video notes by following [[Wikiversity:Research ethics]] and "Do no harm". How to do that can be tricky. "Do no harm" links to Wikipedia because we don't have our own resource where we help you how to do that. === method of interacting with draft and other pages on Wikiversity === "AI Mode" by Google can be used to get inspired by what kind of things to focus on, including if one thinks they started "blathering" and the text started to grow 'for no apparent reason' because the user landed in a "non-productive behavior" and the repeating themselves kept going on and on. Prompts that generate questions and other things could be added into a subsection of this draft research === Video Notes before the creation of a more 'stable' method that adheres to "Do no harm" === * [[/Method_development_through_video_notes|Video Notes]] == Questions that might encourage the development of this idea and its methodology == Questions that encouraged certain dates of time: * June 16: Two months after the creation of the project some 'basic form' of a/- hypothes-is/-es should exist. ie. measuring rapport. How rapport can/will be measured. What tools can be used. How one makes "first contact" with people. Challenges and work to be done regarding the "main hypothesis". This is a prediction. Time will tell if a "main hypothesis" will be ready by June 16. ** 2026-06-18 This plan failed. Suggestions from the Colloquium were also that no deadlines should be set. Opinions change over time. Perhaps one day deadlines will be suggested. '''The New Way: 100% human generated questions and answers''': If a student/contributor figures out their own questions without the help of an AI/"AI"/LLM please contribute in this section. Otherwise go to "The Old Way" section and contribute there. Questions in no particular order: * Is this "idea" testable? - It is in the plans for it to be testable '''but not testable yet'''. * Is it possible to create one or more hypotheses based on this "idea"? - Multiple hypotheses have been made but never published. Plan is to publish hypotheses related to the "idea" in the future. * Is there a hypothesis or a number of hypotheses related to the idea? ** Can this idea be proven false? - It can not yet be proven false considering a hypothesis is not currently described. Goal is to have a hypothesis at least before June 16, 2026 which is 2 months roughly after the first creation of the page. A basic hypothesis structure may/should exist by then but a complete hypothesis being created that is "perfect in some ways" that can't be predicted yet. '''The Old Way''' This is for any AI/LLM that is available to you. Please document the output here and if you want to contribute the input prompts and more specific details go to the subpage [[Draft:The_Neurodiversity-inspired_Idea/AI_Prompt_History_for_Questions]] Questions and 'follow up'-/improved questions generated by Google "AI Mode": * What is missing right now? ** "Improved" version: "What key sections are missing from this research draft to meet Wikiversity standards?" * How will we know if the idea is working? == Naming Suggestions == Feel free to edit/modify or remove content in this section. Example name ideas: * Pre-research: Observations made inside psychiatry spectrums * Psychiatric Spectrum Specifications ''These names should probably not be used but were kept for inspiring questions in readers'': * Next-Gen Brain Types * Next-Gen Sociology * Next-Gen Neurotypes == Potential best-case uses of "idea" if fully developed == This section is for expected best-case uses for the "idea" if fully developed. Probably best to keep this section short. Assumptions may develop from it otherwise. Maybe max 5 items? * Could assist in better matchmaking in relationships where "compatibility"/"rapport" is important. Where collaboration is important and there is a desire for communication with less friction than any random picking of co-workers. * Better matchmaking in romantic relationships. "Better" in the way of a more "exact" match. The way of thinking here is more similar rather than less in the way the brain is structured. How the personality of the 2 people changed over the years, that is probably not much affected by this. * Companies may get a better idea on what education to send some of their employees where they can develop their skills the most. == References/in-Wikiversity-Wikilinks == This section is for linking to specific references: * [2] - [[Draft:The_Neurodiversity-inspired_Idea/AI_Prompt_History_for_Questions#Reference_2]] == Future references to this draft == In the event that other publications start referring to this draft in the future, the template "findsources" is added: {{findsources}} fp1nxmp01ktcvm4oixcudozwm1ef6wb 0 329829 2816250 2816016 2026-06-18T20:21:17Z Scogdill 1331941 /* Irish Aristocrats at the Duchess of Devonshire's 1897 Fancy-dress Ball */ 2816250 wikitext text/x-wiki = The Irish Aristocracy at the End of the 19th Century = == The Irish Peerage == Minus the people who attended the ball, which are in [[Social Victorians/Irish Aristocracy#Irish Aristocrats at the Duchess of Devonshire's 1897 Fancy-dress Ball|this section, below]]. === Dukes and Duchesses === ==== Duke of Leinster ==== Irish peerage * Gerald FitzGerald, 5th Duke of Leinster (16 August 1851 – 1 December 1893)<ref>{{Cite web|url=https://www.thepeerage.com/p1207.htm#i12063|title=Gerald FitzGerald, 5th Duke of Leinster|website=www.thepeerage.com|access-date=2026-05-24}}</ref> * Maurice FitzGerald, 6th Duke of Leinster, 6 years old when he succeeded to the dukedom<ref>{{Cite web|url=https://www.thepeerage.com/p2767.htm#i27667|title=Maurice FitzGerald, 6th Duke of Leinster|website=www.thepeerage.com|access-date=2026-05-24}}</ref> * Subsidiary Titles # Marquess of Kildare (Irish peerage), did not attend the ball. # Earl of Kildare (Irish peerage), did not attend the ball. # Earl of Offaly (Irish peerage) # Viscount Leinster of Taplow (GB peerage) # Baron Offaly (Irish peerage) # Baron Kildare of Kildare (UK peerage) === Marquesses and Marchionesses === ==== Marquess Conyngham<ref>{{Cite journal|date=2026-01-13|title=Marquess Conyngham|url=https://en.wikipedia.org/w/index.php?title=Marquess_Conyngham&oldid=1332742873|journal=Wikipedia|language=en}}</ref> ==== * Did not attend the ball but did attend a number of social events about this time. * Pronounced "''Cunn''ingum."<ref>{{Cite journal|date=2026-01-13|title=Marquess Conyngham|url=https://en.wikipedia.org/w/index.php?title=Marquess_Conyngham&oldid=1332742873|journal=Wikipedia|language=en}}</ref> * Henry Francis Conyngham, 4th Marquess Conyngham (1857–1897)<ref>"Henry Francis Conyngham, 4th Marquess Conyngham." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 7198 https://www.thepeerage.com/p7199.htm#i71982.</ref> * Victor George Henry Francis Conyngham, 5th Marquess Conyngham (1883–1918)<ref>"Victor George Henry Francis Conyngham, 5th Marquess Conyngham." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 7198 https://www.thepeerage.com/p7199.htm#i71983.</ref> * Subsidiary Titles ** Earl of Conyngham ** Viscount Conyngham ** Viscount Mount Charles ==== Marquess of Donegall ==== * Did not attend the ball. * Subsidiary Titles ** Earl of Donegall, did not attend the ball. ** Viscount Chichester — did not attend the ball; some Chichesters attended social events at about this time. ==== Marquess and Marchioness of Downshire ==== * Arthur Wills John Wellington Trumbull Blundell Hill, 6th Marquess of Downshire (2 July 1871 – 29 May 1918) in 1893 married Katherine Mary ("Kitty") Hare (1872–1959)<ref>{{Cite journal|date=2025-02-10|title=Arthur Hill, 6th Marquess of Downshire|url=https://en.wikipedia.org/w/index.php?title=Arthur_Hill,_6th_Marquess_of_Downshire&oldid=1274976272|journal=Wikipedia|language=en}}</ref> * Did not attend the ball. * Subsidiary Titles ** Earl of Hillsborough, did not attend the ball, also not at any social events described so far. ** Viscount Kilwarlin — 6th, Arthur Wills John Wellington Trumbull Hill (31 March 1874 – 29 May 1918)<ref>"Arthur Wills John Wellington Trumbull '''Hill''', 6th Marquess of Downshire." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page #3810 https://www.thepeerage.com/p3811.htm#i38104.</ref> ==== Marquess of Ely ==== * Did not attend the ball, but members of the Loftus family attended a number of social events at about this time. * 4th Marquess: John Henry Wellington Graham Loftus (15 July 1857 – 3 April 1889)<ref>"John Henry Wellington Graham Loftus, 4th Marquess of Ely." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 8545 https://www.thepeerage.com/p8545.htm#i85450.</ref> * 5th Marquess: John Henry Loftus (3 April 1889 – 18 December 1925)<ref>"John Henry Loftus, 5th Marquess of Ely." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 8546 https://www.thepeerage.com/p8546.htm#i85459.</ref> * Subsidiary Titles ** Earl of Ely — did not attend the ball. ** Viscount Loftus ==== [[Social Victorians/People/Bective|Marquess and Marchioness of Headfort]] ==== * Did not attend the ball, but a number of people in this family attended many social events at about this time. * Subsidiary Titles ** [[Social Victorians/People/Bective|Earl of Bective]] ** Viscount Headfort<ref name=":1" /> *** 4th: Thomas Taylour (6 December 1870 – 22 July 1894) *** 5th: Geoffrey Thomas Taylour (22 July 1894 – 29 January 1943) *Papers ==== Marquess of Sligo ==== * Did not attend the ball, but many people with the surname Browne attended a number of social events at about this time. * Subsidiary Titles ** Earl of Altamont. Did not attend the ball; did not attend any social events analyzed so far. ** Earl of Clanricarde — Did not attend the ball but did attend a few social events about this time. ** Viscount of Westport<ref name=":1">"Index to Viscounts and Viscountesses." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. https://www.thepeerage.com/index_viscount.htm.</ref> *** 5th: George John Browne (26 January 1845 – 30 December 1896), 5th Marquess *** 6th: John Thomas Browne (30 December 1896 – 30 December 1903), 6th Marquess ==== Marquess of Waterford ==== * John Henry de La Poer Beresford, 5th Marquess of Waterford (1844–1895) * Henry de La Poer Beresford, 6th Marquess of Waterford (1875–1911)<ref>{{Cite journal|date=2026-02-10|title=Henry Beresford, 6th Marquess of Waterford|url=https://en.wikipedia.org/w/index.php?title=Henry_Beresford,_6th_Marquess_of_Waterford&oldid=1337565707|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but members of the Beresford family were prominent socially at about this time. * Subsidiary Titles ** Viscount Tyrone === Earls and Countesses === ==== Earl of Annesley ==== * Did not attend the ball but did attend a number of social events in the 1890s. * Subsidiary Title ** Viscount Glerawly<ref name=":1" />: 6th: Hugh Annesley (10 August 1874 – 15 December 1908), 6th Earl of Annesley ==== Earl of Bessborough ==== * Frederick George Brabazon Ponsonby, 6th Earl of Bessborough (1815–1895) * Walter William Brabazon Ponsonby, 7th Earl of Bessborough (1821–1906), would have been Viscount Duncannon 1880–1895 * Edward Ponsonby, 8th Earl of Bessborough (1851–1920), would have been Viscount Duncannon 1895–1906 * Did not attend the ball, but the [[Social Victorians/People/Ponsonby|Ponsonby]] family attended many social events at about this time, including mention of Lady Duncannon's school that taught fabric arts. * Subsidiary Titles ** Viscount Duncannon ==== Earl of Caledon ==== * Did not attend the ball but did attend a number of social events about this time. * James Alexander, 4th Earl of Caledon (1846–1898)<ref>{{Cite journal|date=2025-11-21|title=James Alexander, 4th Earl of Caledon|url=https://en.wikipedia.org/w/index.php?title=James_Alexander,_4th_Earl_of_Caledon&oldid=1323312651|journal=Wikipedia|language=en}}</ref> * Eric James Desmond Alexander, 5th Earl of Caledon (1885–1968), succeeded as earl in 1898.<ref>{{Cite journal|date=2025-11-21|title=Eric Alexander, 5th Earl of Caledon|url=https://en.wikipedia.org/w/index.php?title=Eric_Alexander,_5th_Earl_of_Caledon&oldid=1323313583|journal=Wikipedia|language=en}}</ref> * Subsidiary Title ** Viscount Caledon ==== Earl of Carrick ==== * Did not attend the ball. ==== Earl Castle Stewart ==== * Did not attend the ball. * 5th Earl: Henry James Stuart-Richardson (12 September 1874 – 5 June 1914)<ref>"Henry James Stuart-Richardson, 5th Earl Castle Stewart." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 2412 https://www.thepeerage.com/p12413.htm#i124125.</ref> * Subsidiary Title ** Viscount Castle Stewart ==== Earl of Cavan ==== * Did not attend the ball. ==== Earl of Clancarty ==== * Did not attend the ball and attended few social events researched so far. * Richard Somerset Le Poer Trench, 4th Earl of Clancarty (1834–1891)<ref>{{Cite journal|date=2026-01-10|title=Richard Trench, 4th Earl of Clancarty|url=https://en.wikipedia.org/w/index.php?title=Richard_Trench,_4th_Earl_of_Clancarty&oldid=1332219771|journal=Wikipedia|language=en}}</ref> * William Frederick Le Poer Trench, 5th Earl of Clancarty (1868–1929)<ref>{{Cite journal|date=2025-11-05|title=William Trench, 5th Earl of Clancarty|url=https://en.wikipedia.org/w/index.php?title=William_Trench,_5th_Earl_of_Clancarty&oldid=1320532351|journal=Wikipedia|language=en}}</ref> * Subsidiary Title ** Viscount Dunlo ==== [[Social Victorians/People/Clanwilliam|Earl and Countess of Clanwilliam]] ==== * Did not attend the ball. * Subsidiary Title ** Viscount Clanwilliam<ref name=":1" />: 4th: Richard James Meade (7 October 1879 – 4 August 1907), 4th Earl ==== Earl of Cork, Earl of Orrery ==== * Cork and Orrery, did attend the ball. ==== Earl of Courtown ==== * Did not attend the ball. ==== Earl of Darnley ==== * John Bligh, 6th Earl of Darnley (1827–1896), British<ref>{{Cite journal|date=2026-02-07|title=John Bligh, 6th Earl of Darnley|url=https://en.wikipedia.org/w/index.php?title=John_Bligh,_6th_Earl_of_Darnley&oldid=1337113925|journal=Wikipedia|language=en}}</ref> * Edward Bligh, 7th Earl of Darnley (1851–1900), Lord Clifton much of his adult life, "English"<ref>{{Cite journal|date=2026-05-05|title=Edward Bligh, 7th Earl of Darnley|url=https://en.wikipedia.org/w/index.php?title=Edward_Bligh,_7th_Earl_of_Darnley&oldid=1352607379|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but the Bligh family attended some social events from about this time. * Subsidiary Titles: ** Viscount Darnley ==== Earl of Desmond ==== * Did not attend the ball. ==== [[Social Victorians/People/Donoughmore|Earl of Donoughmore]] ==== * Did not attend the ball but did attend a number of social events about this time. * John Luke George Hely-Hutchinson, 5th Earl of Donoughmore (1848–1900)<ref>{{Cite journal|date=2025-05-01|title=John Hely-Hutchinson, 5th Earl of Donoughmore|url=https://en.wikipedia.org/w/index.php?title=John_Hely-Hutchinson,_5th_Earl_of_Donoughmore&oldid=1288332715|journal=Wikipedia|language=en}}</ref> * Subsidiary Title ** Viscount Donoughmore ==== Earl of Drogheda ==== * Did not attend the ball. * Subsidiary Titles ** Viscount Moore — no evidence of the Viscount or Viscountess Moore at social events at about this time. ==== Earl of Granard ==== * Did not attend the ball. * Bernard Arthur William Patrick Hastings Forbes, 8th Earl of Granard (17 September 1874 – 10 September 1948)[https://en.wikipedia.org/wiki/Bernard_Forbes,_8th_Earl_of_Granard] * Anglo-Irish * Subsidiary Titles ** Bernard Arthur William Patrick Hastings Forbes, styled Viscount Forbes from 1874 to 1889 ==== Earl of Kingston ==== * Did not attend the ball. * Subsidiary Title ** Viscount Kingsborough (of Viscount Kingston of Kingborough, co. Sligo)<ref name=":1" /> *** 8th: Henry Newcomen King-Tenison (21 June 1871 – 13 January 1896) *** 9th: Henry Edwyn King-Tenison (13 January 1896 – 11 January 1946) **Viscount Lorton ==== Earl of Lisburne ==== * Did not attend the ball. * Ernest Augustus Malet Vaughan, 5th Earl of Lisburne (1836–1888)<ref>{{Cite journal|date=2025-12-03|title=Ernest Augustus Malet Vaughan, 5th Earl of Lisburne|url=https://en.wikipedia.org/w/index.php?title=Ernest_Augustus_Malet_Vaughan,_5th_Earl_of_Lisburne&oldid=1325511612|journal=Wikipedia|language=en}}</ref> ** Owned a lot of land in Cardiganshire, Wales ** Conservative, but withdrew from politics * George Henry Arthur Vaughan, 6th Earl of Lisburne (1862–1899) * Ernest Edmund Henry Malet Vaughan, 7th Earl of Lisburne (1892–1965) ** Welsh nobleman, of Trawsgoed, Cardiganshire. 7 years old when he succeeded to the earldom ==== Earl of Longford ==== * Did not attend the ball. ==== Earl and Countess of Meath ==== * Did not attend the ball. ==== Earl of Mexborough ==== * Did not attend the ball ==== Earl of Mornington ==== * Subsidiary title of the Duke of Wellington (in the peerage of the UK). ==== Earl of Normanton ==== * Did not attend the ball, but did attend some social events in the 1880s and 1890s. * James Charles Herbert Welbore Ellis Agar, 3rd Earl of Normanton (1818–1896)<ref>{{Cite journal|date=2025-10-06|title=James Agar, 3rd Earl of Normanton|url=https://en.wikipedia.org/w/index.php?title=James_Agar,_3rd_Earl_of_Normanton&oldid=1315461436|journal=Wikipedia|language=en}}</ref> * Sidney James Agar, 4th Earl of Normanton (1865–1933)<ref>{{Cite journal|date=2026-05-19|title=Sidney James Agar, 4th Earl of Normanton|url=https://en.wikipedia.org/w/index.php?title=Sidney_James_Agar,_4th_Earl_of_Normanton&oldid=1355064165|journal=Wikipedia|language=en}}</ref> * Subsidiary Title ** Viscount Somerton ==== Earl of Portarlington ==== * Did not attend the ball. Members of this family attended a few social events at about this time. * Subsidiary Title ** Viscount Carlow<ref name=":1" /> *** 5th: Lionel Seymour William Dawson-Damer (1 March 1889 – 17 December 1892), Earl of Portarlington *** 6th: Lionel George Henry Seymour Dawson-Damer (17 December 1892 – 31 August 1900) ==== Earl of Roden ==== * Did not attend the ball. * Subsidiary Title ** Viscount Jocelyn<ref name=":1" /> *** 6th: John Strange Jocelyn (9 January 1880 – 3 July 1897) *** 7th: William Henry Jocelyn (3 July 1897 – 23 January 1910) ==== Earl of Shannon ==== * Did not attend the ball. ==== Earl of Shelburne ==== * Subsidiary title of the Marquess of Lansdowne (in the peerage of Great Britain). * Did not attend the ball, and did not attend any social events analyzed so far. ==== Earl of Tyrone ==== * Did not attend ==== Earl of Waterford ==== * Not a subsidiary title of the Marquess of Waterford but of the Earl of Shrewsbury in the peerage of England. ==== Earl of Westmeath ==== * Did not attend the ball. ==== Earl of Winterton ==== * Did not attend the ball. === Viscounts and Viscountesses === ==== Viscount Ashbrook ==== * William Spencer Flower, 7th Viscount Ashbrook (1830–1906)<ref>{{Cite journal|date=2025-12-01|title=Viscount Ashbrook|url=https://en.wikipedia.org/w/index.php?title=Viscount_Ashbrook&oldid=1325071512|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, has no social presence at about this time. ==== Viscount Banger ==== * Did not attend the ball but attended a few social events at about this time. * Edward Ward, 4th Viscount Bangor (1827–1881)<ref>{{Cite journal|date=2026-03-16|title=Edward Ward, 4th Viscount Bangor|url=https://en.wikipedia.org/w/index.php?title=Edward_Ward,_4th_Viscount_Bangor&oldid=1343882576|journal=Wikipedia|language=en}}</ref> * Henry William Crosbie Ward, 5th Viscount Bangor (1828–1911)<ref>{{Cite journal|date=2026-03-02|title=Henry Ward, 5th Viscount Bangor|url=https://en.wikipedia.org/w/index.php?title=Henry_Ward,_5th_Viscount_Bangor&oldid=1341354058|journal=Wikipedia|language=en}}</ref> ==== Viscount Boyne ==== * Did not attend the ball, but did attend a number of events at about this time. ==== Viscount Callan ==== * Did not attend the ball, and does not have much if any social presence at about this time. * The Viscount Callan is a subsidiary title of the Earl of Denbigh in the Peerage of England. ==== Viscount Charlemont ==== * Did not attend the ball. * Colonel James Alfred Caulfeild, 7th Viscount Charlemont (20 March 1830 – 4 July 1913), Irish<ref>{{Cite journal|date=2026-05-02|title=James Caulfeild, 7th Viscount Charlemont|url=https://en.wikipedia.org/w/index.php?title=James_Caulfeild,_7th_Viscount_Charlemont&oldid=1352129469|journal=Wikipedia|language=en}}</ref> * Unionist ==== Viscount Chetwynd ==== * Does not seem to have attended the ball, but Chetwynds were socially very active at about this time. * Godfrey John Boyle Chetwynd, 8th Viscount Chetwynd (1863–1936), British<ref>{{Cite journal|date=2026-05-24|title=Godfrey Chetwynd, 8th Viscount Chetwynd|url=https://en.wikipedia.org/w/index.php?title=Godfrey_Chetwynd,_8th_Viscount_Chetwynd&oldid=1355878192|journal=Wikipedia|language=en}}</ref> ==== Viscount de Vesci ==== * Did not attend the ball but attended several social events at about this time. * 4th Viscount de Vesci: John Robert William Vesey (23 December 1875 – 6 July 1903)<ref name=":1" /> * "The family seat was Abbeyleix House, near Abbeyleix, County Laois."<ref>{{Cite journal|date=2026-02-09|title=Viscount de Vesci|url=https://en.wikipedia.org/w/index.php?title=Viscount_de_Vesci&oldid=1337491855|journal=Wikipedia|language=en}}</ref> ==== Viscount Dillon ==== * Did not attend the ball, but several Dillons attended other social events at about this time. ==== Viscount Doneraile<ref>{{Cite journal|date=2026-01-16|title=Viscount Doneraile|url=https://en.wikipedia.org/w/index.php?title=Viscount_Doneraile&oldid=1333262628|journal=Wikipedia|language=en}}</ref> ==== * Did not attend the ball, but did attend the Warwick Bal Poudré and few other social events at about this time. * Hayes St Leger, 4th Viscount Doneraile (1818–1887) * Richard Arthur St Leger, 5th Viscount Doneraile (1825–1891) * Edward St Leger, 6th Viscount Doneraile (1866–1941) ==== [[Social Victorians/People/Downe|Viscount Downe]] ==== * Did not attend the ball but attended many social events at about this time. * Major-General Hugh Richard Dawnay, 8th Viscount Downe (20 July 1844 – 21 January 1924)<ref>{{Cite journal|date=2026-03-24|title=Hugh Dawnay, 8th Viscount Downe|url=https://en.wikipedia.org/w/index.php?title=Hugh_Dawnay,_8th_Viscount_Downe&oldid=1345146095|journal=Wikipedia|language=en}}</ref> * British Army general ==== Viscount Ferrard ==== * See Viscount Massereene, below. By the end of the century, it was the Viscount and Viscountess of Massereene and Ferrard. ==== Viscount Fitzmaurice ==== * A subsidiary title of the Marquess of Lansdowne (in the Peerage of Great Britain). * 6th Viscount Fitzmaurice, Henry Charles Keith Petty-FitzMaurice (5 July 1866 – 3 June 1927)<ref>"Henry Charles Keith Petty-FitzMaurice, 5th Marquess of Lansdowne." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 958 https://www.thepeerage.com/p959.htm#i9586.</ref> ==== Viscount Gage ==== * Henry Charles Gage, 5th Viscount Gage (1854–1912)<ref>{{Cite journal|date=2025-06-21|title=Viscount Gage|url=https://en.wikipedia.org/w/index.php?title=Viscount_Gage&oldid=1296646030|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but members of this family attended a number of social events at about this time. ==== Viscount Galway ==== * George Edmund Milnes Monckton-Arundell, 7th Viscount Galway (1844–1931), British conservative<ref>{{Cite journal|date=2025-08-08|title=George Monckton-Arundell, 7th Viscount Galway|url=https://en.wikipedia.org/w/index.php?title=George_Monckton-Arundell,_7th_Viscount_Galway&oldid=1304770631|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but Viscount and Viscountess Galway attended many social events at about this time. * Subsidiary Title ** Baron Monckton (in the Peerage of the United Kingdom) ==== Viscount Gormanston ==== * Did not attend the ball, has no social presence in the late 19th-century newspapers at this time. ==== [[Social Victorians/People/Gort|Viscount Gort]] ==== * Did not attend the ball, but attended some social events at about this time. * Standish Prendergast Vereker, 4th Viscount Gort (1819–1900)<ref>"Standish Prendergast Vereker, 4th Viscount Gort." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 4626 https://www.thepeerage.com/p4627.htm#i46261.</ref> * John Gage Prendergast Vereker, 5th Viscount Gort (1849–1902)<ref>{{Cite journal|date=2025-05-28|title=John Vereker, 5th Viscount Gort|url=https://en.wikipedia.org/w/index.php?title=John_Vereker,_5th_Viscount_Gort&oldid=1292670203|journal=Wikipedia|language=en}}</ref> ==== Viscount Grandison ==== * Did not attend the ball, has no social presence in the late 19th-century newspapers at this time. * The Viscount Grandison is a subsidiary title of the Earl of Jersey in the Peerage of England. ==== Viscount Grimston ==== * Subsidiary title of the Earl of Verulam (in the Peerage of the United Kingdom) * Did not attend the ball, but a number of members of this family attended social events at about this time. ==== Viscount Harberton ==== * Did not attend the ball; Viscountess Harberton is mentioned once in social events at about this time so far. * James Spencer Pomeroy, 6th Viscount Harberton (1836–1912)<ref>"James Spencer Pomeroy, 6th Viscount Harberton." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person Page 4315 https://www.thepeerage.com/p43151.htm#i431502.</ref> * Florence Wallace Pomeroy, Viscountess Harberton (1843–1911), suffragette, cyclist, President of the Rational Dress Society<ref>{{Cite journal|date=2026-03-12|title=Florence Wallace Pomeroy, Viscountess Harberton|url=https://en.wikipedia.org/w/index.php?title=Florence_Wallace_Pomeroy,_Viscountess_Harberton&oldid=1343082631|journal=Wikipedia|language=en}}</ref> ==== Viscount Lifford ==== * Did not attend the ball; the only social event at about this time so far is the Queen's Diamond Jubilee garden party. * James Hewitt, 4th Viscount Lifford (1811–1887)<ref>{{Cite journal|date=2025-09-11|title=James Hewitt, 4th Viscount Lifford|url=https://en.wikipedia.org/w/index.php?title=James_Hewitt,_4th_Viscount_Lifford&oldid=1310741456|journal=Wikipedia|language=en}}</ref> * James Wilfrid Hewitt, 5th Viscount Lifford (12 October 1837 – 20 March 1913)<ref>"James Wilfrid Hewitt, 5th Viscount Lifford." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person Page 2227 https://www.thepeerage.com/p22271.htm#i222701.</ref> ==== Earl of Listowel ==== * Pronounced "Lish-''toe''-ell."<ref>{{Cite journal|date=2024-10-15|title=Earl of Listowel|url=https://en.wikipedia.org/w/index.php?title=Earl_of_Listowel&oldid=1251322273|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but hosted and attended social events at about this time. * William Hare, 3rd Earl of Listowel (1833–1924)<ref>{{Cite journal|date=2026-04-17|title=William Hare, 3rd Earl of Listowel|url=https://en.wikipedia.org/w/index.php?title=William_Hare,_3rd_Earl_of_Listowel&oldid=1349570352|journal=Wikipedia|language=en}}</ref>, Irish peer * Subsidiary Title ** Viscount Ennismore and Listowel ** Baron Ennismore ==== Viscount Massereene ==== * Did not attend the ball but did attend a few events at about this time. See Viscount Ferrard, above. By the end of the century, it was the Viscount and Viscountess of Massereene and Ferrard. * Anglo-Irish * Clotworthy John Eyre Skeffington, 11th Viscount Massereene (9 October 1842 – 26 June 1905)<ref>{{Cite journal|date=2024-11-23|title=Clotworthy Skeffington, 11th Viscount Massereene|url=https://en.wikipedia.org/w/index.php?title=Clotworthy_Skeffington,_11th_Viscount_Massereene&oldid=1259199982|journal=Wikipedia|language=en}}</ref> and 4th Viscount Ferrard (28 April 1863 – 26 June 1905) ==== Viscount Molesworth ==== * Did not attend the ball, but attended the Warwick Bal Poudré and a number of other social events at about this time. * Samuel Molesworth, 8th Viscount Molesworth (1829–1906), may have been a Quaker ==== Viscount Monck ==== * Did not attend the ball, but attended a number of social events at about this time. * Charles Stanley Monck, 4th Viscount Monck (1819–1894)<ref>{{Cite journal|date=2026-04-05|title=Charles Monck, 4th Viscount Monck|url=https://en.wikipedia.org/w/index.php?title=Charles_Monck,_4th_Viscount_Monck&oldid=1347311992|journal=Wikipedia|language=en}}</ref>, British * Henry Power Charles Stanley Monck, 5th Viscount Monck (1849–1927)<ref>"Henry Power Charles Stanley Monck, 5th Viscount Monck of Ballytrammon." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 3880 https://www.thepeerage.com/p3881.htm#i38802.</ref> ==== Viscount Mountgarret ==== * Did not attend the ball, has no social presence in the late 19th-century newspapers at this time. ==== [[Social Victorians/People/Powerscourt|Viscount Powerscourt]] ==== * Mervyn Wingfield, 7th Viscount Powerscourt (1836–1904)<ref name=":0">{{Cite journal|date=2026-02-18|title=Mervyn Wingfield, 7th Viscount Powerscourt|url=https://en.wikipedia.org/w/index.php?title=Mervyn_Wingfield,_7th_Viscount_Powerscourt&oldid=1339057453|journal=Wikipedia|language=en}}</ref> * Did not attend the ball, but members of this family attended a number of social events at about this time. * Subsidiary Title ** Baron Powerscourt (in the Peerage of the United Kingdom), 1885<ref name=":0" /> ==== Viscount Southwell ==== * Did not attend the ball, though the Viscount and Viscountess attended a few social events at about this time. * 5th<ref name=":1" />: Arthur Robert Pyers Southwell (26 April 1878 – 5 October 1944)<ref>"Arthur Robert Pyers Southwell, 5th Viscount Southwell of Castle Mattress." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page https://www.thepeerage.com/p7550.htm#i75497.</ref> ==== Viscount Valentia ==== * Did not attend the ball, attended some social events at about this time. Was on the Welcome Council for the 1887 American Exhibition. === Barons and Baronesses === Not all the barons extant at the end of the 19th century and listed on the Wikipedia [[wikipedia:Peerage_of_Ireland|Peerage of Ireland]] page are here — only the ones who were active socially. ==== Baron Conway and Killultagh ==== * Did not attend the ball, but people from the Conway and Seymour families attended a number of social events at about this time. * Subsidiary title of the Marquess of Hertford (in the Peerage of England and Great Britain). * Francis Hugh George Seymour, 5th Marquess of Hertford (1812–1884)<ref>{{Cite journal|date=2026-04-05|title=Francis Seymour, 5th Marquess of Hertford|url=https://en.wikipedia.org/w/index.php?title=Francis_Seymour,_5th_Marquess_of_Hertford&oldid=1347294689|journal=Wikipedia|language=en}}</ref> * Hugh de Grey Seymour, 6th Marquess of Hertford (1843–1912)<ref>{{Cite journal|date=2026-04-05|title=Hugh Seymour, 6th Marquess of Hertford|url=https://en.wikipedia.org/w/index.php?title=Hugh_Seymour,_6th_Marquess_of_Hertford&oldid=1347303090|journal=Wikipedia|language=en}}</ref> ==== Baron Digby ==== * Did not attend the ball, but people from this family attended a number of social events at about this time. * Edward St Vincent Digby, 9th and 3rd Baron Digby (1809–1889)<ref>{{Cite journal|date=2025-12-15|title=Edward Digby, 9th Baron Digby|url=https://en.wikipedia.org/w/index.php?title=Edward_Digby,_9th_Baron_Digby&oldid=1327712265|journal=Wikipedia|language=en}}</ref> * Edward Henry Trafalgar Digby, 10th and 4th Baron Digby (1846–1920)<ref>{{Cite journal|date=2026-01-26|title=Edward Digby, 10th Baron Digby|url=https://en.wikipedia.org/w/index.php?title=Edward_Digby,_10th_Baron_Digby&oldid=1334892957|journal=Wikipedia|language=en}}</ref> ==== Baron Inchiquin ==== * Did not attend the ball, but people from this family attended a number of social events at about this time. * Edward Donough O'Brien, 14th Baron Inchiquin (1839–1900)<ref>{{Cite journal|date=2026-04-28|title=Edward O'Brien, 14th Baron Inchiquin|url=https://en.wikipedia.org/w/index.php?title=Edward_O%27Brien,_14th_Baron_Inchiquin&oldid=1351543832|journal=Wikipedia|language=en}}</ref> == Peerage of the United Kingdom of Great Britain and Ireland == After the forced 1801 Act of Union. === Earls and Countesses === ==== Earl of Limerick ==== * Did not attend the ball, but did attend a number of events at about this time. ==== Earl of Norbury ==== * Did not attend the ball, but attended some social events at about this time. * Subsidiary Title ** Baron Norbury ==== Earl of Ranfurly ==== * Did not attend the ball, and they have a small social presence in the newspapers in the 1880s and 1890s. ==== Earl of Rosse ==== * Did not attend the ball, but did attend a few events at about this time. == Peerage of the United Kingdom == * Lurgan == Irish Nationalists == == Irish Unionists == == Irish Aristocrats at the Duchess of Devonshire's 1897 Fancy-dress Ball == ==== [[Social Victorians/People/Abercorn|Duke and Duchess of Abercorn]] ==== * This dukedom is in the peerage of the United Kingdom of Great Britain and Ireland * James Hamilton, 1st Duke of Abercorn (1811–1885), elder son of Lord Hamilton, "styled Viscount Hamilton from 1814 to 1818 and The Marquess of Abercorn from 1818 to 1868, was a Conservative statesman who twice served as Lord Lieutenant of Ireland."<ref>{{Cite journal|date=2026-04-05|title=James Hamilton, 1st Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_1st_Duke_of_Abercorn&oldid=1347253763|journal=Wikipedia|language=en}}</ref> * James Hamilton, 2nd Duke of Abercorn (1838–1913), eldest son of the 1st Duke, "styled Viscount Hamilton until 1868 and Marquess of Hamilton from 1868 to 1885, was a British nobleman, courtier, and diplomat."<ref>{{Cite journal|date=2026-01-25|title=James Hamilton, 2nd Duke of Abercorn|url=https://en.wikipedia.org/w/index.php?title=James_Hamilton,_2nd_Duke_of_Abercorn&oldid=1334676058|journal=Wikipedia|language=en}}</ref> * The Hamilton who became the 3rd duke attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did a few other members of this family. * Subsidiary Titles ** Marquess of Abercorn ** Viscount Hamilton ** Viscount Strabane, county Tyrone *Papers **PRONI for the Abercorn papers [GB 0255 PRONI/D623] **Some individuals' papers (the Tighe Hamilton Howard papers, https://iar.ie/archive/tighe-hamilton-howard-papers) from the Hamilton family are in the National Library of Ireland. "An item level catalogue is available online. These papers form part of the Wicklow Papers (Collection List 69) that are held in the Department of Manuscripts at the National Library of Ireland." ***VII. Sarah Howard Papers, 1830-1887. ****[***] VII.ii. Letters from Sarah Howard to her husband the Hon. Rev. Francis Howard, [n.d.] Call number: '''MS 38,639/2/2''' ****[***] VII.iii. Correspondence between Sarah Howard and her daughter Lady Caroline Howard, ca. 1851 - ca. 1891. Call number: '''MS 38,639/2/3''' ****VII.iv. Correspondence between Sarah Howard and her son Charles Howard, 5th Earl of Wicklow, 1853-ca.1870. Call number: '''MS 38,639/2/4''' *****VII.v. Correpondence between Sarah Howard and her son Cecil Howard, 6th Earl of Wicklow, ca. 1855-1876. Call number: '''MS 38,639/2/5''' ******** VII.vi. Correspondence between Sarah Howard and her daughters Lady Louisa and Lady Alice Howard, 1855-ca. 1877. Call number: '''MS 38,639/2/6''' ******** VII.vix. Additional correspondence of Sarah Howard of Wingfield, Bray Co. Wicklow, 1865-1887. Call number: '''MS 38,639/2/9''' ***VIII. Lady Caroline Howard Papers, 1852-1919. ****VIII.i. Correspondence between Lady Caroline Howard and her brother Charles, Earl of Wicklow, 1852-1880. Call number: '''MS 38,639/2/11''' ****VIII.iv. Additional correspondence of Lady Caroline Howard, 1868-1919. Call number: '''MS 38,639/2/14''' ****VIII.v. Additional papers of Lady Caroline Howard, 1900. Call number: '''MS 38,639/2/15''' ***IX. Additional Howard family correspondence, 1773-1900. ****IX.vii. Correspondence and papers of Lady Louisa Howard, 1856-1907. Call number: '''MS 38,639/2/22''' ******* IX.viii. Correspondence and papers of Lady Alice Howard, [n.d.] Call number: '''MS 38,639/2/23''' ***XI. Other papers, 1737-1913. ****XI.i. Miscellaneous correspondence, 1753-1891. Call number: '''MS 38,639/2/27''' ***Wicklow Papers ****** Journals of Lady Caroline Howard including some accounts of her tours abroad, 1873 Jan. - March, 1875 Aug. - Sept., & 1882 Jan. - April. Call number: '''MS 3586-3588''' ****** Diaries of Lady Louisa Howard including accounts of her travels on the Continent, 1862 Oct. - 1869 June, 1871 April - 1873 April and 1877 Oct. - 1883 July. Call number: '''MS 3589-3593''' ****Diaries of Lady Caroline Howard, 1862 Oct. - 1870 May. Call number: '''MS 3594-3599''' ******* Diaries of Lady Alice Howard, Shelton Abbey and Bray, Co. Wicklow, 1874-1922. Call number: '''MS 3600-3625''' ***** Journals of Lady Alice Howard, including account of tours on the Continent, 1860 June - Oct, 1865 Aug. - 1866 Feb., 1869 Nov. - 1870 Nov. Call number: '''MS 4793-4795''' *** ==== [[Social Victorians/People/Londonderry|Marquess and Marchioness of Londonderry]] ==== * The Marquess and Marchioness attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House, she led one of the courts as Maria Thérèse, plus two of their children attended, one of whom is Viscount Castlereagh. * Charles Stewart Vane-Tempest-Stewart, 6th Marquess of Londonderry<ref>"Charles Stewart Vane-Tempest-Stewart, 6th Marquess of Londonderry." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 1277 https://www.thepeerage.com/p1278.htm#i12772.</ref> * Lady Theresa Susey Helen Chetwynd-Talbot, Marchioness of Londonderry<ref>"Lady Theresa Susey Helen Chetwynd-Talbot." ''The Peerage: A Genealogical Survey of the Peerage of Britain as Well as the Royal Families of Europe''. Person page 1277 https://www.thepeerage.com/p1278.htm#i12771.</ref> * Subsidiary Titles ** [[Social Victorians/People/Londonderry|Earl of Londonderry]] ** Viscount Castlereagh — Charles Stewart Henry Vane-Tempest-Stewart (6 November 1884 – 8 February 1915) *Papers **In PRONI [GB 0255 PRONI/D2846]: "The Theresa, Lady Londonderry Papers comprise c.4,600 papers and 15 volumes of diaries, scrapbooks, etc, 1858-1919, mainly of Theresa, Marchioness of Londonderry (1856-1919), wife/widow of the 6th Marquess, but including some papers of the 6th Marquess himself, of and about his mother, Mary Cornelia, widow of the 5th Marquess, and of his brothers Lords Henry and Herbert Vane-Tempest."<ref>{{Cite web|url=https://iar.ie/archive/theresa-lady-londonderry-papers/|title=Theresa, Lady Londonderry Papers|website=Irish Archives Resource|language=en-US|access-date=2026-06-06}}</ref> **In PRONI [GB 0255 PRONI/D3099]: the "Papers of the 7th Marquess of Londonderry and his wife Edith" collection also hold the papers of Edith's father, [[Social Victorians/People/Henry Chaplin|Henry, 1st Viscount Chaplin]], who attended the ball, as did she and a brother. [D3099/1 Henry, 1st Viscount Chaplin, father-in-law of 7th Marquess of Londonderry. Political and personal papers; D3099/3 Edith Helen Chaplin, wife of 7th Marquess of Londonderry. Personal letters and papers]<ref>{{Cite web|url=https://iar.ie/archive/papers-7th-marquess-londonderry-wife-edith/|title=Papers of the 7th Marquess of Londonderry and his wife Edith|website=Irish Archives Resource|language=en-US|access-date=2026-06-06}}</ref> ==== [[Social Victorians/People/Lucan|Earl of Lucan]] ==== * Some members of the family attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House, and the family attended a number of social events at this time. * Papers: Irish Archives Resource has one listing for Lucan, but it doesn't seem to be relevant: too late and not about the family. ==== [[Social Victorians/People/Ormonde|Marquess and Marchioness of Ormonde]] ==== * The marchioness and her daughters attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House, though nobody mentions the Marquess. * James Edward Butler, 3rd Marquess of Ormonde and 21st Earl of Ormonde (1844–1919)<ref>{{Cite journal|date=2026-05-03|title=Earl of Ormond (Ireland)|url=https://en.wikipedia.org/w/index.php?title=Earl_of_Ormond_(Ireland)&oldid=1352334266|journal=Wikipedia|language=en}}</ref> Now extinct; earldom dormant. Castle Kilkenny was their manor, but they don't appear to have any papers. * Subsidiary Titles * Papers: Irish Archives Resource has one listing, but it's not about the family, the name of a road uses the word ''Ormonde''. ==== [[Social Victorians/People/Antrim|Earl of Antrim]] ==== * The earl and countess did not attend the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House, but two of his brothers did. * Papers ** [https://iar.ie/archive/earl-antrim-estate-papers/ Estate papers of the Earls of Antrim] [GB 0255 PRONI/D2977] are in PRONI. I don't see personal papers listed, but the collection has 50,000 documents 1603–1967. ** Also "D4091 Papers of Sir Schomberg MacDonnell, Louisa Countess of Antrim and the Stuart family of Dalness. MIC615 The diaries of Louisa, Countess of Antrim."<ref>{{Cite web|url=https://iar.ie/archive/earl-antrim-estate-papers/|title=Earl of Antrim Estate Papers|website=Irish Archives Resource|language=en-US|access-date=2026-06-06}}</ref> ==== [[Social Victorians/People/Arran|Earl of Arran]] ==== * Attended the ball. * Subsidiary Titles ** Viscount Sudley: 5th: Arthur Saunders William Charles Fox Gore (25 Jun 1884-14 Mar 1901), 5th Earl of Arran<ref name=":1" /> *Papers ==== [[Social Victorians/People/Belmore|Earl Belmore]] ==== * Did not attend the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House, although [[Social Victorians/People/Rowton|Montagu Lowry-Corry, 1st Baron Rowton]] did, but did attend a number of social events about this time. * 4th Earl: Somerset Richard Lowry-Corry (17 Dec 1845-6 Apr 1913)<ref>{{Cite journal|date=2026-04-17|title=Somerset Lowry-Corry, 4th Earl Belmore|url=https://en.wikipedia.org/w/index.php?title=Somerset_Lowry-Corry,_4th_Earl_Belmore&oldid=1349375684|journal=Wikipedia|language=en}}</ref> * Subsidiary Title ** Viscount Belmore (though the subsidiary title for the heir apparent is Viscount Corry?) *Papers: Belmore Papers [GB 0255 PRONI/D3007]<ref>{{Cite web|url=https://iar.ie/archive/belmore-papers/|title=Belmore Papers|website=Irish Archives Resource|language=en-US|access-date=2026-06-07}}</ref> **D3007/B Rentals and account books (estate, household and personal papers) **D3007/F Curiosa and personal ephemera **D3007/I Private and family letters to Honoria Gladstone, Countess Belmore **D3007/Y Letters and papers of Viscount Corry and the Hon. Cecil Corry, later 5th and 6th Earls Belmore respectively **D3007/Z Family and other photographs ==== [[Social Victorians/People/Dunraven|Earl of Dunraven and Mount-Earl]] ==== * The [[Social Victorians/People/Dunraven|Earl of Dunraven and Mount-Earl]] and Countess of Dunraven, and their daughter Lady Aileen May Wyndham-Quin attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. * Windham Wyndham-Quin, 4th Earl of Dunraven and Mount-Earl (1841–1926)<ref>{{Cite journal|date=2026-05-22|title=Windham Wyndham-Quin, 4th Earl of Dunraven and Mount-Earl|url=https://en.wikipedia.org/w/index.php?title=Windham_Wyndham-Quin,_4th_Earl_of_Dunraven_and_Mount-Earl&oldid=1355461019|journal=Wikipedia|language=en}}</ref>, Anglo-Irish * Papers ==== [[Social Victorians/People/Cole|Earl and Countess of Enniskillen]] ==== * The Earl and Countess and a daughter attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. Papers in PRONI. * Subsidiary Title ** 4th Viscount Enniskillen: Lowry Egerton Cole (12 November 1886 – 28 April 1924)<ref name=":1" /> *Papers ==== [[Social Victorians/People/Crichton|Earl of Erne]] ==== * Some members of the family attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. * The newspapers were very inconsistent in the spelling of the family name Crichton. * Subsidiary Title ** Viscount Erne<ref name=":1" /> *** 3rd Earl of Erne: John Crichton (10 June 1842 – 3 October 1885) *** 4th Earl of Erne: John Henry Crichton (3 October 1885 – 2 December 1914) *Papers: in PRONI. ==== [[Social Victorians/People/Gosford|Earl of Gosford]] ==== * The Earl and Countess of Gosford attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House, as did a son and a daughter. They attended many social events at about this time. * Subsidiary Title ** Viscount Gosford of Market Hill, co. Armagh<ref name=":1" /> *** 5th Earl of Gosford: Archibald Brabazon Sparrow Acheson (15 June 1864 – 11 April 1922) *Papers ==== Earl of Kerry ==== * Subsidiary title of the [[Social Victorians/People/Lansdowne|Marquess of Lansdowne]] (in the peerage of Great Britain). Attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. * Subsidiary Titles ** Viscount Clanmaurice *Papers ==== [[Social Victorians/People/Kilmorey|Earl of Kilmorey]] ==== * Anglo-Irish * Nellie Countess of Kilmorey attended the ball; Francis, 3rd Earl was alive at the time, did he attend? Both he and she attended a number of social events from about this time. * Papers ==== [[Social Victorians/People/Mayo|Earl of Mayo]] ==== * Some members of the family attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. * Viscount Mayo of Monycrower, co. Mayo<ref name=":1" /> ** 7th Earl of Mayo: Dermot Robert Wyndham Bourke (8 February 1872 – 31 December 1927) *Papers ==== [[Social Victorians/People/Midleton|Viscount Midleton]] ==== * Some people from this family seem to have attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House as well as many other social events at about this time. * William Brodrick, 8th Viscount Midleton (6 January 1830 – 18 April 1907), "Irish peer, landowner and Conservative politician in both Houses of Parliament"<ref>{{Cite journal|date=2025-01-05|title=William Brodrick, 8th Viscount Midleton|url=https://en.wikipedia.org/w/index.php?title=William_Brodrick,_8th_Viscount_Midleton&oldid=1267418489|journal=Wikipedia|language=en}}</ref> * Sight and hearing disabilities caused by intermarriage. A daughter became a Republican. * Papers ==== [[Social Victorians/People/Lurgan|Baron Lurgan]] ==== * The Baron, his wife and probably his uncle attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. ** Emily Lady Lurgan ** William Brownlow, Baron Lurgan ** Hon. Cecil Brownlow * Papers, PRONI<ref>{{Cite web|url=https://iar.ie/archive/brownlow-papers/|title=Brownlow Papers|website=Irish Archives Resource|language=en-US|access-date=2026-06-07}}</ref> ==== Baron Carrington ==== * [[Social Victorians/People/Carrington|Charles Robert Wynn-Carington, 1st Marquess of Lincolnshire]] (1843–1928) attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. * Baron Carrington is a subsidiary title of the Marquess of Lincolnshire (created in 1912; Earl Carrington created in 1895).<ref>{{Cite journal|date=2026-05-20|title=Baron Carrington|url=https://en.wikipedia.org/w/index.php?title=Baron_Carrington&oldid=1355207880|journal=Wikipedia|language=en}}</ref> * Papers ==== Baron Dufferin and Claneboye<ref>{{Cite journal|date=2026-02-07|title=Baron Dufferin and Claneboye|url=https://en.wikipedia.org/w/index.php?title=Baron_Dufferin_and_Claneboye&oldid=1337113957|journal=Wikipedia|language=en}}</ref> ==== * Members of this family did attend the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House as well as many social events at about this time. * [[Social Victorians/People/Hamilton Temple Blackwood|Frederick Temple Hamilton-Temple-Blackwood]], 1st Marquess of Dufferin and Ava (1826–1902)<ref>{{Cite journal|date=2026-05-27|title=Frederick Hamilton-Temple-Blackwood, 1st Marquess of Dufferin and Ava|url=https://en.wikipedia.org/w/index.php?title=Frederick_Hamilton-Temple-Blackwood,_1st_Marquess_of_Dufferin_and_Ava&oldid=1356387854|journal=Wikipedia|language=en}}</ref> * Papers ==== Baron Garvagh ==== * [[Social Victorians/People/Garvagh|Florence Canning, Lady Garvagh]] attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball]] at Devonshire House. * Charles John Spencer George Canning, 3rd Baron Garvagh (1852–1915)<ref>{{Cite journal|date=2026-02-06|title=Baron Garvagh|url=https://en.wikipedia.org/w/index.php?title=Baron_Garvagh&oldid=1336941309|journal=Wikipedia|language=en}}</ref> * Papers ==== Baron Rossmore of Monaghan ==== * A [[Social Victorians/People/Naylor|Miss Naylor]] (Lady Rossmore's sister) of this family attended the ball. * Derrick Warner William Westenra, 5th Baron Rossmore (1853–1921)<ref>{{Cite journal|date=2024-08-27|title=Derrick Westenra, 5th Baron Rossmore|url=https://en.wikipedia.org/w/index.php?title=Derrick_Westenra,_5th_Baron_Rossmore&oldid=1242602083|journal=Wikipedia|language=en}}</ref> * Papers == References == {{reflist}} 7zmorgvmht59dgdugierv7u9qgxpydz 4 330132 2816213 2815121 2026-06-18T13:45:24Z AUBSTRAWBS 3060598 2816213 wikitext text/x-wiki Hello and welcome to the Wikiversity '''[[anime]]''' '''[[userbox]]''' page. This page currenly has a comrehensive list of all the anime/manga related userboxes on wikiversity. We are currently bringing over userboxes from wikipedia. ==Various== ===Anime=== {| !Adding this to your page!!Creates this |- |<nowiki>{{User animanga userbox}}</nowiki>||{{User animanga userbox}} |- |<nowiki>{{User anime user anime}}</nowiki>||{{User anime user anime}} |- |<nowiki>{{User anime User anime-0}}</nowiki>||{{User anime User anime-0}} |- |<nowiki>{{User anime User anime-1}}</nowiki>||{{User anime User anime-1}} |- |<nowiki>{{User anime User anime-2}}</nowiki>||{{User anime User anime-2}} |- |<nowiki>{{User anime User anime-3}}</nowiki>||{{User anime User anime-3}} |- |<nowiki>{{User anime User anime-4}}</nowiki>||{{User anime User anime-4}} |- |<nowiki>{{User/anime/User anime-N}}</nowiki>||{{User/anime/User anime-N}} |- |<nowiki>{{Template:User/anime/Better actors}}</nowiki>||{{Template:User/anime/Better actors}} |} === Manga === {| !Adding this to your page!!Creates this |- |<nowiki>{{User/manga/User manga}}</nowiki>||{{User/manga/User manga}} |- |<nowiki>{{User/manga/User manga-}}</nowiki>||{{User/manga/User manga-}} |- |<nowiki>{{User/manga/User manga-o}}</nowiki>||{{User/manga/User manga-o}} |- |<nowiki>{{User/manga/User manga-1}}</nowiki>||{{User/manga/User manga-1}} |- |<nowiki>{{User/manga/User manga-2}}</nowiki>||{{User/manga/User manga-2}} |} == Anime before it gets moved to better page == {| !Adding this to your page!!Creates this |- |<nowiki>{{User gate anime}}</nowiki>||{{User gate anime}} |- |<nowiki>{{User apothecary diaries}}</nowiki>||{{User apothecary diaries}} |} 6n4r9jfmc6zglp70332do6irzj2buny 0 330146 2816276 2815427 2026-06-19T02:07:23Z DavidMCEddy 218607 typo 2816276 wikitext text/x-wiki :''This discusses a 2026-06-18 interview with Dimitar Zagorski<ref name=Dimi><!--Dimitar Zagorski, 3rd-->{{cite Q|Q130719781}}</ref> about the European Commission's targeted public consultation on copyright law, including a video and 29:00 mm:ss podcast excerpted from the interview. The podcast is released 2026-06-27 to the fortnightly "Media & Democracy" show<ref name=M&D><!--Media & Democracy-->{{cite Q|Q127839818}}</ref> syndicated for the [[w:Pacifica Foundation|Pacifica Radio]]<ref><!--Pacifica Radio Network-->{{cite Q|Q2045587}}</ref> Network of [[w:List of Pacifica Radio stations and affiliates|over 200 community radio stations]].''<ref><!--list of Pacifica Radio stations and affiliates-->{{cite Q|Q6593294}}</ref> :''It is posted here to invite others to contribute other perspectives, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] while [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV>The rules of writing from a neutral point of view citing credible sources may not be enforced on other parts of Wikiversity. However, they can facilitate dialog between people with dramatically different beliefs.</ref> and treating others with respect.''<ref name=AGF>[[Wikiversity:Assume good faith|Wikiversity asks contributors to assume good faith]], similar to Wikipedia. The rule in [[w:Wikinews|Wikinews]] was different: Contributors there were asked to [[Wikinews:Never assume|"Don't assume things; be skeptical about everything."]] That's wise. However, we should still treat others with respect while being skeptical.</ref> <!--[[File:Wikimedia concerns with European copyright rules including AI and scientific research.webm|thumb|2026-06-18 interview with Dimi Zagorski about Wikimedia concerns with European copyright rules including AI and scientific research.]]--> <!--[[File:Wikimedia concerns with European copyright rules including AI and scientific research.ogg|thumb|29:00 mm:ss excerpts from a 2026-06-18 interview with Dimi Zagorski about Wikimedia concerns with European copyright rules including AI and scientific research.]]--> Dimitar ("Dimi") Zagorski, 3rd.,<ref name=Dimi/> discusses the European Commission's targeted public consultation on copyright law and other issues that concern the Wikimedia Foundation including AI, scientific research, and the statutory review of the 2019 [[w:Directive on Copyright in the Digital Single Market|Directive on Copyright in the Digital Single Market]]. Zagorski is Policy Director for Wikimedia Europe in Brussels. He is interviewed by Spencer Graves.<ref><!--Spencer Graves-->{{cite Q|Q56452480}}</ref> As background for this interview, we review the structure of the government of the [[w:European Union|European Union]] (EU). == European Union == The European Union (EU) is a political and economic union of 27 member states located primarily in Europe with [[w:Special territories of members of the European Economic Area|32 special territories]] or subnational units of EU member states stretching from [[w:Greenland|Greenland]] to the southern Indian ocean and the South Pacific. The EU has a population of over 450 million and nominal gross domestic product (GDP) of around €19 trillion in 2025; this makes it roughly one sixth of the global economy. The [[w:Institutions of the European Union|Institutions of the European Union]] encompass seven principal decision-making bodies: # [[w:European Parliament|European Parliament]], whose approval is required for proposed legislation to become law. It currently has 720 members (MEPs). It functions roughly like the [[w:United States House of Representatives|House of Representatives in the US]], in that MEPs can amend or reject proposed legislation, though they cannot initiate legislation. # [[w:European Council|European Council]] of heads of state or government. # [[w:Council of the European Union|Council of the European Union]], often referred to simply as the Council and less formally as the "Council of Ministers". This is a legislative body, which works with the European Parliament to amend and approve or veto proposals of the European Commission, which holds the right of initiative; neither the Council nor the Parliament can initiate legislation. The Presidency of the council is not a single post, but is held by a member state's government and rotates every six months. It functions roughly like the [[w:United States Senate|Senate in the US]] but cannot initiate legislation. # [[w:European Commission|European Commission]] (EC), the executive cabinet of the European Union. Currently, there is one Commissioner per member state, including the president (currently [[w:Ursula von der Leyen|Ursula von der Leyen]]), but members are bound by their oath of office to represent the interest of the EU as a whole rather than their home state. All EU legislation must be initiated by the Commission, to be amended and approved or vetoed by the European Parliament and the Council. # [[w:Court of Justice of the European Union|Court of Justice of the European Union]]. # [[w:European Central Bank|European Central Bank]]. # [[w:European Court of Auditors|European Court of Auditors]]. == The need for media reform to improve democracy == This article is part of [[:category:Media reform to improve democracy]]. A summary of episodes to 2025-11-15 is available in [[Media & Democracy lessons for the future]]. ==Discussion == :''[Interested readers are invited to comment here, subject to the Wikimedia rules of [[w:Wikipedia:Neutral point of view|writing from a neutral point of view]] [[w:Wikipedia:Citing sources|citing credible sources]]<ref name=NPOV/> and treating others with respect.<ref name=AGF/>]'' == Notes == {{reflist}} <!--== Bibliography ==--> [[Category:Media]] [[Category:News]] [[Category:Democracy]] [[Category:Politics]] [[Category:Media literacy]] [[Category:Wikimedia]] [[Category:Wikimedia Foundation]] [[Category:Wikimedia Foundation staff]] [[Category:Media reform to improve democracy]] <!--list of categories https://en.wikiversity.org/wiki/Wikiversity:Category_Review [[Wikiversity:Category Review]]--> anwdb3ndk4bzufols88kwko8uk5hfx1 1 330243 2816204 2816148 2026-06-18T12:08:24Z MathXplore 2888076 Reset talk page with [[:w:simple:User:DannyS712/Reset talk|reset talk]] (version 1.1) 2816204 wikitext text/x-wiki {{Talk header}} 6ujz0t3lkt6jsf7d1r360l6l7wj3njb 0 330245 2816203 2816197 2026-06-18T12:07:53Z MathXplore 2888076 Added {{[[Template:BookCat|BookCat]]}} using [[User:1234qwer1234qwer4/BookCat.js|BookCat.js]] 2816203 wikitext text/x-wiki <gallery mode=packed heights=200> Ammophila inaturalist 62890466 02.jpg Ammophila vulcania inaturalist 190579369 4.jpg Chalybion spinolae inaturalist 266865342.jpg Chalybion spinolae inaturalist 35158124.jpg Chlorion maxillosum inaturalist 11094810.jpg Isodontia inaturalist 117326575.jpg Podalonia canescens inaturalist 36851303.jpg Prionyx kirbii inaturalist 144918989.jpg Prionyx inaturalist 24434071.jpg Sphex decipiens inaturalist 72628134 01.jpg </gallery> {{BookCat}} n1bugg7hy95iwpuq9bij4k5adut6wwf 2816241 2816203 2026-06-18T19:41:25Z Alandmanson 1669821 2816241 wikitext text/x-wiki <gallery mode=packed heights=200> Ammophila inaturalist 62890466 02.jpg Ammophila vulcania inaturalist 190579369 4.jpg Chalybion spinolae inaturalist 266865342.jpg Chalybion spinolae inaturalist 35158124.jpg Chalybion 2019 12 02 2314.jpg Chlorion maxillosum inaturalist 11094810.jpg Isodontia inaturalist 117326575.jpg Podalonia canescens inaturalist 36851303.jpg Prionyx kirbii inaturalist 144918989.jpg Prionyx inaturalist 24434071.jpg Sphex decipiens inaturalist 72628134 01.jpg </gallery> {{BookCat}} 74uh2m42dr2mogpzbx07esv0lia54e1 3 330246 2816205 2026-06-18T12:09:32Z MathXplore 2888076 vandalism1 ([[m:User:ZbVl/VD|Vandoom]]) 2816205 wikitext text/x-wiki == 2026-06-18 == [[File:Information.svg|25px|alt=Information icon]] Hello, I’m letting you know that one or more of your recent contributions have been reverted because they did not appear constructive. If you would like to experiment, please use the [[Wikiversity:Sandbox|sandbox]] or ask for assistance at the [[Wikiversity:Colloquium|Colloquium]]. Thank you.<!-- Glow-vandalism1 @ 1781784573267.5s --><nowiki></nowiki> [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 12:09, 18 June 2026 (UTC) nk7mc6bimeqp74f3hj05uxun4tgqnzq 10 330247 2816208 2026-06-18T13:12:59Z AUBSTRAWBS 3060598 Created page with "{{userbox | border-c = #77E0E8 | id = [[:Category:Wikipedians interested in anime and manga|^^]] | id-c = #77E0E8 | id-s = 14 | info = This user is a moderate fan of manga. | info-c = #D0F8FF | info-lh = 1.25em | info-s = 8 }}" 2816208 wikitext text/x-wiki {{userbox | border-c = #77E0E8 | id = [[:Category:Wikipedians interested in anime and manga|^^]] | id-c = #77E0E8 | id-s = 14 | info = This user is a moderate fan of manga. | info-c = #D0F8FF | info-lh = 1.25em | info-s = 8 }} si8iu2zrpkqinve1h2rd7l3frw8il95 2816210 2816208 2026-06-18T13:43:42Z AUBSTRAWBS 3060598 2816210 wikitext text/x-wiki {{userbox | border-c = #77E0E8 | id = [[:Category:ians interested in anime and manga|^^]] | id-c = #77E0E8 | id-s = 14 | info = This user is a moderate fan of manga. | info-c = #D0F8FF | info-lh = 1.25em | info-s = 8 }} 3pu5xa5ccs1ngm731u309rcmfo674ki 2816211 2816210 2026-06-18T13:45:06Z AUBSTRAWBS 3060598 AUBSTRAWBS moved page [[User/manga/User manga-2]] to [[Template:User/manga/User manga-2]] 2816210 wikitext text/x-wiki {{userbox | border-c = #77E0E8 | id = [[:Category:ians interested in anime and manga|^^]] | id-c = #77E0E8 | id-s = 14 | info = This user is a moderate fan of manga. | info-c = #D0F8FF | info-lh = 1.25em | info-s = 8 }} 3pu5xa5ccs1ngm731u309rcmfo674ki 0 330248 2816212 2026-06-18T13:45:06Z AUBSTRAWBS 3060598 AUBSTRAWBS moved page [[User/manga/User manga-2]] to [[Template:User/manga/User manga-2]] 2816212 wikitext text/x-wiki #REDIRECT [[Template:User/manga/User manga-2]] cd2lzf6jk8luqd8dpw405e8zgxhbgbe 6 330249 2816215 2026-06-18T13:52:13Z Young1lim 21186 {{Information |Description=Carry Lookahead Adders 2A traditional (20260618 - 20260617) |Source={{own|Young1lim}} |Date=2026-06-18 |Author=Young W. 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Thanks. </div><!-- Glow-advert1 @ 1781821021798s --><nowiki></nowiki> [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 22:16, 18 June 2026 (UTC) 06t2sswv5p28hn1y05u19eatfn7u7un 3 330259 2816255 2026-06-18T22:18:47Z MathXplore 2888076 advert1 ([[m:User:ZbVl/VD|Vandoom]]) 2816255 wikitext text/x-wiki == 2026-06-18 == <div class="mw-content-ltr" dir="ltr" style="text-align: left" lang="en">[[File:Information.svg|25px|alt=Information icon]] Hello. Apologies for writing this in English, but I wanted to let you know that one or more of [[Special:Contributions/CIAMobility|your recent contributions]] have been undone because they appeared to be promotional. [[:m:en:WP:SOAPBOX|Advertising or using <span style="white-space:nowrap">Wikiversity</span> as a "soapbox"]] are not permitted. Take a look at the welcome pages to learn more about <span style="white-space:nowrap">Wikiversity</span>. Thanks. </div><!-- Glow-advert1 @ 1781821131833.5s --><nowiki></nowiki> [[User:MathXplore|MathXplore]] ([[User talk:MathXplore|discuss]] • [[Special:Contributions/MathXplore|contribs]]) 22:18, 18 June 2026 (UTC) 3mnjqltfwyzydy6oqa79xescdae02cg 0 330260 2816261 2026-06-19T00:24:20Z Howie2024 2995240 Write in subpage: PDT and Decoherence 2816261 wikitext text/x-wiki = Probability Dilation Theory / Decoherence Analogy and Simulation = ''This page is a subpage of [[Probability Dilation Theory]] and develops a speculative but mathematically well‑defined analogy between iterative probability dilation and quantum decoherence. It includes definitions, motivation, and a reproducible Python simulation.'' == 1. Overview == This page explores how Probability Dilation Theory (PDT) can be used to construct toy models that resemble certain statistical aspects of quantum decoherence. This work is: mathematical, not physical exploratory, not authoritative analogical, not a claim about quantum mechanics open for critique and improvement The goal is to provide a clear, reproducible framework for studying how iterative reweighting of probability measures can mimic the probability‑flow behavior seen in decoherence processes. == 2. Background: PDT and Decoherence == === 2.1 Probability Dilation Theory (PDT) === PDT studies how a probability measure :<math>P</math> is transformed by a positive dilation field :<math>D(x) > 0</math> via the operator :<math>P_D(A) = \frac{\int_A D(x)\, dP(x)}{\int_X D(x)\, dP(x)}.</math> Iterating this operator produces a sequence :<math>P_{n+1} = \mathcal{D}(P_n)</math> which may converge to attractors, fixed points, or stable distributions. === 2.2 Decoherence (informal summary) === In quantum mechanics, decoherence describes how a system loses phase coherence due to interaction with an environment. A density matrix :<math>\rho</math> evolves under a completely positive trace‑preserving (CPTP) map :<math>\mathcal{E}(\rho)</math> that typically: leaves diagonal probabilities unchanged or slowly biased exponentially suppresses off‑diagonal terms (coherences) This produces an effectively classical mixture. === 2.3 Why compare them? === Although PDT is purely classical, the diagonal part of many decoherence channels behaves like a dilation step, and the off‑diagonal decay can be modeled as a simple contraction. This makes PDT a useful toy model for exploring: probability concentration pointer‑state attractors emergent classicality iterative reweighting dynamics No physical claims are made. == 3. A Minimal Toy Model == We consider a two‑state system with classical probabilities :<math>P_n = (p_0^{(n)}, p_1^{(n)})</math> and a coherence magnitude :<math>r_n = |c_n|.</math> The model consists of two coupled updates: === 3.1 PDT dilation on probabilities === Given a dilation field :<math>D = (D_0, D_1)</math> we update :<math>p_i^{(n+1)} = \frac{D_i\, p_i^{(n)}}{D_0 p_0^{(n)} + D_1 p_1^{(n)}}.</math> This biases the system toward states with larger <math>D_i</math>. === 3.2 Coherence decay === We model decoherence by :<math>r_{n+1} = \alpha\, r_n</math> with :<math>0 \le \alpha < 1.</math> This is not quantum mechanical — it is a simple exponential decay rule. === 3.3 Combined update === Each iteration applies: PDT dilation on the diagonal Exponential decay on the coherence magnitude This produces a system that: flows toward a classical attractor loses coherence over time resembles decoherence in its statistical behavior == 4. Numerical Simulation (Python) == Below is a complete, runnable Python script that simulates the toy model and produces probability and coherence plots. <syntaxhighlight lang="python"> import numpy as np import matplotlib.pyplot as plt def simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=20 ): p0 = p0_init p1 = 1.0 - p0_init coh = coherence_init p0_hist = [p0] p1_hist = [p1] coh_hist = [coh] for n in range(steps): # PDT dilation step Z = D0 * p0 + D1 * p1 p0 = (D0 * p0) / Z p1 = (D1 * p1) / Z Coherence decay coh = alpha * coh p0_hist.append(p0) p1_hist.append(p1) coh_hist.append(coh) return np.array(p0_hist), np.array(p1_hist), np.array(coh_hist) if name == "main": p0_hist, p1_hist, coh_hist = simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=25 ) t = np.arange(len(p0_hist)) fig, ax1 = plt.subplots() ax1.set_xlabel("Iteration") ax1.set_ylabel("Probability", color="tab:blue") ax1.plot(t, p0_hist, label="p0", color="tab:blue") ax1.plot(t, p1_hist, label="p1", color="tab:cyan", linestyle="--") ax1.tick_params(axis="y", labelcolor="tab:blue") ax1.legend(loc="upper left") ax2 = ax1.twinx() ax2.set_ylabel("Coherence |c|", color="tab:red") ax2.plot(t, coh_hist, label="|c|", color="tab:red") ax2.tick_params(axis="y", labelcolor="tab:red") fig.tight_layout() plt.title("Iterative Dilation + Decoherence Toy Model") plt.show() </syntaxhighlight> == 5. Interpretation == === 5.1 Probability flow === If <math>D_0 > D_1</math>, the system flows toward :<math>p_0 \to 1.</math> This is analogous to a pointer state in decoherence. === 5.2 Coherence decay === The coherence magnitude :<math>r_n</math> decays exponentially, mimicking the suppression of off‑diagonal terms in a density matrix. === 5.3 Combined effect === The system becomes: more classical (probabilities concentrate) less coherent (off‑diagonal terms vanish) This mirrors the qualitative behavior of decoherence. == 6. Limitations == This is not a quantum model. No physical claims are made about dilation fields. The analogy is structural, not ontological. Decoherence involves entanglement; PDT does not. The model is intended for intuition and exploration only. == 7. Open Questions == Can more general decoherence channels be embedded in PDT? What are the fixed points of multi‑state dilation systems? Can continuous‑time dilation flows be defined? Are there PDE analogues of iterative dilation? Can this framework be useful in machine learning or statistical physics? == 8. See Also == [[Probability Dilation Theory]] [[Quantum decoherence]] [[Density matrix]] [[Bayesian updating]] == 9. Invitation for Collaboration == This page is part of an ongoing exploration of PDT and its possible mathematical analogies. Feedback, critique, and contributions from mathematicians, physicists, and computer scientists are welcome. ogzfyiybo9vc088enlgwpomdpyjgi6x 2816266 2816261 2026-06-19T01:17:51Z Howie2024 2995240 /* 8. See Also */ 2816266 wikitext text/x-wiki = Probability Dilation Theory / Decoherence Analogy and Simulation = ''This page is a subpage of [[Probability Dilation Theory]] and develops a speculative but mathematically well‑defined analogy between iterative probability dilation and quantum decoherence. It includes definitions, motivation, and a reproducible Python simulation.'' == 1. Overview == This page explores how Probability Dilation Theory (PDT) can be used to construct toy models that resemble certain statistical aspects of quantum decoherence. This work is: mathematical, not physical exploratory, not authoritative analogical, not a claim about quantum mechanics open for critique and improvement The goal is to provide a clear, reproducible framework for studying how iterative reweighting of probability measures can mimic the probability‑flow behavior seen in decoherence processes. == 2. Background: PDT and Decoherence == === 2.1 Probability Dilation Theory (PDT) === PDT studies how a probability measure :<math>P</math> is transformed by a positive dilation field :<math>D(x) > 0</math> via the operator :<math>P_D(A) = \frac{\int_A D(x)\, dP(x)}{\int_X D(x)\, dP(x)}.</math> Iterating this operator produces a sequence :<math>P_{n+1} = \mathcal{D}(P_n)</math> which may converge to attractors, fixed points, or stable distributions. === 2.2 Decoherence (informal summary) === In quantum mechanics, decoherence describes how a system loses phase coherence due to interaction with an environment. A density matrix :<math>\rho</math> evolves under a completely positive trace‑preserving (CPTP) map :<math>\mathcal{E}(\rho)</math> that typically: leaves diagonal probabilities unchanged or slowly biased exponentially suppresses off‑diagonal terms (coherences) This produces an effectively classical mixture. === 2.3 Why compare them? === Although PDT is purely classical, the diagonal part of many decoherence channels behaves like a dilation step, and the off‑diagonal decay can be modeled as a simple contraction. This makes PDT a useful toy model for exploring: probability concentration pointer‑state attractors emergent classicality iterative reweighting dynamics No physical claims are made. == 3. A Minimal Toy Model == We consider a two‑state system with classical probabilities :<math>P_n = (p_0^{(n)}, p_1^{(n)})</math> and a coherence magnitude :<math>r_n = |c_n|.</math> The model consists of two coupled updates: === 3.1 PDT dilation on probabilities === Given a dilation field :<math>D = (D_0, D_1)</math> we update :<math>p_i^{(n+1)} = \frac{D_i\, p_i^{(n)}}{D_0 p_0^{(n)} + D_1 p_1^{(n)}}.</math> This biases the system toward states with larger <math>D_i</math>. === 3.2 Coherence decay === We model decoherence by :<math>r_{n+1} = \alpha\, r_n</math> with :<math>0 \le \alpha < 1.</math> This is not quantum mechanical — it is a simple exponential decay rule. === 3.3 Combined update === Each iteration applies: PDT dilation on the diagonal Exponential decay on the coherence magnitude This produces a system that: flows toward a classical attractor loses coherence over time resembles decoherence in its statistical behavior == 4. Numerical Simulation (Python) == Below is a complete, runnable Python script that simulates the toy model and produces probability and coherence plots. <syntaxhighlight lang="python"> import numpy as np import matplotlib.pyplot as plt def simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=20 ): p0 = p0_init p1 = 1.0 - p0_init coh = coherence_init p0_hist = [p0] p1_hist = [p1] coh_hist = [coh] for n in range(steps): # PDT dilation step Z = D0 * p0 + D1 * p1 p0 = (D0 * p0) / Z p1 = (D1 * p1) / Z Coherence decay coh = alpha * coh p0_hist.append(p0) p1_hist.append(p1) coh_hist.append(coh) return np.array(p0_hist), np.array(p1_hist), np.array(coh_hist) if name == "main": p0_hist, p1_hist, coh_hist = simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=25 ) t = np.arange(len(p0_hist)) fig, ax1 = plt.subplots() ax1.set_xlabel("Iteration") ax1.set_ylabel("Probability", color="tab:blue") ax1.plot(t, p0_hist, label="p0", color="tab:blue") ax1.plot(t, p1_hist, label="p1", color="tab:cyan", linestyle="--") ax1.tick_params(axis="y", labelcolor="tab:blue") ax1.legend(loc="upper left") ax2 = ax1.twinx() ax2.set_ylabel("Coherence |c|", color="tab:red") ax2.plot(t, coh_hist, label="|c|", color="tab:red") ax2.tick_params(axis="y", labelcolor="tab:red") fig.tight_layout() plt.title("Iterative Dilation + Decoherence Toy Model") plt.show() </syntaxhighlight> == 5. Interpretation == === 5.1 Probability flow === If <math>D_0 > D_1</math>, the system flows toward :<math>p_0 \to 1.</math> This is analogous to a pointer state in decoherence. === 5.2 Coherence decay === The coherence magnitude :<math>r_n</math> decays exponentially, mimicking the suppression of off‑diagonal terms in a density matrix. === 5.3 Combined effect === The system becomes: more classical (probabilities concentrate) less coherent (off‑diagonal terms vanish) This mirrors the qualitative behavior of decoherence. == 6. Limitations == This is not a quantum model. No physical claims are made about dilation fields. The analogy is structural, not ontological. Decoherence involves entanglement; PDT does not. The model is intended for intuition and exploration only. == 7. Open Questions == Can more general decoherence channels be embedded in PDT? What are the fixed points of multi‑state dilation systems? Can continuous‑time dilation flows be defined? Are there PDE analogues of iterative dilation? Can this framework be useful in machine learning or statistical physics? == 8. See Also == [[Probability Dilation Theory]] [[w:Quantum decoherence|Quantum decoherence]] [[w:Density matrix|Density matrix]] [[w:Bayesian inference|Bayesian inference]] [[w:Bayes' theorem|Bayes' theorem]] == 9. Invitation for Collaboration == This page is part of an ongoing exploration of PDT and its possible mathematical analogies. Feedback, critique, and contributions from mathematicians, physicists, and computer scientists are welcome. cpyg1kxmhq31yyyocbykl8ch4tnyebg 2816267 2816266 2026-06-19T01:20:38Z Howie2024 2995240 /* 8. See Also */ 2816267 wikitext text/x-wiki = Probability Dilation Theory / Decoherence Analogy and Simulation = ''This page is a subpage of [[Probability Dilation Theory]] and develops a speculative but mathematically well‑defined analogy between iterative probability dilation and quantum decoherence. It includes definitions, motivation, and a reproducible Python simulation.'' == 1. Overview == This page explores how Probability Dilation Theory (PDT) can be used to construct toy models that resemble certain statistical aspects of quantum decoherence. This work is: mathematical, not physical exploratory, not authoritative analogical, not a claim about quantum mechanics open for critique and improvement The goal is to provide a clear, reproducible framework for studying how iterative reweighting of probability measures can mimic the probability‑flow behavior seen in decoherence processes. == 2. Background: PDT and Decoherence == === 2.1 Probability Dilation Theory (PDT) === PDT studies how a probability measure :<math>P</math> is transformed by a positive dilation field :<math>D(x) > 0</math> via the operator :<math>P_D(A) = \frac{\int_A D(x)\, dP(x)}{\int_X D(x)\, dP(x)}.</math> Iterating this operator produces a sequence :<math>P_{n+1} = \mathcal{D}(P_n)</math> which may converge to attractors, fixed points, or stable distributions. === 2.2 Decoherence (informal summary) === In quantum mechanics, decoherence describes how a system loses phase coherence due to interaction with an environment. A density matrix :<math>\rho</math> evolves under a completely positive trace‑preserving (CPTP) map :<math>\mathcal{E}(\rho)</math> that typically: leaves diagonal probabilities unchanged or slowly biased exponentially suppresses off‑diagonal terms (coherences) This produces an effectively classical mixture. === 2.3 Why compare them? === Although PDT is purely classical, the diagonal part of many decoherence channels behaves like a dilation step, and the off‑diagonal decay can be modeled as a simple contraction. This makes PDT a useful toy model for exploring: probability concentration pointer‑state attractors emergent classicality iterative reweighting dynamics No physical claims are made. == 3. A Minimal Toy Model == We consider a two‑state system with classical probabilities :<math>P_n = (p_0^{(n)}, p_1^{(n)})</math> and a coherence magnitude :<math>r_n = |c_n|.</math> The model consists of two coupled updates: === 3.1 PDT dilation on probabilities === Given a dilation field :<math>D = (D_0, D_1)</math> we update :<math>p_i^{(n+1)} = \frac{D_i\, p_i^{(n)}}{D_0 p_0^{(n)} + D_1 p_1^{(n)}}.</math> This biases the system toward states with larger <math>D_i</math>. === 3.2 Coherence decay === We model decoherence by :<math>r_{n+1} = \alpha\, r_n</math> with :<math>0 \le \alpha < 1.</math> This is not quantum mechanical — it is a simple exponential decay rule. === 3.3 Combined update === Each iteration applies: PDT dilation on the diagonal Exponential decay on the coherence magnitude This produces a system that: flows toward a classical attractor loses coherence over time resembles decoherence in its statistical behavior == 4. Numerical Simulation (Python) == Below is a complete, runnable Python script that simulates the toy model and produces probability and coherence plots. <syntaxhighlight lang="python"> import numpy as np import matplotlib.pyplot as plt def simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=20 ): p0 = p0_init p1 = 1.0 - p0_init coh = coherence_init p0_hist = [p0] p1_hist = [p1] coh_hist = [coh] for n in range(steps): # PDT dilation step Z = D0 * p0 + D1 * p1 p0 = (D0 * p0) / Z p1 = (D1 * p1) / Z Coherence decay coh = alpha * coh p0_hist.append(p0) p1_hist.append(p1) coh_hist.append(coh) return np.array(p0_hist), np.array(p1_hist), np.array(coh_hist) if name == "main": p0_hist, p1_hist, coh_hist = simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=25 ) t = np.arange(len(p0_hist)) fig, ax1 = plt.subplots() ax1.set_xlabel("Iteration") ax1.set_ylabel("Probability", color="tab:blue") ax1.plot(t, p0_hist, label="p0", color="tab:blue") ax1.plot(t, p1_hist, label="p1", color="tab:cyan", linestyle="--") ax1.tick_params(axis="y", labelcolor="tab:blue") ax1.legend(loc="upper left") ax2 = ax1.twinx() ax2.set_ylabel("Coherence |c|", color="tab:red") ax2.plot(t, coh_hist, label="|c|", color="tab:red") ax2.tick_params(axis="y", labelcolor="tab:red") fig.tight_layout() plt.title("Iterative Dilation + Decoherence Toy Model") plt.show() </syntaxhighlight> == 5. Interpretation == === 5.1 Probability flow === If <math>D_0 > D_1</math>, the system flows toward :<math>p_0 \to 1.</math> This is analogous to a pointer state in decoherence. === 5.2 Coherence decay === The coherence magnitude :<math>r_n</math> decays exponentially, mimicking the suppression of off‑diagonal terms in a density matrix. === 5.3 Combined effect === The system becomes: more classical (probabilities concentrate) less coherent (off‑diagonal terms vanish) This mirrors the qualitative behavior of decoherence. == 6. Limitations == This is not a quantum model. No physical claims are made about dilation fields. The analogy is structural, not ontological. Decoherence involves entanglement; PDT does not. The model is intended for intuition and exploration only. == 7. Open Questions == Can more general decoherence channels be embedded in PDT? What are the fixed points of multi‑state dilation systems? Can continuous‑time dilation flows be defined? Are there PDE analogues of iterative dilation? Can this framework be useful in machine learning or statistical physics? == 8. See Also == * [[Probability Dilation Theory]] * [[w:Quantum decoherence|Quantum decoherence]] * [[w:Density matrix|Density matrix]] * [[w:Quantum channel|Quantum channel]] * [[w:Completely positive map|Completely positive map]] * [[w:Open quantum system|Open quantum system]] * [[w:Bayesian inference|Bayesian inference]] * [[w:Bayes' theorem|Bayes' theorem]] * [[w:Probability measure|Probability measure]] * [[w:Markov chain|Markov chain]] * [[w:Fixed point (mathematics)|Fixed point]] * [[w:Shannon entropy|Shannon entropy]] * [[w:Information theory|Information theory]] == 9. Invitation for Collaboration == This page is part of an ongoing exploration of PDT and its possible mathematical analogies. Feedback, critique, and contributions from mathematicians, physicists, and computer scientists are welcome. nlatffpyfuyvcm56bwyr2er0klc0e3t 2816269 2816267 2026-06-19T01:31:26Z Howie2024 2995240 /* 2.3 Why compare them? */ Scope of Analogy 2816269 wikitext text/x-wiki = Probability Dilation Theory / Decoherence Analogy and Simulation = ''This page is a subpage of [[Probability Dilation Theory]] and develops a speculative but mathematically well‑defined analogy between iterative probability dilation and quantum decoherence. It includes definitions, motivation, and a reproducible Python simulation.'' == 1. Overview == This page explores how Probability Dilation Theory (PDT) can be used to construct toy models that resemble certain statistical aspects of quantum decoherence. This work is: mathematical, not physical exploratory, not authoritative analogical, not a claim about quantum mechanics open for critique and improvement The goal is to provide a clear, reproducible framework for studying how iterative reweighting of probability measures can mimic the probability‑flow behavior seen in decoherence processes. == 2. Background: PDT and Decoherence == === 2.1 Probability Dilation Theory (PDT) === PDT studies how a probability measure :<math>P</math> is transformed by a positive dilation field :<math>D(x) > 0</math> via the operator :<math>P_D(A) = \frac{\int_A D(x)\, dP(x)}{\int_X D(x)\, dP(x)}.</math> Iterating this operator produces a sequence :<math>P_{n+1} = \mathcal{D}(P_n)</math> which may converge to attractors, fixed points, or stable distributions. === 2.2 Decoherence (informal summary) === In quantum mechanics, decoherence describes how a system loses phase coherence due to interaction with an environment. A density matrix :<math>\rho</math> evolves under a completely positive trace‑preserving (CPTP) map :<math>\mathcal{E}(\rho)</math> that typically: leaves diagonal probabilities unchanged or slowly biased exponentially suppresses off‑diagonal terms (coherences) This produces an effectively classical mixture. === 2.3 Why compare them? === Although PDT is purely classical, the diagonal part of many decoherence channels behaves like a dilation step, and the off‑diagonal decay can be modeled as a simple contraction. This makes PDT a useful toy model for exploring: probability concentration pointer‑state attractors emergent classicality iterative reweighting dynamics No physical claims are made. === Scope of the Analogy === The purpose of this analogy is not to reproduce quantum mechanics. Instead, it isolates one particular aspect of decoherence: the evolution of probability weights toward stable classical outcomes. PDT models probability reweighting through iterative dilation. Quantum decoherence additionally involves phase information, density matrices, and entanglement with an environment. Consequently, the present framework should be viewed as a mathematical toy model that captures certain statistical features of decoherence while omitting many essential quantum-mechanical structures. == 3. A Minimal Toy Model == We consider a two‑state system with classical probabilities :<math>P_n = (p_0^{(n)}, p_1^{(n)})</math> and a coherence magnitude :<math>r_n = |c_n|.</math> The model consists of two coupled updates: === 3.1 PDT dilation on probabilities === Given a dilation field :<math>D = (D_0, D_1)</math> we update :<math>p_i^{(n+1)} = \frac{D_i\, p_i^{(n)}}{D_0 p_0^{(n)} + D_1 p_1^{(n)}}.</math> This biases the system toward states with larger <math>D_i</math>. === 3.2 Coherence decay === We model decoherence by :<math>r_{n+1} = \alpha\, r_n</math> with :<math>0 \le \alpha < 1.</math> This is not quantum mechanical — it is a simple exponential decay rule. === 3.3 Combined update === Each iteration applies: PDT dilation on the diagonal Exponential decay on the coherence magnitude This produces a system that: flows toward a classical attractor loses coherence over time resembles decoherence in its statistical behavior == 4. Numerical Simulation (Python) == Below is a complete, runnable Python script that simulates the toy model and produces probability and coherence plots. <syntaxhighlight lang="python"> import numpy as np import matplotlib.pyplot as plt def simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=20 ): p0 = p0_init p1 = 1.0 - p0_init coh = coherence_init p0_hist = [p0] p1_hist = [p1] coh_hist = [coh] for n in range(steps): # PDT dilation step Z = D0 * p0 + D1 * p1 p0 = (D0 * p0) / Z p1 = (D1 * p1) / Z Coherence decay coh = alpha * coh p0_hist.append(p0) p1_hist.append(p1) coh_hist.append(coh) return np.array(p0_hist), np.array(p1_hist), np.array(coh_hist) if name == "main": p0_hist, p1_hist, coh_hist = simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=25 ) t = np.arange(len(p0_hist)) fig, ax1 = plt.subplots() ax1.set_xlabel("Iteration") ax1.set_ylabel("Probability", color="tab:blue") ax1.plot(t, p0_hist, label="p0", color="tab:blue") ax1.plot(t, p1_hist, label="p1", color="tab:cyan", linestyle="--") ax1.tick_params(axis="y", labelcolor="tab:blue") ax1.legend(loc="upper left") ax2 = ax1.twinx() ax2.set_ylabel("Coherence |c|", color="tab:red") ax2.plot(t, coh_hist, label="|c|", color="tab:red") ax2.tick_params(axis="y", labelcolor="tab:red") fig.tight_layout() plt.title("Iterative Dilation + Decoherence Toy Model") plt.show() </syntaxhighlight> == 5. Interpretation == === 5.1 Probability flow === If <math>D_0 > D_1</math>, the system flows toward :<math>p_0 \to 1.</math> This is analogous to a pointer state in decoherence. === 5.2 Coherence decay === The coherence magnitude :<math>r_n</math> decays exponentially, mimicking the suppression of off‑diagonal terms in a density matrix. === 5.3 Combined effect === The system becomes: more classical (probabilities concentrate) less coherent (off‑diagonal terms vanish) This mirrors the qualitative behavior of decoherence. == 6. Limitations == This is not a quantum model. No physical claims are made about dilation fields. The analogy is structural, not ontological. Decoherence involves entanglement; PDT does not. The model is intended for intuition and exploration only. == 7. Open Questions == Can more general decoherence channels be embedded in PDT? What are the fixed points of multi‑state dilation systems? Can continuous‑time dilation flows be defined? Are there PDE analogues of iterative dilation? Can this framework be useful in machine learning or statistical physics? == 8. See Also == * [[Probability Dilation Theory]] * [[w:Quantum decoherence|Quantum decoherence]] * [[w:Density matrix|Density matrix]] * [[w:Quantum channel|Quantum channel]] * [[w:Completely positive map|Completely positive map]] * [[w:Open quantum system|Open quantum system]] * [[w:Bayesian inference|Bayesian inference]] * [[w:Bayes' theorem|Bayes' theorem]] * [[w:Probability measure|Probability measure]] * [[w:Markov chain|Markov chain]] * [[w:Fixed point (mathematics)|Fixed point]] * [[w:Shannon entropy|Shannon entropy]] * [[w:Information theory|Information theory]] == 9. Invitation for Collaboration == This page is part of an ongoing exploration of PDT and its possible mathematical analogies. Feedback, critique, and contributions from mathematicians, physicists, and computer scientists are welcome. lk8iwdcm5hndoa313hjguhwwe6cf5tv 2816270 2816269 2026-06-19T01:38:25Z Howie2024 2995240 /* Scope of the Analogy */ Diagonal Channel Approximation 2816270 wikitext text/x-wiki = Probability Dilation Theory / Decoherence Analogy and Simulation = ''This page is a subpage of [[Probability Dilation Theory]] and develops a speculative but mathematically well‑defined analogy between iterative probability dilation and quantum decoherence. It includes definitions, motivation, and a reproducible Python simulation.'' == 1. Overview == This page explores how Probability Dilation Theory (PDT) can be used to construct toy models that resemble certain statistical aspects of quantum decoherence. This work is: mathematical, not physical exploratory, not authoritative analogical, not a claim about quantum mechanics open for critique and improvement The goal is to provide a clear, reproducible framework for studying how iterative reweighting of probability measures can mimic the probability‑flow behavior seen in decoherence processes. == 2. Background: PDT and Decoherence == === 2.1 Probability Dilation Theory (PDT) === PDT studies how a probability measure :<math>P</math> is transformed by a positive dilation field :<math>D(x) > 0</math> via the operator :<math>P_D(A) = \frac{\int_A D(x)\, dP(x)}{\int_X D(x)\, dP(x)}.</math> Iterating this operator produces a sequence :<math>P_{n+1} = \mathcal{D}(P_n)</math> which may converge to attractors, fixed points, or stable distributions. === 2.2 Decoherence (informal summary) === In quantum mechanics, decoherence describes how a system loses phase coherence due to interaction with an environment. A density matrix :<math>\rho</math> evolves under a completely positive trace‑preserving (CPTP) map :<math>\mathcal{E}(\rho)</math> that typically: leaves diagonal probabilities unchanged or slowly biased exponentially suppresses off‑diagonal terms (coherences) This produces an effectively classical mixture. === 2.3 Why compare them? === Although PDT is purely classical, the diagonal part of many decoherence channels behaves like a dilation step, and the off‑diagonal decay can be modeled as a simple contraction. This makes PDT a useful toy model for exploring: probability concentration pointer‑state attractors emergent classicality iterative reweighting dynamics No physical claims are made. === Scope of the Analogy === The purpose of this analogy is not to reproduce quantum mechanics. Instead, it isolates one particular aspect of decoherence: the evolution of probability weights toward stable classical outcomes. PDT models probability reweighting through iterative dilation. Quantum decoherence additionally involves phase information, density matrices, and entanglement with an environment. Consequently, the present framework should be viewed as a mathematical toy model that captures certain statistical features of decoherence while omitting many essential quantum-mechanical structures. === 2.4 Diagonal Channel Approximation === A useful mathematical connection between decoherence-inspired dynamics and Probability Dilation Theory arises from considering only the diagonal components of a quantum channel. Let <math> P_n(x) </math> denote the diagonal probability associated with state <math> x. </math> Suppose the diagonal probabilities evolve according to <math> P_{n+1}(x) = \sum_y \mathcal E_{xy}P_n(y), </math> where <math> \mathcal E </math> is a stochastic kernel derived from a decoherence channel. If the channel becomes strongly diagonal-dominant in a preferred basis, then one may approximate <math> \mathcal E_{xy} \approx D(x)\delta_{xy}, </math> where <math> \delta_{xy} </math> is the Kronecker delta and <math> D(x)>0 </math> acts as a state-dependent weighting factor. Under this approximation, <math> P_{n+1}(x) \approx D(x)P_n(x). </math> Normalizing the resulting probabilities gives <math> P_{n+1}(x) = \frac{D(x)P_n(x)} {\sum_z D(z)P_n(z)}, </math> which is precisely the Probability Dilation Theory update rule. This observation does not imply that PDT is equivalent to quantum decoherence. Rather, it shows that under a diagonal-dominant approximation, certain probability-flow aspects of decoherence channels may be represented by normalized probability dilation. The approximation therefore provides a mathematical bridge between decoherence-inspired probability evolution and iterative PDT dynamics. == 3. A Minimal Toy Model == We consider a two‑state system with classical probabilities :<math>P_n = (p_0^{(n)}, p_1^{(n)})</math> and a coherence magnitude :<math>r_n = |c_n|.</math> The model consists of two coupled updates: === 3.1 PDT dilation on probabilities === Given a dilation field :<math>D = (D_0, D_1)</math> we update :<math>p_i^{(n+1)} = \frac{D_i\, p_i^{(n)}}{D_0 p_0^{(n)} + D_1 p_1^{(n)}}.</math> This biases the system toward states with larger <math>D_i</math>. === 3.2 Coherence decay === We model decoherence by :<math>r_{n+1} = \alpha\, r_n</math> with :<math>0 \le \alpha < 1.</math> This is not quantum mechanical — it is a simple exponential decay rule. === 3.3 Combined update === Each iteration applies: PDT dilation on the diagonal Exponential decay on the coherence magnitude This produces a system that: flows toward a classical attractor loses coherence over time resembles decoherence in its statistical behavior == 4. Numerical Simulation (Python) == Below is a complete, runnable Python script that simulates the toy model and produces probability and coherence plots. <syntaxhighlight lang="python"> import numpy as np import matplotlib.pyplot as plt def simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=20 ): p0 = p0_init p1 = 1.0 - p0_init coh = coherence_init p0_hist = [p0] p1_hist = [p1] coh_hist = [coh] for n in range(steps): # PDT dilation step Z = D0 * p0 + D1 * p1 p0 = (D0 * p0) / Z p1 = (D1 * p1) / Z Coherence decay coh = alpha * coh p0_hist.append(p0) p1_hist.append(p1) coh_hist.append(coh) return np.array(p0_hist), np.array(p1_hist), np.array(coh_hist) if name == "main": p0_hist, p1_hist, coh_hist = simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=25 ) t = np.arange(len(p0_hist)) fig, ax1 = plt.subplots() ax1.set_xlabel("Iteration") ax1.set_ylabel("Probability", color="tab:blue") ax1.plot(t, p0_hist, label="p0", color="tab:blue") ax1.plot(t, p1_hist, label="p1", color="tab:cyan", linestyle="--") ax1.tick_params(axis="y", labelcolor="tab:blue") ax1.legend(loc="upper left") ax2 = ax1.twinx() ax2.set_ylabel("Coherence |c|", color="tab:red") ax2.plot(t, coh_hist, label="|c|", color="tab:red") ax2.tick_params(axis="y", labelcolor="tab:red") fig.tight_layout() plt.title("Iterative Dilation + Decoherence Toy Model") plt.show() </syntaxhighlight> == 5. Interpretation == === 5.1 Probability flow === If <math>D_0 > D_1</math>, the system flows toward :<math>p_0 \to 1.</math> This is analogous to a pointer state in decoherence. === 5.2 Coherence decay === The coherence magnitude :<math>r_n</math> decays exponentially, mimicking the suppression of off‑diagonal terms in a density matrix. === 5.3 Combined effect === The system becomes: more classical (probabilities concentrate) less coherent (off‑diagonal terms vanish) This mirrors the qualitative behavior of decoherence. == 6. Limitations == This is not a quantum model. No physical claims are made about dilation fields. The analogy is structural, not ontological. Decoherence involves entanglement; PDT does not. The model is intended for intuition and exploration only. == 7. Open Questions == Can more general decoherence channels be embedded in PDT? What are the fixed points of multi‑state dilation systems? Can continuous‑time dilation flows be defined? Are there PDE analogues of iterative dilation? Can this framework be useful in machine learning or statistical physics? == 8. See Also == * [[Probability Dilation Theory]] * [[w:Quantum decoherence|Quantum decoherence]] * [[w:Density matrix|Density matrix]] * [[w:Quantum channel|Quantum channel]] * [[w:Completely positive map|Completely positive map]] * [[w:Open quantum system|Open quantum system]] * [[w:Bayesian inference|Bayesian inference]] * [[w:Bayes' theorem|Bayes' theorem]] * [[w:Probability measure|Probability measure]] * [[w:Markov chain|Markov chain]] * [[w:Fixed point (mathematics)|Fixed point]] * [[w:Shannon entropy|Shannon entropy]] * [[w:Information theory|Information theory]] == 9. Invitation for Collaboration == This page is part of an ongoing exploration of PDT and its possible mathematical analogies. Feedback, critique, and contributions from mathematicians, physicists, and computer scientists are welcome. eeh5kpzjtib1g6gjhxlvtmhn200svhw 2816272 2816270 2026-06-19T01:40:52Z Howie2024 2995240 /* 7. Open Questions */ added when do diagonal-dominant quantum channels reduce to normalized PDT operators 2816272 wikitext text/x-wiki = Probability Dilation Theory / Decoherence Analogy and Simulation = ''This page is a subpage of [[Probability Dilation Theory]] and develops a speculative but mathematically well‑defined analogy between iterative probability dilation and quantum decoherence. It includes definitions, motivation, and a reproducible Python simulation.'' == 1. Overview == This page explores how Probability Dilation Theory (PDT) can be used to construct toy models that resemble certain statistical aspects of quantum decoherence. This work is: mathematical, not physical exploratory, not authoritative analogical, not a claim about quantum mechanics open for critique and improvement The goal is to provide a clear, reproducible framework for studying how iterative reweighting of probability measures can mimic the probability‑flow behavior seen in decoherence processes. == 2. Background: PDT and Decoherence == === 2.1 Probability Dilation Theory (PDT) === PDT studies how a probability measure :<math>P</math> is transformed by a positive dilation field :<math>D(x) > 0</math> via the operator :<math>P_D(A) = \frac{\int_A D(x)\, dP(x)}{\int_X D(x)\, dP(x)}.</math> Iterating this operator produces a sequence :<math>P_{n+1} = \mathcal{D}(P_n)</math> which may converge to attractors, fixed points, or stable distributions. === 2.2 Decoherence (informal summary) === In quantum mechanics, decoherence describes how a system loses phase coherence due to interaction with an environment. A density matrix :<math>\rho</math> evolves under a completely positive trace‑preserving (CPTP) map :<math>\mathcal{E}(\rho)</math> that typically: leaves diagonal probabilities unchanged or slowly biased exponentially suppresses off‑diagonal terms (coherences) This produces an effectively classical mixture. === 2.3 Why compare them? === Although PDT is purely classical, the diagonal part of many decoherence channels behaves like a dilation step, and the off‑diagonal decay can be modeled as a simple contraction. This makes PDT a useful toy model for exploring: probability concentration pointer‑state attractors emergent classicality iterative reweighting dynamics No physical claims are made. === Scope of the Analogy === The purpose of this analogy is not to reproduce quantum mechanics. Instead, it isolates one particular aspect of decoherence: the evolution of probability weights toward stable classical outcomes. PDT models probability reweighting through iterative dilation. Quantum decoherence additionally involves phase information, density matrices, and entanglement with an environment. Consequently, the present framework should be viewed as a mathematical toy model that captures certain statistical features of decoherence while omitting many essential quantum-mechanical structures. === 2.4 Diagonal Channel Approximation === A useful mathematical connection between decoherence-inspired dynamics and Probability Dilation Theory arises from considering only the diagonal components of a quantum channel. Let <math> P_n(x) </math> denote the diagonal probability associated with state <math> x. </math> Suppose the diagonal probabilities evolve according to <math> P_{n+1}(x) = \sum_y \mathcal E_{xy}P_n(y), </math> where <math> \mathcal E </math> is a stochastic kernel derived from a decoherence channel. If the channel becomes strongly diagonal-dominant in a preferred basis, then one may approximate <math> \mathcal E_{xy} \approx D(x)\delta_{xy}, </math> where <math> \delta_{xy} </math> is the Kronecker delta and <math> D(x)>0 </math> acts as a state-dependent weighting factor. Under this approximation, <math> P_{n+1}(x) \approx D(x)P_n(x). </math> Normalizing the resulting probabilities gives <math> P_{n+1}(x) = \frac{D(x)P_n(x)} {\sum_z D(z)P_n(z)}, </math> which is precisely the Probability Dilation Theory update rule. This observation does not imply that PDT is equivalent to quantum decoherence. Rather, it shows that under a diagonal-dominant approximation, certain probability-flow aspects of decoherence channels may be represented by normalized probability dilation. The approximation therefore provides a mathematical bridge between decoherence-inspired probability evolution and iterative PDT dynamics. == 3. A Minimal Toy Model == We consider a two‑state system with classical probabilities :<math>P_n = (p_0^{(n)}, p_1^{(n)})</math> and a coherence magnitude :<math>r_n = |c_n|.</math> The model consists of two coupled updates: === 3.1 PDT dilation on probabilities === Given a dilation field :<math>D = (D_0, D_1)</math> we update :<math>p_i^{(n+1)} = \frac{D_i\, p_i^{(n)}}{D_0 p_0^{(n)} + D_1 p_1^{(n)}}.</math> This biases the system toward states with larger <math>D_i</math>. === 3.2 Coherence decay === We model decoherence by :<math>r_{n+1} = \alpha\, r_n</math> with :<math>0 \le \alpha < 1.</math> This is not quantum mechanical — it is a simple exponential decay rule. === 3.3 Combined update === Each iteration applies: PDT dilation on the diagonal Exponential decay on the coherence magnitude This produces a system that: flows toward a classical attractor loses coherence over time resembles decoherence in its statistical behavior == 4. Numerical Simulation (Python) == Below is a complete, runnable Python script that simulates the toy model and produces probability and coherence plots. <syntaxhighlight lang="python"> import numpy as np import matplotlib.pyplot as plt def simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=20 ): p0 = p0_init p1 = 1.0 - p0_init coh = coherence_init p0_hist = [p0] p1_hist = [p1] coh_hist = [coh] for n in range(steps): # PDT dilation step Z = D0 * p0 + D1 * p1 p0 = (D0 * p0) / Z p1 = (D1 * p1) / Z Coherence decay coh = alpha * coh p0_hist.append(p0) p1_hist.append(p1) coh_hist.append(coh) return np.array(p0_hist), np.array(p1_hist), np.array(coh_hist) if name == "main": p0_hist, p1_hist, coh_hist = simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=25 ) t = np.arange(len(p0_hist)) fig, ax1 = plt.subplots() ax1.set_xlabel("Iteration") ax1.set_ylabel("Probability", color="tab:blue") ax1.plot(t, p0_hist, label="p0", color="tab:blue") ax1.plot(t, p1_hist, label="p1", color="tab:cyan", linestyle="--") ax1.tick_params(axis="y", labelcolor="tab:blue") ax1.legend(loc="upper left") ax2 = ax1.twinx() ax2.set_ylabel("Coherence |c|", color="tab:red") ax2.plot(t, coh_hist, label="|c|", color="tab:red") ax2.tick_params(axis="y", labelcolor="tab:red") fig.tight_layout() plt.title("Iterative Dilation + Decoherence Toy Model") plt.show() </syntaxhighlight> == 5. Interpretation == === 5.1 Probability flow === If <math>D_0 > D_1</math>, the system flows toward :<math>p_0 \to 1.</math> This is analogous to a pointer state in decoherence. === 5.2 Coherence decay === The coherence magnitude :<math>r_n</math> decays exponentially, mimicking the suppression of off‑diagonal terms in a density matrix. === 5.3 Combined effect === The system becomes: more classical (probabilities concentrate) less coherent (off‑diagonal terms vanish) This mirrors the qualitative behavior of decoherence. == 6. Limitations == This is not a quantum model. No physical claims are made about dilation fields. The analogy is structural, not ontological. Decoherence involves entanglement; PDT does not. The model is intended for intuition and exploration only. == 7. Open Questions == Can more general decoherence channels be embedded in PDT? What are the fixed points of multi‑state dilation systems? Can continuous‑time dilation flows be defined? Are there PDE analogues of iterative dilation? Can this framework be useful in machine learning or statistical physics? Under what conditions do diagonal-dominant quantum channels reduce to normalized probability dilation operators? == 8. See Also == * [[Probability Dilation Theory]] * [[w:Quantum decoherence|Quantum decoherence]] * [[w:Density matrix|Density matrix]] * [[w:Quantum channel|Quantum channel]] * [[w:Completely positive map|Completely positive map]] * [[w:Open quantum system|Open quantum system]] * [[w:Bayesian inference|Bayesian inference]] * [[w:Bayes' theorem|Bayes' theorem]] * [[w:Probability measure|Probability measure]] * [[w:Markov chain|Markov chain]] * [[w:Fixed point (mathematics)|Fixed point]] * [[w:Shannon entropy|Shannon entropy]] * [[w:Information theory|Information theory]] == 9. Invitation for Collaboration == This page is part of an ongoing exploration of PDT and its possible mathematical analogies. Feedback, critique, and contributions from mathematicians, physicists, and computer scientists are welcome. ofrefdyfcohvmcs4ag1xka19pjv9sto 2816274 2816272 2026-06-19T01:47:22Z Howie2024 2995240 /* 4. Numerical Simulation (Python) */ 2816274 wikitext text/x-wiki = Probability Dilation Theory / Decoherence Analogy and Simulation = ''This page is a subpage of [[Probability Dilation Theory]] and develops a speculative but mathematically well‑defined analogy between iterative probability dilation and quantum decoherence. It includes definitions, motivation, and a reproducible Python simulation.'' == 1. Overview == This page explores how Probability Dilation Theory (PDT) can be used to construct toy models that resemble certain statistical aspects of quantum decoherence. This work is: mathematical, not physical exploratory, not authoritative analogical, not a claim about quantum mechanics open for critique and improvement The goal is to provide a clear, reproducible framework for studying how iterative reweighting of probability measures can mimic the probability‑flow behavior seen in decoherence processes. == 2. Background: PDT and Decoherence == === 2.1 Probability Dilation Theory (PDT) === PDT studies how a probability measure :<math>P</math> is transformed by a positive dilation field :<math>D(x) > 0</math> via the operator :<math>P_D(A) = \frac{\int_A D(x)\, dP(x)}{\int_X D(x)\, dP(x)}.</math> Iterating this operator produces a sequence :<math>P_{n+1} = \mathcal{D}(P_n)</math> which may converge to attractors, fixed points, or stable distributions. === 2.2 Decoherence (informal summary) === In quantum mechanics, decoherence describes how a system loses phase coherence due to interaction with an environment. A density matrix :<math>\rho</math> evolves under a completely positive trace‑preserving (CPTP) map :<math>\mathcal{E}(\rho)</math> that typically: leaves diagonal probabilities unchanged or slowly biased exponentially suppresses off‑diagonal terms (coherences) This produces an effectively classical mixture. === 2.3 Why compare them? === Although PDT is purely classical, the diagonal part of many decoherence channels behaves like a dilation step, and the off‑diagonal decay can be modeled as a simple contraction. This makes PDT a useful toy model for exploring: probability concentration pointer‑state attractors emergent classicality iterative reweighting dynamics No physical claims are made. === Scope of the Analogy === The purpose of this analogy is not to reproduce quantum mechanics. Instead, it isolates one particular aspect of decoherence: the evolution of probability weights toward stable classical outcomes. PDT models probability reweighting through iterative dilation. Quantum decoherence additionally involves phase information, density matrices, and entanglement with an environment. Consequently, the present framework should be viewed as a mathematical toy model that captures certain statistical features of decoherence while omitting many essential quantum-mechanical structures. === 2.4 Diagonal Channel Approximation === A useful mathematical connection between decoherence-inspired dynamics and Probability Dilation Theory arises from considering only the diagonal components of a quantum channel. Let <math> P_n(x) </math> denote the diagonal probability associated with state <math> x. </math> Suppose the diagonal probabilities evolve according to <math> P_{n+1}(x) = \sum_y \mathcal E_{xy}P_n(y), </math> where <math> \mathcal E </math> is a stochastic kernel derived from a decoherence channel. If the channel becomes strongly diagonal-dominant in a preferred basis, then one may approximate <math> \mathcal E_{xy} \approx D(x)\delta_{xy}, </math> where <math> \delta_{xy} </math> is the Kronecker delta and <math> D(x)>0 </math> acts as a state-dependent weighting factor. Under this approximation, <math> P_{n+1}(x) \approx D(x)P_n(x). </math> Normalizing the resulting probabilities gives <math> P_{n+1}(x) = \frac{D(x)P_n(x)} {\sum_z D(z)P_n(z)}, </math> which is precisely the Probability Dilation Theory update rule. This observation does not imply that PDT is equivalent to quantum decoherence. Rather, it shows that under a diagonal-dominant approximation, certain probability-flow aspects of decoherence channels may be represented by normalized probability dilation. The approximation therefore provides a mathematical bridge between decoherence-inspired probability evolution and iterative PDT dynamics. == 3. A Minimal Toy Model == We consider a two‑state system with classical probabilities :<math>P_n = (p_0^{(n)}, p_1^{(n)})</math> and a coherence magnitude :<math>r_n = |c_n|.</math> The model consists of two coupled updates: === 3.1 PDT dilation on probabilities === Given a dilation field :<math>D = (D_0, D_1)</math> we update :<math>p_i^{(n+1)} = \frac{D_i\, p_i^{(n)}}{D_0 p_0^{(n)} + D_1 p_1^{(n)}}.</math> This biases the system toward states with larger <math>D_i</math>. === 3.2 Coherence decay === We model decoherence by :<math>r_{n+1} = \alpha\, r_n</math> with :<math>0 \le \alpha < 1.</math> This is not quantum mechanical — it is a simple exponential decay rule. === 3.3 Combined update === Each iteration applies: PDT dilation on the diagonal Exponential decay on the coherence magnitude This produces a system that: flows toward a classical attractor loses coherence over time resembles decoherence in its statistical behavior == 4. Numerical Simulation (Python) == Below is a complete, runnable Python script that simulates the toy model and produces probability and coherence plots. <syntaxhighlight lang="python"> import numpy as np import matplotlib.pyplot as plt def simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=20 ): p0 = p0_init p1 = 1.0 - p0_init coh = coherence_init p0_hist = [p0] p1_hist = [p1] coh_hist = [coh] for n in range(steps): # PDT dilation step Z = D0 * p0 + D1 * p1 p0 = (D0 * p0) / Z p1 = (D1 * p1) / Z # Coherence decay coh = alpha * coh p0_hist.append(p0) p1_hist.append(p1) coh_hist.append(coh) return np.array(p0_hist), np.array(p1_hist), np.array(coh_hist) if __name__ == "__main__": p0_hist, p1_hist, coh_hist = simulate_iterative_dilation_decoherence( p0_init=0.5, coherence_init=0.5, D0=2.0, D1=1.0, alpha=0.8, steps=25 ) t = np.arange(len(p0_hist)) fig, ax1 = plt.subplots() ax1.set_xlabel("Iteration") ax1.set_ylabel("Probability", color="tab:blue") ax1.plot(t, p0_hist, label="p0", color="tab:blue") ax1.plot(t, p1_hist, label="p1", color="tab:cyan", linestyle="--") ax1.tick_params(axis="y", labelcolor="tab:blue") ax1.legend(loc="upper left") ax2 = ax1.twinx() ax2.set_ylabel("Coherence |c|", color="tab:red") ax2.plot(t, coh_hist, label="|c|", color="tab:red") ax2.tick_params(axis="y", labelcolor="tab:red") fig.tight_layout() plt.title("Iterative Dilation + Decoherence Toy Model") plt.show() </syntaxhighlight> == 5. Interpretation == === 5.1 Probability flow === If <math>D_0 > D_1</math>, the system flows toward :<math>p_0 \to 1.</math> This is analogous to a pointer state in decoherence. === 5.2 Coherence decay === The coherence magnitude :<math>r_n</math> decays exponentially, mimicking the suppression of off‑diagonal terms in a density matrix. === 5.3 Combined effect === The system becomes: more classical (probabilities concentrate) less coherent (off‑diagonal terms vanish) This mirrors the qualitative behavior of decoherence. == 6. Limitations == This is not a quantum model. No physical claims are made about dilation fields. The analogy is structural, not ontological. Decoherence involves entanglement; PDT does not. The model is intended for intuition and exploration only. == 7. Open Questions == Can more general decoherence channels be embedded in PDT? What are the fixed points of multi‑state dilation systems? Can continuous‑time dilation flows be defined? Are there PDE analogues of iterative dilation? Can this framework be useful in machine learning or statistical physics? Under what conditions do diagonal-dominant quantum channels reduce to normalized probability dilation operators? == 8. See Also == * [[Probability Dilation Theory]] * [[w:Quantum decoherence|Quantum decoherence]] * [[w:Density matrix|Density matrix]] * [[w:Quantum channel|Quantum channel]] * [[w:Completely positive map|Completely positive map]] * [[w:Open quantum system|Open quantum system]] * [[w:Bayesian inference|Bayesian inference]] * [[w:Bayes' theorem|Bayes' theorem]] * [[w:Probability measure|Probability measure]] * [[w:Markov chain|Markov chain]] * [[w:Fixed point (mathematics)|Fixed point]] * [[w:Shannon entropy|Shannon entropy]] * [[w:Information theory|Information theory]] == 9. Invitation for Collaboration == This page is part of an ongoing exploration of PDT and its possible mathematical analogies. Feedback, critique, and contributions from mathematicians, physicists, and computer scientists are welcome. 8i5114mep1lw2s2srsmdv2sdj2fpe40 2 330261 2816281 2026-06-19T07:25:57Z Alandmanson 1669821 Created page with "== Are changes in southern African rangelands contributing to the insect apocalypse? == Rangeland scientists have long considered threats to biodiversity, but the consistent decline in insect biomass has only become a significant field of study in recent years. Estimates suggest that insect populations are shrinking at rates of between one and two percent annually in many areas (Wagner et al., 2021). Why is there a decline in insect abundance? In places like the grasslan..." 2816281 wikitext text/x-wiki == Are changes in southern African rangelands contributing to the insect apocalypse? == Rangeland scientists have long considered threats to biodiversity, but the consistent decline in insect biomass has only become a significant field of study in recent years. Estimates suggest that insect populations are shrinking at rates of between one and two percent annually in many areas (Wagner et al., 2021). Why is there a decline in insect abundance? In places like the grassland biome of southern Africa, the reasons are clear – large areas have been converted to crop, timber, or ornamental production; these are systems designed to maximize productivity for human use – this has obviously resulted in systems that minimize insect productivity. Of course, there is a complex interplay of many reasons; key drivers include habitat destruction and fragmentation, agricultural intensification (including widespread pesticide use), climate change (altered temperature and rainfall patterns), light pollution, chemical pollution, and invasive species. (Halsch et al., 2025) Rangelands comprise the home of many insects; increased pressure from humans and their livestock has had widespread effects on rangelands: • Soil loss – many insects use topsoil for at least part of their life cycle, including termites and ants, the dominant forms of terrestrial biomass. • Drier topsoils as a result of the loss of vegetative cover, both live aerial cover and dead plant (mostly leaf) matter. • Loss of wetlands and reduced water quality – many insects evolved to spend at least part of their lives in streams, rivers, or other wetlands; most of these habitats have been transformed in the last 50 years. • Declining quality of food for insect herbivores – rangelands appear resilient, but management for livestock production invariably changes species composition, often with a loss of perennial forbs and grass species. Loss of consistent sources of high-quality insect food is, however, not confined to local effects. In a groundbreaking study at the Konza Prairie Long Term Ecological Research Site, Welti et al. (2020) found that grasshopper biomass fluctuates with herbage quality, and that nutrient dilution appears to be causing a long-term decline in grasshoppers. This nutrient dilution is linked to CO2 fertilization, which increases plant production (when rainfall is sufficient) and lowers the concentrations of many nutrients in plant material (Kaspari & Welti, 2024). Insect decline is probably already affecting insectivores; small mammals, Botha’s Lark, Yellow-breasted Pipit and small raptors come to mind. Feedback effects on plant species are also likely as specialist herbivores are affected differently. So the warning signs are clear, and it seems unlikely that effective action to reverse the loss of insect biomass is possible in these times, where even responses to the needs and wants of the majority of our own species appear inadequate. References Halsch, C. A., Elphick, C. S., Bahlai, C. A., Forister, M. L., Wagner, D. L., Ware, J. L., & Grames, E. M. (2025). Meta-synthesis reveals interconnections among apparent drivers of insect biodiversity loss. BioScience, biaf034. Kaspari, M., & Welti, E. A. (2024). Nutrient dilution and the future of herbivore populations. Trends in Ecology & Evolution, 39(9), 809-820. Kaspari, M., & Welti, E. A. (2025). Building plant diversity into mechanisms of nutrient dilution. Trends in Ecology & Evolution. Wagner, D. L., Grames, E. M., Forister, M. L., Berenbaum, M. R., & Stopak, D. (2021). Insect decline in the Anthropocene: Death by a thousand cuts. Proceedings of the National Academy of Sciences, 118(2), e2023989118. Welti, E. A., Roeder, K. A., De Beurs, K. M., Joern, A., & Kaspari, M. (2020). Nutrient dilution and climate cycles underlie declines in a dominant insect herbivore. Proceedings of the National Academy of Sciences, 117(13), 7271-7275. mbao5x3pbgc7tc65gt7jmyn5n3fs4wq 2816282 2816281 2026-06-19T07:27:32Z Alandmanson 1669821 2816282 wikitext text/x-wiki == Are changes in southern African rangelands contributing to the insect apocalypse? == Rangeland scientists have long considered threats to biodiversity, but the consistent decline in insect biomass has only become a significant field of study in recent years. Estimates suggest that insect populations are shrinking at rates of between one and two percent annually in many areas (Wagner et al., 2021). Why is there a decline in insect abundance? In places like the grassland biome of southern Africa, the reasons are clear – large areas have been converted to crop, timber, or ornamental production; these are systems designed to maximize productivity for human use – this has obviously resulted in systems that minimize insect productivity. Of course, there is a complex interplay of many reasons; key drivers include habitat destruction and fragmentation, agricultural intensification (including widespread pesticide use), climate change (altered temperature and rainfall patterns), light pollution, chemical pollution, and invasive species. (Halsch et al., 2025) Rangelands comprise the home of many insects; increased pressure from humans and their livestock has had widespread effects on rangelands: *Soil loss – many insects use topsoil for at least part of their life cycle, including termites and ants, the dominant forms of terrestrial biomass. *Drier topsoils as a result of the loss of vegetative cover, both live aerial cover and dead plant (mostly leaf) matter. *Loss of wetlands and reduced water quality – many insects evolved to spend at least part of their lives in streams, rivers, or other wetlands; most of these habitats have been transformed in the last 50 years. *Declining quality of food for insect herbivores – rangelands appear resilient, but management for livestock production invariably changes species composition, often with a loss of perennial forbs and grass species.<br> Loss of consistent sources of high-quality insect food is, however, not confined to local effects. In a groundbreaking study at the Konza Prairie Long Term Ecological Research Site, Welti et al. (2020) found that grasshopper biomass fluctuates with herbage quality, and that nutrient dilution appears to be causing a long-term decline in grasshoppers. This nutrient dilution is linked to CO2 fertilization, which increases plant production (when rainfall is sufficient) and lowers the concentrations of many nutrients in plant material (Kaspari & Welti, 2024). Insect decline is probably already affecting insectivores; small mammals, Botha’s Lark, Yellow-breasted Pipit and small raptors come to mind. Feedback effects on plant species are also likely as specialist herbivores are affected differently. So the warning signs are clear, and it seems unlikely that effective action to reverse the loss of insect biomass is possible in these times, where even responses to the needs and wants of the majority of our own species appear inadequate.<br> == References == Halsch, C. A., Elphick, C. S., Bahlai, C. A., Forister, M. L., Wagner, D. L., Ware, J. L., & Grames, E. M. (2025). Meta-synthesis reveals interconnections among apparent drivers of insect biodiversity loss. BioScience, biaf034. Kaspari, M., & Welti, E. A. (2024). Nutrient dilution and the future of herbivore populations. Trends in Ecology & Evolution, 39(9), 809-820. Kaspari, M., & Welti, E. A. (2025). Building plant diversity into mechanisms of nutrient dilution. Trends in Ecology & Evolution. Wagner, D. L., Grames, E. M., Forister, M. L., Berenbaum, M. R., & Stopak, D. (2021). Insect decline in the Anthropocene: Death by a thousand cuts. Proceedings of the National Academy of Sciences, 118(2), e2023989118. Welti, E. A., Roeder, K. A., De Beurs, K. M., Joern, A., & Kaspari, M. (2020). Nutrient dilution and climate cycles underlie declines in a dominant insect herbivore. Proceedings of the National Academy of Sciences, 117(13), 7271-7275. c4pl5zzfwnoctylqndvcttoss96iimj 2816283 2816282 2026-06-19T10:47:28Z Alandmanson 1669821 /* Are changes in southern African rangelands contributing to the insect apocalypse? */ 2816283 wikitext text/x-wiki == Are changes in southern African rangelands contributing to the insect apocalypse? == Rangeland scientists have long considered threats to biodiversity, but the consistent decline in insect biomass has only become a significant field of study in recent years. Estimates suggest that insect populations are shrinking at rates of between one and two percent annually in many areas (Wagner et al., 2021).<br> Why is there a decline in insect abundance? In places like the grassland biome of southern Africa, the reasons are clear – large areas have been converted to crop, timber, or ornamental production; these are systems designed to maximize productivity for human use – this has obviously resulted in systems that minimize insect productivity. Of course, there is a complex interplay of many reasons; key drivers include habitat destruction and fragmentation, agricultural intensification (including widespread pesticide use), climate change (altered temperature and rainfall patterns), light pollution, chemical pollution, and invasive species. (Halsch et al., 2025)<br> Rangelands comprise the home of many insects; increased pressure from humans and their livestock has had widespread effects on rangelands: *Soil loss – many insects use topsoil for at least part of their life cycle, including termites and ants, the dominant forms of terrestrial biomass. *Drier topsoils as a result of the loss of vegetative cover, both live aerial cover and dead plant (mostly leaf) matter. *Loss of wetlands and reduced water quality – many insects evolved to spend at least part of their lives in streams, rivers, or other wetlands; most of these habitats have been transformed in the last 50 years. *Declining quality of food for insect herbivores – rangelands appear resilient, but management for livestock production invariably changes species composition, often with a loss of perennial forbs and grass species.<br> Loss of consistent sources of high-quality insect food is, however, not confined to local effects. In a groundbreaking study at the Konza Prairie Long Term Ecological Research Site, Welti et al. (2020) found that grasshopper biomass fluctuates with herbage quality, and that nutrient dilution appears to be causing a long-term decline in grasshoppers. This nutrient dilution is linked to CO2 fertilization, which increases plant production (when rainfall is sufficient) and lowers the concentrations of many nutrients in plant material (Kaspari & Welti, 2024). Insect decline is probably already affecting insectivores; small mammals, Botha’s Lark, Yellow-breasted Pipit and small raptors come to mind. Feedback effects on plant species are also likely as specialist herbivores are affected differently.<br> So the warning signs are clear, and it seems unlikely that effective action to reverse the loss of insect biomass is possible in these times, where even responses to the needs and wants of the majority of our own species appear inadequate.<br> == References == Halsch, C. A., Elphick, C. S., Bahlai, C. A., Forister, M. L., Wagner, D. L., Ware, J. L., & Grames, E. M. (2025). Meta-synthesis reveals interconnections among apparent drivers of insect biodiversity loss. BioScience, biaf034. Kaspari, M., & Welti, E. A. (2024). Nutrient dilution and the future of herbivore populations. Trends in Ecology & Evolution, 39(9), 809-820. Kaspari, M., & Welti, E. A. (2025). Building plant diversity into mechanisms of nutrient dilution. Trends in Ecology & Evolution. Wagner, D. L., Grames, E. M., Forister, M. L., Berenbaum, M. R., & Stopak, D. (2021). Insect decline in the Anthropocene: Death by a thousand cuts. Proceedings of the National Academy of Sciences, 118(2), e2023989118. Welti, E. A., Roeder, K. A., De Beurs, K. M., Joern, A., & Kaspari, M. (2020). Nutrient dilution and climate cycles underlie declines in a dominant insect herbivore. Proceedings of the National Academy of Sciences, 117(13), 7271-7275. 8jvous34iij28ew4gvw1yjsx98yl0re 2816284 2816283 2026-06-19T10:48:18Z Alandmanson 1669821 /* References */ 2816284 wikitext text/x-wiki == Are changes in southern African rangelands contributing to the insect apocalypse? == Rangeland scientists have long considered threats to biodiversity, but the consistent decline in insect biomass has only become a significant field of study in recent years. Estimates suggest that insect populations are shrinking at rates of between one and two percent annually in many areas (Wagner et al., 2021).<br> Why is there a decline in insect abundance? In places like the grassland biome of southern Africa, the reasons are clear – large areas have been converted to crop, timber, or ornamental production; these are systems designed to maximize productivity for human use – this has obviously resulted in systems that minimize insect productivity. Of course, there is a complex interplay of many reasons; key drivers include habitat destruction and fragmentation, agricultural intensification (including widespread pesticide use), climate change (altered temperature and rainfall patterns), light pollution, chemical pollution, and invasive species. (Halsch et al., 2025)<br> Rangelands comprise the home of many insects; increased pressure from humans and their livestock has had widespread effects on rangelands: *Soil loss – many insects use topsoil for at least part of their life cycle, including termites and ants, the dominant forms of terrestrial biomass. *Drier topsoils as a result of the loss of vegetative cover, both live aerial cover and dead plant (mostly leaf) matter. *Loss of wetlands and reduced water quality – many insects evolved to spend at least part of their lives in streams, rivers, or other wetlands; most of these habitats have been transformed in the last 50 years. *Declining quality of food for insect herbivores – rangelands appear resilient, but management for livestock production invariably changes species composition, often with a loss of perennial forbs and grass species.<br> Loss of consistent sources of high-quality insect food is, however, not confined to local effects. In a groundbreaking study at the Konza Prairie Long Term Ecological Research Site, Welti et al. (2020) found that grasshopper biomass fluctuates with herbage quality, and that nutrient dilution appears to be causing a long-term decline in grasshoppers. This nutrient dilution is linked to CO2 fertilization, which increases plant production (when rainfall is sufficient) and lowers the concentrations of many nutrients in plant material (Kaspari & Welti, 2024). Insect decline is probably already affecting insectivores; small mammals, Botha’s Lark, Yellow-breasted Pipit and small raptors come to mind. Feedback effects on plant species are also likely as specialist herbivores are affected differently.<br> So the warning signs are clear, and it seems unlikely that effective action to reverse the loss of insect biomass is possible in these times, where even responses to the needs and wants of the majority of our own species appear inadequate.<br> == References == Halsch, C. A., Elphick, C. S., Bahlai, C. A., Forister, M. L., Wagner, D. L., Ware, J. L., & Grames, E. M. (2025). Meta-synthesis reveals interconnections among apparent drivers of insect biodiversity loss. BioScience, biaf034.<br> Kaspari, M., & Welti, E. A. (2024). Nutrient dilution and the future of herbivore populations. Trends in Ecology & Evolution, 39(9), 809-820.<br> Kaspari, M., & Welti, E. A. (2025). Building plant diversity into mechanisms of nutrient dilution. Trends in Ecology & Evolution.<br> Wagner, D. L., Grames, E. M., Forister, M. L., Berenbaum, M. R., & Stopak, D. (2021). Insect decline in the Anthropocene: Death by a thousand cuts. Proceedings of the National Academy of Sciences, 118(2), e2023989118.<br> Welti, E. A., Roeder, K. A., De Beurs, K. M., Joern, A., & Kaspari, M. (2020). Nutrient dilution and climate cycles underlie declines in a dominant insect herbivore. Proceedings of the National Academy of Sciences, 117(13), 7271-7275.<br> k9935do9t05243qdk5fdptmnjgrs6k9 2816285 2816284 2026-06-19T10:48:31Z Alandmanson 1669821 /* References */ 2816285 wikitext text/x-wiki == Are changes in southern African rangelands contributing to the insect apocalypse? == Rangeland scientists have long considered threats to biodiversity, but the consistent decline in insect biomass has only become a significant field of study in recent years. Estimates suggest that insect populations are shrinking at rates of between one and two percent annually in many areas (Wagner et al., 2021).<br> Why is there a decline in insect abundance? In places like the grassland biome of southern Africa, the reasons are clear – large areas have been converted to crop, timber, or ornamental production; these are systems designed to maximize productivity for human use – this has obviously resulted in systems that minimize insect productivity. Of course, there is a complex interplay of many reasons; key drivers include habitat destruction and fragmentation, agricultural intensification (including widespread pesticide use), climate change (altered temperature and rainfall patterns), light pollution, chemical pollution, and invasive species. (Halsch et al., 2025)<br> Rangelands comprise the home of many insects; increased pressure from humans and their livestock has had widespread effects on rangelands: *Soil loss – many insects use topsoil for at least part of their life cycle, including termites and ants, the dominant forms of terrestrial biomass. *Drier topsoils as a result of the loss of vegetative cover, both live aerial cover and dead plant (mostly leaf) matter. *Loss of wetlands and reduced water quality – many insects evolved to spend at least part of their lives in streams, rivers, or other wetlands; most of these habitats have been transformed in the last 50 years. *Declining quality of food for insect herbivores – rangelands appear resilient, but management for livestock production invariably changes species composition, often with a loss of perennial forbs and grass species.<br> Loss of consistent sources of high-quality insect food is, however, not confined to local effects. In a groundbreaking study at the Konza Prairie Long Term Ecological Research Site, Welti et al. (2020) found that grasshopper biomass fluctuates with herbage quality, and that nutrient dilution appears to be causing a long-term decline in grasshoppers. This nutrient dilution is linked to CO2 fertilization, which increases plant production (when rainfall is sufficient) and lowers the concentrations of many nutrients in plant material (Kaspari & Welti, 2024). Insect decline is probably already affecting insectivores; small mammals, Botha’s Lark, Yellow-breasted Pipit and small raptors come to mind. Feedback effects on plant species are also likely as specialist herbivores are affected differently.<br> So the warning signs are clear, and it seems unlikely that effective action to reverse the loss of insect biomass is possible in these times, where even responses to the needs and wants of the majority of our own species appear inadequate.<br> == References == Halsch, C. A., Elphick, C. S., Bahlai, C. A., Forister, M. L., Wagner, D. L., Ware, J. L., & Grames, E. M. (2025). Meta-synthesis reveals interconnections among apparent drivers of insect biodiversity loss. BioScience, biaf034.<br> Kaspari, M., & Welti, E. A. (2024). Nutrient dilution and the future of herbivore populations. Trends in Ecology & Evolution, 39(9), 809-820.<br> Kaspari, M., & Welti, E. A. (2025). Building plant diversity into mechanisms of nutrient dilution. Trends in Ecology & Evolution.<br> Wagner, D. L., Grames, E. M., Forister, M. L., Berenbaum, M. R., & Stopak, D. (2021). Insect decline in the Anthropocene: Death by a thousand cuts. Proceedings of the National Academy of Sciences, 118(2), e2023989118.<br> Welti, E. A., Roeder, K. A., De Beurs, K. M., Joern, A., & Kaspari, M. (2020). Nutrient dilution and climate cycles underlie declines in a dominant insect herbivore. Proceedings of the National Academy of Sciences, 117(13), 7271-7275.<br> qw6ndh8v8foa0gr499sdpprhcq1qlfq 2816286 2816285 2026-06-19T10:49:48Z Alandmanson 1669821 /* Are changes in southern African rangelands contributing to the insect apocalypse? */ 2816286 wikitext text/x-wiki == Are changes in southern African rangelands contributing to the insect apocalypse? == Rangeland scientists have long considered threats to biodiversity, but the consistent decline in insect biomass has only become a significant field of study in recent years. Estimates suggest that insect populations are shrinking at rates of between one and two percent annually in many areas (Wagner et al., 2021).<br> Why is there a decline in insect abundance? In places like the grassland biome of southern Africa, the reasons are clear – large areas have been converted to crop, timber, or ornamental production; these are systems designed to maximize productivity for human use – this has obviously resulted in systems that minimize insect productivity. Of course, there is a complex interplay of many reasons; key drivers include habitat destruction and fragmentation, agricultural intensification (including widespread pesticide use), climate change (altered temperature and rainfall patterns), light pollution, chemical pollution, and invasive species (Halsch et al., 2025).<br> Rangelands comprise the home of many insects; increased pressure from humans and their livestock has had widespread effects on rangelands: *Soil loss – many insects use topsoil for at least part of their life cycle, including termites and ants, the dominant forms of terrestrial biomass. *Drier topsoils as a result of the loss of vegetative cover, both live aerial cover and dead plant (mostly leaf) matter. *Loss of wetlands and reduced water quality – many insects evolved to spend at least part of their lives in streams, rivers, or other wetlands; most of these habitats have been transformed in the last 50 years. *Declining quality of food for insect herbivores – rangelands appear resilient, but management for livestock production invariably changes species composition, often with a loss of perennial forbs and grass species.<br> Loss of consistent sources of high-quality insect food is, however, not confined to local effects. In a groundbreaking study at the Konza Prairie Long Term Ecological Research Site, Welti et al. (2020) found that grasshopper biomass fluctuates with herbage quality, and that nutrient dilution appears to be causing a long-term decline in grasshoppers. This nutrient dilution is linked to CO2 fertilization, which increases plant production (when rainfall is sufficient) and lowers the concentrations of many nutrients in plant material (Kaspari & Welti, 2024). Insect decline is probably already affecting insectivores; small mammals, Botha’s Lark, Yellow-breasted Pipit and small raptors come to mind. Feedback effects on plant species are also likely as specialist herbivores are affected differently.<br> So the warning signs are clear, and it seems unlikely that effective action to reverse the loss of insect biomass is possible in these times, where even responses to the needs and wants of the majority of our own species appear inadequate.<br> == References == Halsch, C. A., Elphick, C. S., Bahlai, C. A., Forister, M. L., Wagner, D. L., Ware, J. L., & Grames, E. M. (2025). Meta-synthesis reveals interconnections among apparent drivers of insect biodiversity loss. BioScience, biaf034.<br> Kaspari, M., & Welti, E. A. (2024). Nutrient dilution and the future of herbivore populations. Trends in Ecology & Evolution, 39(9), 809-820.<br> Kaspari, M., & Welti, E. A. (2025). Building plant diversity into mechanisms of nutrient dilution. Trends in Ecology & Evolution.<br> Wagner, D. L., Grames, E. M., Forister, M. L., Berenbaum, M. R., & Stopak, D. (2021). Insect decline in the Anthropocene: Death by a thousand cuts. Proceedings of the National Academy of Sciences, 118(2), e2023989118.<br> Welti, E. A., Roeder, K. A., De Beurs, K. M., Joern, A., & Kaspari, M. (2020). Nutrient dilution and climate cycles underlie declines in a dominant insect herbivore. Proceedings of the National Academy of Sciences, 117(13), 7271-7275.<br> 9oiw3dc079ml36pu40gdfxofv7ti0vq 2816287 2816286 2026-06-19T11:57:49Z Alandmanson 1669821 /* References */ 2816287 wikitext text/x-wiki == Are changes in southern African rangelands contributing to the insect apocalypse? == Rangeland scientists have long considered threats to biodiversity, but the consistent decline in insect biomass has only become a significant field of study in recent years. Estimates suggest that insect populations are shrinking at rates of between one and two percent annually in many areas (Wagner et al., 2021).<br> Why is there a decline in insect abundance? In places like the grassland biome of southern Africa, the reasons are clear – large areas have been converted to crop, timber, or ornamental production; these are systems designed to maximize productivity for human use – this has obviously resulted in systems that minimize insect productivity. Of course, there is a complex interplay of many reasons; key drivers include habitat destruction and fragmentation, agricultural intensification (including widespread pesticide use), climate change (altered temperature and rainfall patterns), light pollution, chemical pollution, and invasive species (Halsch et al., 2025).<br> Rangelands comprise the home of many insects; increased pressure from humans and their livestock has had widespread effects on rangelands: *Soil loss – many insects use topsoil for at least part of their life cycle, including termites and ants, the dominant forms of terrestrial biomass. *Drier topsoils as a result of the loss of vegetative cover, both live aerial cover and dead plant (mostly leaf) matter. *Loss of wetlands and reduced water quality – many insects evolved to spend at least part of their lives in streams, rivers, or other wetlands; most of these habitats have been transformed in the last 50 years. *Declining quality of food for insect herbivores – rangelands appear resilient, but management for livestock production invariably changes species composition, often with a loss of perennial forbs and grass species.<br> Loss of consistent sources of high-quality insect food is, however, not confined to local effects. In a groundbreaking study at the Konza Prairie Long Term Ecological Research Site, Welti et al. (2020) found that grasshopper biomass fluctuates with herbage quality, and that nutrient dilution appears to be causing a long-term decline in grasshoppers. This nutrient dilution is linked to CO2 fertilization, which increases plant production (when rainfall is sufficient) and lowers the concentrations of many nutrients in plant material (Kaspari & Welti, 2024). Insect decline is probably already affecting insectivores; small mammals, Botha’s Lark, Yellow-breasted Pipit and small raptors come to mind. Feedback effects on plant species are also likely as specialist herbivores are affected differently.<br> So the warning signs are clear, and it seems unlikely that effective action to reverse the loss of insect biomass is possible in these times, where even responses to the needs and wants of the majority of our own species appear inadequate.<br> == References == Halsch, C. A., Elphick, C. S., Bahlai, C. A., Forister, M. L., Wagner, D. L., Ware, J. L., & Grames, E. M. (2025). Meta-synthesis reveals interconnections among apparent drivers of insect biodiversity loss. BioScience, biaf034. https://doi.org/10.1093/biosci/biaf034<br> Kaspari, M., & Welti, E. A. (2024). Nutrient dilution and the future of herbivore populations. Trends in Ecology & Evolution, 39(9), 809-820. https://doi.org/10.1016/j.tree.2025.02.011<br> Kaspari, M., & Welti, E. A. (2025). Building plant diversity into mechanisms of nutrient dilution. Trends in Ecology & Evolution. https://par.nsf.gov/servlets/purl/10640767<br> Wagner, D. L., Grames, E. M., Forister, M. L., Berenbaum, M. R., & Stopak, D. (2021). Insect decline in the Anthropocene: Death by a thousand cuts. Proceedings of the National Academy of Sciences, 118(2), e2023989118. https://doi.org/10.1073/pnas.2023989118<br> Welti, E. A., Roeder, K. A., De Beurs, K. M., Joern, A., & Kaspari, M. (2020). Nutrient dilution and climate cycles underlie declines in a dominant insect herbivore. Proceedings of the National Academy of Sciences, 117(13), 7271-7275. https://doi.org/10.1016/j.tree.2025.02.011<br> 143m6efc2zb0ssf9q6n6h5bdb1js06m